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Lösungsweg

int((X^2/2-1)*sqrt(x^2+1),x)


int(c*f(x),x) = c*int(f(x),x)
(X^2/2-1)*int(sqrt(x^2+1),x)

Substitution: sqrt(x^2+1) = sqrt(1)*cosh(a)/1, x = (sqrt(1)*sinh(a)-0/2)/1
(X^2/2-1)*int(cosh(a)^2,a)

int(cosh(x)^n,x) = cosh(x)^(n-1)*sinh(x)/n+((n-1)/n)*int(cosh(x)^(n-2),x)
(X^2/2-1)*(cosh(a)^(2-1)*sinh(a)/2+(2-1)/2*int(cosh(a)^(2-2),a))

Vereinfachen
(X^2/2-1)*(cosh(a)*sinh(a)/2+1/2*int(cosh(a)^0,a))

int(f^0,x) = x
(X^2/2-1)*(cosh(a)*sinh(a)/2+1/2*a)

Rücksubstitution: a = arsinh(x)
(X^2/2-1)*(cosh(arsinh(x))*sinh(arsinh(x))+arsinh(x))/2*sqrt(1)/1

Vereinfachen
(X^2-2)*(cosh(arsinh(x))*x+arsinh(x))/4

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