Problem number 68

13.3 Problem number 68

$\int x^{2} \sinh^{- 1} (a + b x)^{2} d x$

Optimal antiderivative $- \frac{4 x}{9 b^{2}} + \frac{2 a^{2} x}{b^{2}} - \frac{a {(b x + a)}^{2}}{2 b^{3}} + \frac{2 {(b x + a)}^{3}}{27 b^{3}} - \frac{a arcsinh {(b x + a)}^{2}}{2 b^{3}} + \frac{a^{3} arcsinh {(b x + a)}^{2}}{3 b^{3}} + \frac{x^{3} arcsinh {(b x + a)}^{2}}{3} + \frac{4 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{9 b^{3}} - \frac{2 a^{2} arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{b^{3}} + \frac{a (b x + a) arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{b^{3}} - \frac{2 {(b x + a)}^{2} arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{9 b^{3}}$

command

int(x^2*arcsinh(b*x+a)^2,x)

Maple 2022.1 output

$\int x^{2} arcsinh {(b x + a)}^{2} d x$

Maple 2021.1 output

$\frac{- \frac{a (2 arcsinh {(b x + a)}^{2} {(b x + a)}^{2} - 2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} (b x + a) + arcsinh {(b x + a)}^{2} + {(b x + a)}^{2} + 1)}{2} - \frac{arcsinh {(b x + a)}^{2} (b x + a)}{3} + \frac{arcsinh {(b x + a)}^{2} (b x + a) (1 + {(b x + a)}^{2})}{3} + \frac{2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{3} - \frac{14 b x}{27} - \frac{14 a}{27} - \frac{2 arcsinh (b x + a) {(1 + {(b x + a)}^{2})}^{\frac{3}{2}}}{9} + \frac{2 (1 + {(b x + a)}^{2}) (b x + a)}{27} + a^{2} (arcsinh {(b x + a)}^{2} (b x + a) - 2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} + 2 b x + 2 a)}{b^{3}}$

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