Problem number 67

13.2 Problem number 67

$\int x^{3} \sinh^{- 1} (a + b x)^{2} d x$

Optimal antiderivative $\frac{4 a x}{3 b^{3}} - \frac{2 a^{3} x}{b^{3}} - \frac{3 {(b x + a)}^{2}}{32 b^{4}} + \frac{3 a^{2} {(b x + a)}^{2}}{4 b^{4}} - \frac{2 a {(b x + a)}^{3}}{9 b^{4}} + \frac{{(b x + a)}^{4}}{32 b^{4}} - \frac{3 arcsinh {(b x + a)}^{2}}{32 b^{4}} + \frac{3 a^{2} arcsinh {(b x + a)}^{2}}{4 b^{4}} - \frac{a^{4} arcsinh {(b x + a)}^{2}}{4 b^{4}} + \frac{x^{4} arcsinh {(b x + a)}^{2}}{4} - \frac{4 a arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{3 b^{4}} + \frac{2 a^{3} arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{b^{4}} + \frac{3 (b x + a) arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{16 b^{4}} - \frac{3 a^{2} (b x + a) arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{2 b^{4}} + \frac{2 a {(b x + a)}^{2} arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{3 b^{4}} - \frac{{(b x + a)}^{3} arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}}}{8 b^{4}}$

command

int(x^3*arcsinh(b*x+a)^2,x)

Maple 2022.1 output

$\int x^{3} arcsinh {(b x + a)}^{2} d x$

Maple 2021.1 output

$\frac{- a^{3} (arcsinh {(b x + a)}^{2} (b x + a) - 2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} + 2 b x + 2 a) + \frac{3 a^{2} (2 arcsinh {(b x + a)}^{2} {(b x + a)}^{2} - 2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} (b x + a) + arcsinh {(b x + a)}^{2} + {(b x + a)}^{2} + 1)}{4} - \frac{a (9 arcsinh {(b x + a)}^{2} {(b x + a)}^{3} - 6 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} {(b x + a)}^{2} + 27 arcsinh {(b x + a)}^{2} (b x + a) + 2 {(b x + a)}^{3} - 42 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} + 42 b x + 42 a)}{9} + \frac{arcsinh {(b x + a)}^{2} {(1 + {(b x + a)}^{2})}^{2}}{4} - \frac{arcsinh (b x + a) (b x + a) {(1 + {(b x + a)}^{2})}^{\frac{3}{2}}}{8} + \frac{5 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} (b x + a)}{16} + \frac{5 arcsinh {(b x + a)}^{2}}{32} + \frac{{(1 + {(b x + a)}^{2})}^{2}}{32} - \frac{5 {(b x + a)}^{2}}{32} - \frac{5}{32} + 3 a (arcsinh {(b x + a)}^{2} (b x + a) - 2 arcsinh (b x + a) \sqrt{1 + {(b x + a)}^{2}} + 2 b x + 2 a) - \frac{arcsinh {(b x + a)}^{2} (1 + {(b x + a)}^{2})}{2}}{b^{4}}$

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