Optimal. Leaf size=243 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^2}-\frac{2 x^2 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^4}-\frac{40 x \sqrt{a x-1}}{9 a^3 \sqrt{1-a x}}-\frac{2 x \sqrt{a x-1} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a x}}-\frac{40 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^4}-\frac{2 x^3 \sqrt{a x-1}}{27 a \sqrt{1-a x}}-\frac{x^3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a x}} \]
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Rubi [A] time = 1.02626, antiderivative size = 329, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5798, 5759, 5718, 5654, 8, 5662, 30} \[ -\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 x \sqrt{a x-1} \sqrt{a x+1}}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5759
Rule 5718
Rule 5654
Rule 8
Rule 5662
Rule 30
Rubi steps
\begin{align*} \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \cosh ^{-1}(a x)^2 \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \cosh ^{-1}(a x)^2 \, dx}{a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{a^2 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \, dx}{9 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{40 x \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{27 a \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a^2 x^2}}-\frac{x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.155989, size = 140, normalized size = 0.58 \[ \frac{\sqrt{1-a^2 x^2} \left (2 a x \left (a^2 x^2+60\right )-9 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^3+9 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)^2-6 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+20\right ) \cosh ^{-1}(a x)\right )}{27 a^4 \sqrt{a x-1} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.198, size = 375, normalized size = 1.5 \begin{align*} -{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )-2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}+4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}-3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+18\,{\rm arccosh} \left (ax\right )-18}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \sqrt{ax+1}\sqrt{ax-1}ax+{a}^{2}{x}^{2}-1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+18\,{\rm arccosh} \left (ax\right )+18}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}-4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}+3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.80281, size = 177, normalized size = 0.73 \begin{align*} -\frac{1}{3} \,{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname{arcosh}\left (a x\right )^{3} + \frac{2}{27} \, a{\left (\frac{3 \,{\left (-i \, \sqrt{a^{2} x^{2} - 1} x^{2} - \frac{20 i \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname{arcosh}\left (a x\right )}{a^{3}} + \frac{i \, a^{2} x^{3} + 60 i \, x}{a^{4}}\right )} + \frac{{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname{arcosh}\left (a x\right )^{2}}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20353, size = 447, normalized size = 1.84 \begin{align*} -\frac{9 \,{\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 9 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 6 \,{\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (a^{3} x^{3} + 60 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1}}{27 \,{\left (a^{6} x^{2} - a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{acosh}^{3}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.33952, size = 200, normalized size = 0.82 \begin{align*} \frac{{\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3}}{3 \, a^{4}} + \frac{-2 i \, a^{2} x^{3} + 9 \,{\left (-i \, a^{2} x^{3} - 6 i \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 120 i \, x - \frac{3 \,{\left (-2 i \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 42 i \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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