Optimal. Leaf size=153 \[ \frac{x^3 \sqrt{a^2 x^2+1}}{32 a^2}-\frac{15 x \sqrt{a^2 x^2+1}}{64 a^4}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}-\frac{x^4 \sinh ^{-1}(a x)}{8 a} \]
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Rubi [A] time = 0.28968, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5758, 5675, 5661, 321, 215} \[ \frac{x^3 \sqrt{a^2 x^2+1}}{32 a^2}-\frac{15 x \sqrt{a^2 x^2+1}}{64 a^4}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}-\frac{x^4 \sinh ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 5758
Rule 5675
Rule 5661
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^4 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}-\frac{3 \int \frac{x^2 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2}-\frac{\int x^3 \sinh ^{-1}(a x) \, dx}{2 a}\\ &=-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{1}{8} \int \frac{x^4}{\sqrt{1+a^2 x^2}} \, dx+\frac{3 \int \frac{\sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{8 a^4}+\frac{3 \int x \sinh ^{-1}(a x) \, dx}{4 a^3}\\ &=\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}-\frac{3 \int \frac{x^2}{\sqrt{1+a^2 x^2}} \, dx}{32 a^2}-\frac{3 \int \frac{x^2}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2}\\ &=-\frac{15 x \sqrt{1+a^2 x^2}}{64 a^4}+\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{3 \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{64 a^4}+\frac{3 \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{16 a^4}\\ &=-\frac{15 x \sqrt{1+a^2 x^2}}{64 a^4}+\frac{x^3 \sqrt{1+a^2 x^2}}{32 a^2}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{8 a^4}+\frac{x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 a^2}+\frac{\sinh ^{-1}(a x)^3}{8 a^5}\\ \end{align*}
Mathematica [A] time = 0.0694461, size = 98, normalized size = 0.64 \[ \frac{a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-15\right )+8 a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-3\right ) \sinh ^{-1}(a x)^2+\left (-8 a^4 x^4+24 a^2 x^2+15\right ) \sinh ^{-1}(a x)+8 \sinh ^{-1}(a x)^3}{64 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 125, normalized size = 0.8 \begin{align*}{\frac{1}{64\,{a}^{5}} \left ( 16\,{a}^{3}{x}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-8\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) +2\,{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}-24\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax\sqrt{{a}^{2}{x}^{2}+1}+24\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) +8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}-15\,ax\sqrt{{a}^{2}{x}^{2}+1}+15\,{\it Arcsinh} \left ( ax \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.12243, size = 297, normalized size = 1.94 \begin{align*} \frac{8 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 8 \, \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} -{\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) +{\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt{a^{2} x^{2} + 1}}{64 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.75894, size = 146, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{x^{4} \operatorname{asinh}{\left (a x \right )}}{8 a} + \frac{x^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac{x^{3} \sqrt{a^{2} x^{2} + 1}}{32 a^{2}} + \frac{3 x^{2} \operatorname{asinh}{\left (a x \right )}}{8 a^{3}} - \frac{3 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{8 a^{4}} - \frac{15 x \sqrt{a^{2} x^{2} + 1}}{64 a^{4}} + \frac{\operatorname{asinh}^{3}{\left (a x \right )}}{8 a^{5}} + \frac{15 \operatorname{asinh}{\left (a x \right )}}{64 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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