Optimal. Leaf size=75 \[ \frac{1}{4} x^4 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{8 a^2}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{8 a^4}+\frac{x^3}{3 a} \]
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Rubi [A] time = 0.0440765, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6336, 30, 266, 47, 51, 63, 208} \[ \frac{1}{4} x^4 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{8 a^2}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{8 a^4}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}(a x)} x^3 \, dx &=\frac{\int x^2 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x^3 \, dx\\ &=\frac{x^3}{3 a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^3} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{x^3}{3 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^2}} x^4-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{8 a^2}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{x^3}{3 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^2}} x^4+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 a^4}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{x^3}{3 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^2}} x^4+\frac{\operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )}{8 a^2}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{x^3}{3 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^2}} x^4-\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.0552299, size = 76, normalized size = 1.01 \[ \frac{a^2 x^2 \left (6 a^2 x^2 \sqrt{\frac{1}{a^2 x^2}+1}+3 \sqrt{\frac{1}{a^2 x^2}+1}+8 a x\right )-3 \log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{24 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.177, size = 109, normalized size = 1.5 \begin{align*} -{\frac{x}{8\,{a}^{4}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( -2\,x \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}{a}^{4}+x\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+\ln \left ( x+\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}}+{\frac{{x}^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973333, size = 144, normalized size = 1.92 \begin{align*} \frac{x^{3}}{3 \, a} + \frac{{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{8 \,{\left (a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} + a^{4}\right )}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{16 \, a^{4}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{16 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.64219, size = 171, normalized size = 2.28 \begin{align*} \frac{8 \, a^{3} x^{3} + 3 \,{\left (2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 3 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{24 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.50681, size = 73, normalized size = 0.97 \begin{align*} \frac{a x^{5}}{4 \sqrt{a^{2} x^{2} + 1}} + \frac{x^{3}}{3 a} + \frac{3 x^{3}}{8 a \sqrt{a^{2} x^{2} + 1}} + \frac{x}{8 a^{3} \sqrt{a^{2} x^{2} + 1}} - \frac{\operatorname{asinh}{\left (a x \right )}}{8 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16368, size = 93, normalized size = 1.24 \begin{align*} \frac{1}{8} \, \sqrt{a^{2} x^{2} + 1}{\left (\frac{2 \, x^{2}{\left | a \right |} \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{{\left | a \right |} \mathrm{sgn}\left (x\right )}{a^{4}}\right )} x + \frac{x^{3}}{3 \, a} + \frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{8 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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