Optimal. Leaf size=85 \[ \frac{x^4 \sqrt{\frac{1}{a^2 x^2}+1}}{2 a}+\frac{2 x^3}{3 a^2}+\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{4 a^3}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{4 a^5}+\frac{x^5}{5} \]
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Rubi [A] time = 0.254138, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6338, 6742, 266, 47, 51, 63, 208} \[ \frac{x^4 \sqrt{\frac{1}{a^2 x^2}+1}}{2 a}+\frac{2 x^3}{3 a^2}+\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{4 a^3}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{4 a^5}+\frac{x^5}{5} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} x^4 \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 x^4 \, dx\\ &=\int \left (\frac{2 x^2}{a^2}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x^3}{a}+x^4\right ) \, dx\\ &=\frac{2 x^3}{3 a^2}+\frac{x^5}{5}+\frac{2 \int \sqrt{1+\frac{1}{a^2 x^2}} x^3 \, dx}{a}\\ &=\frac{2 x^3}{3 a^2}+\frac{x^5}{5}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^3} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{2 x^3}{3 a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^4}{2 a}+\frac{x^5}{5}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^3}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{4 a^3}+\frac{2 x^3}{3 a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^4}{2 a}+\frac{x^5}{5}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{8 a^5}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{4 a^3}+\frac{2 x^3}{3 a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^4}{2 a}+\frac{x^5}{5}+\frac{\operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )}{4 a^3}\\ &=\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{4 a^3}+\frac{2 x^3}{3 a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^4}{2 a}+\frac{x^5}{5}-\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{4 a^5}\\ \end{align*}
Mathematica [A] time = 0.0630435, size = 84, normalized size = 0.99 \[ \frac{a^2 x^2 \left (12 a^3 x^3+30 a^2 x^2 \sqrt{\frac{1}{a^2 x^2}+1}+15 \sqrt{\frac{1}{a^2 x^2}+1}+40 a x\right )-15 \log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{60 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.174, size = 117, normalized size = 1.4 \begin{align*}{\frac{{x}^{5}}{5}}+{\frac{2\,{x}^{3}}{3\,{a}^{2}}}+{\frac{x}{4\,{a}^{5}}\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}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03488, size = 158, normalized size = 1.86 \begin{align*} \frac{1}{5} \, x^{5} + \frac{2 \, x^{3}}{3 \, a^{2}} + \frac{\frac{2 \,{\left ({\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{a^{2} x^{2}} + 1}\right )}}{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}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{a^{4}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{a^{4}}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56241, size = 193, normalized size = 2.27 \begin{align*} \frac{12 \, a^{5} x^{5} + 40 \, a^{3} x^{3} + 15 \,{\left (2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 15 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{60 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.68562, size = 82, normalized size = 0.96 \begin{align*} \frac{x^{5}}{5} + \frac{x^{5}}{2 \sqrt{a^{2} x^{2} + 1}} + \frac{2 x^{3}}{3 a^{2}} + \frac{3 x^{3}}{4 a^{2} \sqrt{a^{2} x^{2} + 1}} + \frac{x}{4 a^{4} \sqrt{a^{2} x^{2} + 1}} - \frac{\operatorname{asinh}{\left (a x \right )}}{4 a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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