Optimal. Leaf size=52 \[ \frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a^3}+\frac{2 x}{a^2}+\frac{x^3}{3} \]
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Rubi [A] time = 0.239738, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6338, 6742, 266, 47, 63, 208} \[ \frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a^3}+\frac{2 x}{a^2}+\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} x^2 \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 x^2 \, dx\\ &=\int \left (\frac{2}{a^2}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x}{a}+x^2\right ) \, dx\\ &=\frac{2 x}{a^2}+\frac{x^3}{3}+\frac{2 \int \sqrt{1+\frac{1}{a^2 x^2}} x \, dx}{a}\\ &=\frac{2 x}{a^2}+\frac{x^3}{3}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{2 x}{a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{a}+\frac{x^3}{3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a^3}\\ &=\frac{2 x}{a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{a}+\frac{x^3}{3}-\frac{\operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )}{a}\\ &=\frac{2 x}{a^2}+\frac{\sqrt{1+\frac{1}{a^2 x^2}} x^2}{a}+\frac{x^3}{3}+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0378865, size = 57, normalized size = 1.1 \[ \frac{a x \left (a^2 x^2+3 a x \sqrt{\frac{1}{a^2 x^2}+1}+6\right )+3 \log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{3 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.171, size = 90, normalized size = 1.7 \begin{align*}{\frac{{x}^{3}}{3}}+2\,{\frac{x}{{a}^{2}}}+{\frac{x}{{a}^{3}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( 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 = 0.988807, size = 120, normalized size = 2.31 \begin{align*} \frac{1}{3} \, x^{3} + \frac{\frac{2 \, \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - a^{2}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{a^{2}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{a^{2}}}{2 \, a} + \frac{2 \, x}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5246, size = 159, normalized size = 3.06 \begin{align*} \frac{a^{3} x^{3} + 3 \, a^{2} x^{2} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 6 \, a x - 3 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.37071, size = 36, normalized size = 0.69 \begin{align*} \frac{x^{3}}{3} + \frac{x \sqrt{a^{2} x^{2} + 1}}{a^{2}} + \frac{2 x}{a^{2}} + \frac{\operatorname{asinh}{\left (a x \right )}}{a^{3}} \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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