Integrand size = 12, antiderivative size = 52 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {2 x}{a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{a}+\frac {x^3}{3}+\frac {\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a^3} \]
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Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6473, 6874, 272, 43, 65, 214} \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 x}{a^2}+\frac {\text {arctanh}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a^3}+\frac {x^3}{3} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 6473
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {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 {\text {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 {\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.10 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {a x \left (6+3 a \sqrt {1+\frac {1}{a^2 x^2}} x+a^2 x^2\right )+3 \log \left (\left (1+\sqrt {1+\frac {1}{a^2 x^2}}\right ) x\right )}{3 a^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(46)=92\).
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\frac {1}{3} a^{2} x^{3}+x}{a^{2}}+\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (x \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right )\right )}{a^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}+\frac {x}{a^{2}}\) | \(98\) |
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Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.38 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 6 \, a x - 3 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{3 \, a^{3}} \]
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Time = 1.61 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {x^{3}}{3} + \frac {x \sqrt {a^{2} x^{2} + 1}}{a^{2}} + \frac {2 x}{a^{2}} + \frac {\operatorname {asinh}{\left (a x \right )}}{a^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.71 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {1}{3} \, x^{3} + \frac {\frac {2 \, \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{a^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - a^{2}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a^{2}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a^{2}}}{2 \, a} + \frac {2 \, x}{a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.19 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {\sqrt {a^{2} x^{2} + 1} x {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {a^{2} x^{3} + 6 \, x}{3 \, a^{2}} - \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) \mathrm {sgn}\left (x\right )}{a^{3}} \]
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Time = 5.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx=\frac {2\,x}{a^2}+\frac {x^3}{3}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^3} \]
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