Integrand size = 12, antiderivative size = 85 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}-\frac {\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^5} \]
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Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6473, 6874, 272, 43, 44, 65, 214} \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {2 x^3}{3 a^2}+\frac {x^4 \sqrt {\frac {1}{a^2 x^2}+1}}{2 a}-\frac {\text {arctanh}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{4 a^5}+\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{4 a^3}+\frac {x^5}{5} \]
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Rule 43
Rule 44
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^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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {a^2 x^2 \left (15 \sqrt {1+\frac {1}{a^2 x^2}}+40 a x+30 a^2 \sqrt {1+\frac {1}{a^2 x^2}} x^2+12 a^3 x^3\right )-15 \log \left (\left (1+\sqrt {1+\frac {1}{a^2 x^2}}\right ) x\right )}{60 a^5} \]
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Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {\frac {1}{5} a^{2} x^{5}+\frac {1}{3} x^{3}}{a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-2 x \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} a^{4}+x \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right )\right )}{4 a^{5} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}+\frac {x^{3}}{3 a^{2}}\) | \(127\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {12 \, a^{5} x^{5} + 40 \, a^{3} x^{3} + 15 \, {\left (2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 15 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{60 \, a^{5}} \]
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Time = 2.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {x^{5}}{5} + \frac {x^{5}}{2 \sqrt {a^{2} x^{2} + 1}} + \frac {2 x^{3}}{3 a^{2}} + \frac {3 x^{3}}{4 a^{2} \sqrt {a^{2} x^{2} + 1}} + \frac {x}{4 a^{4} \sqrt {a^{2} x^{2} + 1}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.38 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {1}{5} \, x^{5} + \frac {2 \, x^{3}}{3 \, a^{2}} + \frac {\frac {2 \, {\left ({\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \sqrt {\frac {1}{a^{2} x^{2}} + 1}\right )}}{a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} + a^{4}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a^{4}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a^{4}}}{8 \, a} \]
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Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {1}{4} \, \sqrt {a^{2} x^{2} + 1} x {\left (\frac {2 \, x^{2} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {{\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{5}}\right )} + \frac {3 \, a^{2} x^{5} + 10 \, x^{3}}{15 \, a^{2}} + \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) \mathrm {sgn}\left (x\right )}{4 \, a^{5}} \]
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Time = 4.84 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx=\frac {x^5}{5}+\frac {2\,x^3}{3\,a^2}+\frac {x^4\,\sqrt {\frac {1}{a^2\,x^2}+1}}{2\,a}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{4\,a^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^5} \]
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