Optimal. Leaf size=50 \[ \frac {x^2}{10 a^3}+\frac {\log \left (1-a^2 x^2\right )}{10 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {x^4}{20 a} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5917, 266, 43} \[ \frac {x^2}{10 a^3}+\frac {\log \left (1-a^2 x^2\right )}{10 a^5}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5917
Rubi steps
\begin {align*} \int x^4 \coth ^{-1}(a x) \, dx &=\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{5} a \int \frac {x^5}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \operatorname {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{10 a^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ \frac {x^2}{10 a^3}+\frac {\log \left (1-a^2 x^2\right )}{10 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {x^4}{20 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 55, normalized size = 1.10 \[ \frac {2 \, a^{5} x^{5} \log \left (\frac {a x + 1}{a x - 1}\right ) + a^{4} x^{4} + 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} - 1\right )}{20 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcoth}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.98 \[ \frac {x^{5} \mathrm {arccoth}\left (a x \right )}{5}+\frac {x^{4}}{20 a}+\frac {x^{2}}{10 a^{3}}+\frac {\ln \left (a x -1\right )}{10 a^{5}}+\frac {\ln \left (a x +1\right )}{10 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 46, normalized size = 0.92 \[ \frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right ) + \frac {1}{20} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 43, normalized size = 0.86 \[ \frac {\frac {\ln \left (a^2\,x^2-1\right )}{10}+\frac {a^2\,x^2}{10}+\frac {a^4\,x^4}{20}}{a^5}+\frac {x^5\,\mathrm {acoth}\left (a\,x\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 54, normalized size = 1.08 \[ \begin {cases} \frac {x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {x^{4}}{20 a} + \frac {x^{2}}{10 a^{3}} + \frac {\log {\left (a x + 1 \right )}}{5 a^{5}} - \frac {\operatorname {acoth}{\left (a x \right )}}{5 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{5}}{10} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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