Optimal. Leaf size=81 \[ -\frac {\coth ^{-1}(a x)^2}{4 a^4}+\frac {x \coth ^{-1}(a x)}{2 a^3}+\frac {x^2}{12 a^2}+\frac {\log \left (1-a^2 x^2\right )}{3 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2+\frac {x^3 \coth ^{-1}(a x)}{6 a} \]
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Rubi [A] time = 0.16, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5917, 5981, 266, 43, 5911, 260, 5949} \[ \frac {x^2}{12 a^2}+\frac {\log \left (1-a^2 x^2\right )}{3 a^4}+\frac {x \coth ^{-1}(a x)}{2 a^3}-\frac {\coth ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2+\frac {x^3 \coth ^{-1}(a x)}{6 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 5911
Rule 5917
Rule 5949
Rule 5981
Rubi steps
\begin {align*} \int x^3 \coth ^{-1}(a x)^2 \, dx &=\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \coth ^{-1}(a x)^2+\frac {\int x^2 \coth ^{-1}(a x) \, dx}{2 a}-\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a}\\ &=\frac {x^3 \coth ^{-1}(a x)}{6 a}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{6} \int \frac {x^3}{1-a^2 x^2} \, dx+\frac {\int \coth ^{-1}(a x) \, dx}{2 a^3}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a^3}\\ &=\frac {x \coth ^{-1}(a x)}{2 a^3}+\frac {x^3 \coth ^{-1}(a x)}{6 a}-\frac {\coth ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{12} \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{2 a^2}\\ &=\frac {x \coth ^{-1}(a x)}{2 a^3}+\frac {x^3 \coth ^{-1}(a x)}{6 a}-\frac {\coth ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{4 a^4}-\frac {1}{12} \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{12 a^2}+\frac {x \coth ^{-1}(a x)}{2 a^3}+\frac {x^3 \coth ^{-1}(a x)}{6 a}-\frac {\coth ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \coth ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.77 \[ \frac {3 \left (a^4 x^4-1\right ) \coth ^{-1}(a x)^2+a^2 x^2+4 \log \left (1-a^2 x^2\right )+2 a x \left (a^2 x^2+3\right ) \coth ^{-1}(a x)}{12 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 81, normalized size = 1.00 \[ \frac {4 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (a^{3} x^{3} + 3 \, a x\right )} \log \left (\frac {a x + 1}{a x - 1}\right ) + 16 \, \log \left (a^{2} x^{2} - 1\right )}{48 \, a^{4}} \]
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 \[ \int x^{3} \operatorname {arcoth}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 176, normalized size = 2.17 \[ \frac {x^{4} \mathrm {arccoth}\left (a x \right )^{2}}{4}+\frac {x^{3} \mathrm {arccoth}\left (a x \right )}{6 a}+\frac {x \,\mathrm {arccoth}\left (a x \right )}{2 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{4 a^{4}}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{4 a^{4}}+\frac {\ln \left (a x -1\right )^{2}}{16 a^{4}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8 a^{4}}+\frac {\ln \left (a x +1\right )^{2}}{16 a^{4}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8 a^{4}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{8 a^{4}}+\frac {x^{2}}{12 a^{2}}+\frac {\ln \left (a x -1\right )}{3 a^{4}}+\frac {\ln \left (a x +1\right )}{3 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 118, normalized size = 1.46 \[ \frac {1}{4} \, x^{4} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{12} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {arcoth}\left (a x\right ) + \frac {4 \, a^{2} x^{2} - 2 \, {\left (3 \, \log \left (a x - 1\right ) - 8\right )} \log \left (a x + 1\right ) + 3 \, \log \left (a x + 1\right )^{2} + 3 \, \log \left (a x - 1\right )^{2} + 16 \, \log \left (a x - 1\right )}{48 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 65, normalized size = 0.80 \[ \frac {x^4\,{\mathrm {acoth}\left (a\,x\right )}^2}{4}+\frac {\frac {\ln \left (a^2\,x^2-1\right )}{3}+\frac {a^2\,x^2}{12}-\frac {{\mathrm {acoth}\left (a\,x\right )}^2}{4}+\frac {a^3\,x^3\,\mathrm {acoth}\left (a\,x\right )}{6}+\frac {a\,x\,\mathrm {acoth}\left (a\,x\right )}{2}}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.44, size = 90, normalized size = 1.11 \[ \begin {cases} \frac {x^{4} \operatorname {acoth}^{2}{\left (a x \right )}}{4} + \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{6 a} + \frac {x^{2}}{12 a^{2}} + \frac {x \operatorname {acoth}{\left (a x \right )}}{2 a^{3}} + \frac {2 \log {\left (a x + 1 \right )}}{3 a^{4}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{4 a^{4}} - \frac {2 \operatorname {acoth}{\left (a x \right )}}{3 a^{4}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{4}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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