Optimal. Leaf size=196 \[ \frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{10 a^5}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\frac {\coth ^{-1}(a x)^3}{5 a^5}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{5 a^5}+\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^2}{20 a^3}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}+\frac {\log \left (1-a^2 x^2\right )}{2 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a} \]
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Rubi [A] time = 0.58, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5917, 5981, 266, 43, 5911, 260, 5949, 5985, 5919, 6059, 6610} \[ \frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{10 a^5}-\frac {3 \coth ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {x^2}{20 a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {\coth ^{-1}(a x)^3}{5 a^5}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 5911
Rule 5917
Rule 5919
Rule 5949
Rule 5981
Rule 5985
Rule 6059
Rule 6610
Rubi steps
\begin {align*} \int x^4 \coth ^{-1}(a x)^3 \, dx &=\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {1}{5} (3 a) \int \frac {x^5 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \coth ^{-1}(a x)^3+\frac {3 \int x^3 \coth ^{-1}(a x)^2 \, dx}{5 a}-\frac {3 \int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 a}\\ &=\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{10} \int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {3 \int x \coth ^{-1}(a x)^2 \, dx}{5 a^3}-\frac {3 \int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 a^3}\\ &=\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \int \frac {\coth ^{-1}(a x)^2}{1-a x} \, dx}{5 a^4}+\frac {3 \int x^2 \coth ^{-1}(a x) \, dx}{10 a^2}-\frac {3 \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{10 a^2}-\frac {3 \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^2}\\ &=\frac {x^3 \coth ^{-1}(a x)}{10 a^2}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {3 \int \coth ^{-1}(a x) \, dx}{10 a^4}-\frac {3 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{10 a^4}+\frac {3 \int \coth ^{-1}(a x) \, dx}{5 a^4}-\frac {3 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^4}+\frac {6 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}-\frac {\int \frac {x^3}{1-a^2 x^2} \, dx}{10 a}\\ &=\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}-\frac {3 \int \frac {x}{1-a^2 x^2} \, dx}{10 a^3}-\frac {3 \int \frac {x}{1-a^2 x^2} \, dx}{5 a^3}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )}{20 a}\\ &=\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {9 \log \left (1-a^2 x^2\right )}{20 a^5}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{10 a^5}-\frac {\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 )}{20 a}\\ &=\frac {x^2}{20 a^3}+\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {\log \left (1-a^2 x^2\right )}{2 a^5}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{10 a^5}\\ \end {align*}
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Mathematica [C] time = 0.59, size = 175, normalized size = 0.89 \[ \frac {8 a^5 x^5 \coth ^{-1}(a x)^3+6 a^4 x^4 \coth ^{-1}(a x)^2+4 a^3 x^3 \coth ^{-1}(a x)+2 a^2 x^2-40 \log \left (\frac {1}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\right )+12 a^2 x^2 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x) \text {Li}_2\left (e^{2 \coth ^{-1}(a x)}\right )+12 \text {Li}_3\left (e^{2 \coth ^{-1}(a x)}\right )+36 a x \coth ^{-1}(a x)+8 \coth ^{-1}(a x)^3-18 \coth ^{-1}(a x)^2-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-i \pi ^3-2}{40 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arcoth}\left (a x\right )^{3}, x\right ) \]
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 \[ \int x^{4} \operatorname {arcoth}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.92, size = 806, normalized size = 4.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a^{5} x^{5} + 1\right )} \log \left (a x + 1\right )^{3} + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} - 2 \, {\left (a^{5} x^{5} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )^{2}}{80 \, a^{5}} + \frac {1}{8} \, \int -\frac {5 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (a x - 1\right )^{3} + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} - 5 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (a^{5} x^{5} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{5 \, {\left (a^{5} x + a^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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