Optimal. Leaf size=205 \[ \frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2 c}-\frac {3 \text {Li}_4\left (1-\frac {2}{a x+1}\right )}{4 a^2 c}-\frac {3 \text {Li}_2\left (1-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^2 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)}{2 a^2 c}+\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {\log \left (\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^2 c}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c} \]
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Rubi [A] time = 0.37, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5930, 5910, 5984, 5918, 5948, 6058, 6610, 6056, 6060} \[ \frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2 c}-\frac {3 \text {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a^2 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a^2 c}+\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {\log \left (\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^2 c}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c} \]
Antiderivative was successfully verified.
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Rule 5910
Rule 5918
Rule 5930
Rule 5948
Rule 5984
Rule 6056
Rule 6058
Rule 6060
Rule 6610
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^3}{c+a c x} \, dx &=-\frac {\int \frac {\tanh ^{-1}(a x)^3}{c+a c x} \, dx}{a}+\frac {\int \tanh ^{-1}(a x)^3 \, dx}{a c}\\ &=\frac {x \tanh ^{-1}(a x)^3}{a c}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a c}+\frac {3 \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c}-\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}+\frac {3 \int \frac {\text {Li}_3\left (1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 a c}+\frac {6 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c}-\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \text {Li}_4\left (1-\frac {2}{1+a x}\right )}{4 a^2 c}+\frac {3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {\tanh ^{-1}(a x)^3}{a^2 c}+\frac {x \tanh ^{-1}(a x)^3}{a c}-\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^2 c}-\frac {3 \text {Li}_4\left (1-\frac {2}{1+a x}\right )}{4 a^2 c}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 126, normalized size = 0.61 \[ \frac {-\frac {3}{2} \left (\tanh ^{-1}(a x)-2\right ) \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {3}{2} \left (\tanh ^{-1}(a x)-1\right ) \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {3}{4} \text {Li}_4\left (-e^{-2 \tanh ^{-1}(a x)}\right )+a x \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-3 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a^2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \operatorname {artanh}\left (a x\right )^{3}}{a c x + c}, x\right ) \]
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 \[ \int \frac {x \operatorname {artanh}\left (a x\right )^{3}}{a c x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 833, normalized size = 4.06 \[ \frac {x \arctanh \left (a x \right )^{3}}{a c}-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x +1\right )}{a^{2} c}+\frac {2 \arctanh \left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} c}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2} c}-\frac {3 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2} c}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{2} c}-\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{2} c}-\frac {3 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{2} c}+\frac {\arctanh \left (a x \right )^{3}}{a^{2} c}-\frac {\arctanh \left (a x \right )^{4}}{2 a^{2} c}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{3} \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2} \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {\ln \relax (2) \arctanh \left (a x \right )^{3}}{a^{2} c}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \arctanh \left (a x \right )^{3}}{a^{2} c}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right ) \arctanh \left (a x \right )^{3}}{2 a^{2} c}+\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2} c} \]
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 \[ -\frac {{\left (a x - \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3}}{8 \, a^{2} c} + \frac {1}{8} \, \int \frac {{\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \, {\left (a^{2} x^{2} + a x + {\left (a^{2} x^{2} - 2 \, a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{a^{3} c x^{2} - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^3}{c+a\,c\,x} \,d x \]
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 \[ \frac {\int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{a x + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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