Optimal. Leaf size=384 \[ -\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {Li}_5\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^3 c}+\frac {6 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}+\frac {6 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}-\frac {6 \text {Li}_3\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{a^3 c}+\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^3 c}+\frac {6 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c} \]
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Rubi [A] time = 0.86, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5930, 5916, 5980, 5910, 5984, 5918, 5948, 6058, 6610, 6062} \[ -\frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {6 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{a^3 c}+\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^3 c}+\frac {6 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c} \]
Antiderivative was successfully verified.
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Rule 5910
Rule 5916
Rule 5918
Rule 5930
Rule 5948
Rule 5980
Rule 5984
Rule 6058
Rule 6062
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^4}{c-a c x} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)^4}{c-a c x} \, dx}{a}-\frac {\int x \tanh ^{-1}(a x)^4 \, dx}{a c}\\ &=-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {\int \frac {\tanh ^{-1}(a x)^4}{c-a c x} \, dx}{a^2}+\frac {2 \int \frac {x^2 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{c}-\frac {\int \tanh ^{-1}(a x)^4 \, dx}{a^2 c}\\ &=-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {2 \int \tanh ^{-1}(a x)^3 \, dx}{a^2 c}+\frac {2 \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2 c}-\frac {4 \int \frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}+\frac {4 \int \frac {x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a c}\\ &=-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {4 \int \frac {\tanh ^{-1}(a x)^3}{1-a x} \, dx}{a^2 c}-\frac {6 \int \frac {\tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}+\frac {6 \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a c}\\ &=-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {4 \tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a^2 c}+\frac {6 \int \frac {\tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}-\frac {12 \int \frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}\\ &=-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {6 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {4 \tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \int \frac {\text {Li}_4\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}-\frac {12 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}-\frac {12 \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}\\ &=-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {6 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {4 \tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {6 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {Li}_5\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}-\frac {6 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}+\frac {6 \int \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}\\ &=-\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {6 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {4 \tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {6 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {Li}_5\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 233, normalized size = 0.61 \[ -\frac {-\frac {1}{2} \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4+2 \left (\tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x)+3\right ) \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x) \text {Li}_4\left (-e^{-2 \tanh ^{-1}(a x)}\right )+3 \left (\tanh ^{-1}(a x)+1\right )^2 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text {Li}_4\left (-e^{-2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \text {Li}_5\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {2}{5} \tanh ^{-1}(a x)^5+a x \tanh ^{-1}(a x)^4-\tanh ^{-1}(a x)^4+2 a x \tanh ^{-1}(a x)^3-2 \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^4 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-4 \tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-6 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a^3 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{2} \operatorname {artanh}\left (a x\right )^{4}}{a c x - c}, x\right ) \]
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 \[ \int -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{4}}{a c x - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.98, size = 496, normalized size = 1.29 \[ -\frac {x^{2} \arctanh \left (a x \right )^{4}}{2 a c}-\frac {x \arctanh \left (a x \right )^{4}}{a^{2} c}-\frac {2 x \arctanh \left (a x \right )^{3}}{a^{2} c}-\frac {\arctanh \left (a x \right )^{4}}{2 a^{3} c}-\frac {2 \arctanh \left (a x \right )^{3}}{a^{3} c}+\frac {\arctanh \left (a x \right )^{4} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {2 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \polylog \left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3} c}+\frac {6 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {6 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c} \]
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 \[ -\frac {4 \, \log \left (-a x + 1\right )^{5} + 5 \, {\left (2 \, \log \left (-a x + 1\right )^{4} - 4 \, \log \left (-a x + 1\right )^{3} + 6 \, \log \left (-a x + 1\right )^{2} - 6 \, \log \left (-a x + 1\right ) + 3\right )} {\left (a x - 1\right )}^{2} + 40 \, {\left (\log \left (-a x + 1\right )^{4} - 4 \, \log \left (-a x + 1\right )^{3} + 12 \, \log \left (-a x + 1\right )^{2} - 24 \, \log \left (-a x + 1\right ) + 24\right )} {\left (a x - 1\right )}}{320 \, a^{3} c} + \frac {1}{16} \, \int -\frac {x^{2} \log \left (a x + 1\right )^{4} - 4 \, x^{2} \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right ) + 6 \, x^{2} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{2} - 4 \, x^{2} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{3}}{a c x - c}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^4}{c-a\,c\,x} \,d x \]
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 \[ - \frac {\int \frac {x^{2} \operatorname {atanh}^{4}{\left (a x \right )}}{a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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