Optimal. Leaf size=384 \[ -\frac {2 \tanh ^{-1}(a x)^3}{a^3 c}-\frac {2 x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {\tanh ^{-1}(a x)^4}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^4}{a^2 c}-\frac {x^2 \tanh ^{-1}(a x)^4}{2 a c}+\frac {6 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {4 \tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {\tanh ^{-1}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {6 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {2 \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {6 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^3 c}-\frac {3 \text {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{2 a^3 c} \]
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Rubi [A]
time = 0.60, 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 = {6077, 6037,
6127, 6021, 6131, 6055, 6095, 6205, 6745, 6209} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 6021
Rule 6037
Rule 6055
Rule 6077
Rule 6095
Rule 6127
Rule 6131
Rule 6205
Rule 6209
Rule 6745
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 = 0.27, size = 233, normalized size = 0.61 \begin {gather*} -\frac {-2 \tanh ^{-1}(a x)^3+2 a x \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^4+a x \tanh ^{-1}(a x)^4-\frac {1}{2} \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4-\frac {2}{5} \tanh ^{-1}(a x)^5-6 \tanh ^{-1}(a x)^2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )-4 \tanh ^{-1}(a x)^3 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^4 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \left (3+3 \tanh ^{-1}(a x)+\tanh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \left (1+\tanh ^{-1}(a x)\right )^2 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text {PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x) \text {PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \text {PolyLog}\left (5,-e^{-2 \tanh ^{-1}(a x)}\right )}{a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.95, size = 442, normalized size = 1.15
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )^{3} \left (a x \arctanh \left (a x \right )+3 \arctanh \left (a x \right )+4\right ) \left (a x -1\right )}{2 c}+\frac {\arctanh \left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {2 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \polylog \left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {4 \arctanh \left (a x \right )^{3}}{c}+\frac {6 \arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {2 \arctanh \left (a x \right )^{4}}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {6 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}}{a^{3}}\) | \(442\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )^{3} \left (a x \arctanh \left (a x \right )+3 \arctanh \left (a x \right )+4\right ) \left (a x -1\right )}{2 c}+\frac {\arctanh \left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {2 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \polylog \left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {4 \arctanh \left (a x \right )^{3}}{c}+\frac {6 \arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {2 \arctanh \left (a x \right )^{4}}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {6 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}+\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}}{a^{3}}\) | \(442\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \frac {\int \frac {x^{2} \operatorname {atanh}^{4}{\left (a x \right )}}{a x - 1}\, dx}{c} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^4}{c-a\,c\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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