Optimal. Leaf size=191 \[ -\frac {3 a \text {Li}_3\left (\frac {2}{a x+1}-1\right )}{2 c}+\frac {3 a \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{4 c}+\frac {3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{2 c}+\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac {3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c} \]
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Rubi [A] time = 0.46, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5934, 5916, 5988, 5932, 5948, 6056, 6610, 6060} \[ -\frac {3 a \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 c}+\frac {3 a \text {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 c}-\frac {3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{c}+\frac {3 a \tanh ^{-1}(a x) \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 c}+\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac {3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 5932
Rule 5934
Rule 5948
Rule 5988
Rule 6056
Rule 6060
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx &=-\left (a \int \frac {\tanh ^{-1}(a x)^3}{x (c+a c x)} \, dx\right )+\frac {\int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}-\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {\left (3 a^2\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 c}-\frac {\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}+\frac {\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^3}{c}-\frac {\tanh ^{-1}(a x)^3}{c x}+\frac {3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 154, normalized size = 0.81 \[ \frac {a \left (-\frac {3}{2} \left (\tanh ^{-1}(a x)-2\right ) \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \left (\tanh ^{-1}(a x)-1\right ) \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )-\frac {3}{4} \text {Li}_4\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \tanh ^{-1}(a x)^4-\frac {\tanh ^{-1}(a x)^3}{a x}-\tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac {\pi ^4}{64}+\frac {i \pi ^3}{8}\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x^{3} + c x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a c x + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.78, size = 1451, normalized size = 7.60 \[ \text {result too large to display} \]
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 \[ -\frac {{\left (a x \log \left (a x + 1\right ) - 1\right )} \log \left (-a x + 1\right )^{3}}{8 \, c x} + \frac {1}{8} \, \int \frac {{\left (a x - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a x - 1\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) - 3 \, {\left (a^{2} x^{2} + a x - {\left (a^{3} x^{3} + a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{a^{2} c x^{4} - c x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,\left (c+a\,c\,x\right )} \,d x \]
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 \[ \frac {\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a x^{3} + x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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