Optimal. Leaf size=380 \[ \frac {2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \]
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Rubi [A]
time = 0.66, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6081, 6037,
6129, 6135, 6079, 6095, 6203, 6745, 6207, 6205, 6209} \begin {gather*} -\frac {3 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )}{c}-\frac {3 a^2 \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{c}-\frac {3 a^2 \text {Li}_5\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {2 a^2 \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^3}{c}-\frac {6 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {3 a^2 \text {Li}_3\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {6 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}-\frac {6 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a^2 \text {Li}_4\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}+\frac {2 a^2 \tanh ^{-1}(a x)^3}{c}+\frac {a^2 \log \left (2-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{c}+\frac {4 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac {6 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}-\frac {2 a \tanh ^{-1}(a x)^3}{c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6037
Rule 6079
Rule 6081
Rule 6095
Rule 6129
Rule 6135
Rule 6203
Rule 6205
Rule 6207
Rule 6209
Rule 6745
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^4}{x^3 (c-a c x)} \, dx &=a \int \frac {\tanh ^{-1}(a x)^4}{x^2 (c-a c x)} \, dx+\frac {\int \frac {\tanh ^{-1}(a x)^4}{x^3} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}+a^2 \int \frac {\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx+\frac {a \int \frac {\tanh ^{-1}(a x)^4}{x^2} \, dx}{c}+\frac {(2 a) \int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {(2 a) \int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}+\frac {\left (4 a^2\right ) \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac {\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{c}-\frac {\left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {\left (4 a^2\right ) \int \frac {\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx}{c}+\frac {\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}-\frac {\left (6 a^3\right ) \int \frac {\tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}+\frac {\left (6 a^3\right ) \int \frac {\tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac {\left (12 a^3\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 {2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {\left (3 a^3\right ) \int \frac {\text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac {\left (12 a^3\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}+\frac {\left (12 a^3\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 {2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \text {Li}_5\left (-1+\frac {2}{1-a x}\right )}{2 c}+\frac {\left (6 a^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}+\frac {\left (6 a^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac {2 a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac {\tanh ^{-1}(a x)^4}{2 c x^2}-\frac {a \tanh ^{-1}(a x)^4}{c x}+\frac {a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a^2 \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {6 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {Li}_5\left (-1+\frac {2}{1-a x}\right )}{2 c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.02, size = 250, normalized size = 0.66 \begin {gather*} -\frac {a^2 \left (-\frac {i \pi ^3}{4}-\frac {\pi ^4}{16}+\frac {i \pi ^5}{160}+2 \tanh ^{-1}(a x)^3+\frac {2 \tanh ^{-1}(a x)^3}{a x}+\frac {1}{2} \tanh ^{-1}(a x)^4+\frac {\tanh ^{-1}(a x)^4}{2 a^2 x^2}+\frac {\tanh ^{-1}(a x)^4}{a x}-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 )\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(778\) vs.
\(2(374)=748\).
time = 13.82, size = 779, normalized size = 2.05 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 \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 {\operatorname {atanh}^{4}{\left (a x \right )}}{a x^{4} - x^{3}}\, 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 {{\mathrm {atanh}\left (a\,x\right )}^4}{x^3\,\left (c-a\,c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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