Optimal. Leaf size=239 \[ \frac {a \tanh ^{-1}(a x)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x)^2 \text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x) \text {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \text {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \]
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Rubi [A]
time = 0.37, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6081, 6037,
6135, 6079, 6095, 6203, 6207, 6745, 6205, 6209} \begin {gather*} -\frac {3 a \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{c}-\frac {3 a \text {Li}_5\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {2 a \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^3}{c}-\frac {6 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {3 a \text {Li}_3\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {6 a \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a \text {Li}_4\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {a \tanh ^{-1}(a x)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \log \left (2-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{c}+\frac {4 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 6037
Rule 6079
Rule 6081
Rule 6095
Rule 6135
Rule 6203
Rule 6205
Rule 6207
Rule 6209
Rule 6745
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^4}{x^2 (c-a c x)} \, dx &=a \int \frac {\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx+\frac {\int \frac {\tanh ^{-1}(a x)^4}{x^2} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {(4 a) \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx}{c}-\frac {\left (4 a^2\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 {a \tanh ^{-1}(a x)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {(4 a) \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 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a \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) \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac {\left (12 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)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {\left (3 a^2\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^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)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a \text {Li}_5\left (-1+\frac {2}{1-a x}\right )}{2 c}+\frac {\left (6 a^2\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a \tanh ^{-1}(a x)^4}{c}-\frac {\tanh ^{-1}(a x)^4}{c x}+\frac {a \tanh ^{-1}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {4 a \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {2 a \tanh ^{-1}(a x)^3 \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \tanh ^{-1}(a x)^2 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {6 a \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \tanh ^{-1}(a x) \text {Li}_4\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {3 a \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a \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 = 0.30, size = 172, normalized size = 0.72 \begin {gather*} -\frac {a \left (-\frac {\pi ^4}{16}+\frac {i \pi ^5}{160}+\tanh ^{-1}(a x)^4+\frac {\tanh ^{-1}(a x)^4}{a x}-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)^2 \left (3+\tanh ^{-1}(a x)\right ) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x) \left (2+\tanh ^{-1}(a x)\right ) \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. \(572\) vs.
\(2(237)=474\).
time = 12.43, size = 573, normalized size = 2.40
method | result | size |
derivativedivides | \(a \left (\frac {\arctanh \left (a x \right )^{4} \left (a x -1\right )}{c a x}+\frac {\arctanh \left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \arctanh \left (a x \right ) \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \polylog \left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\arctanh \left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \arctanh \left (a x \right ) \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \polylog \left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \arctanh \left (a x \right )^{4}}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(573\) |
default | \(a \left (\frac {\arctanh \left (a x \right )^{4} \left (a x -1\right )}{c a x}+\frac {\arctanh \left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \arctanh \left (a x \right ) \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \polylog \left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\arctanh \left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \arctanh \left (a x \right ) \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \polylog \left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \arctanh \left (a x \right )^{4}}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {4 \arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {12 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {24 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {24 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(573\) |
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^{3} - x^{2}}\, 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^2\,\left (c-a\,c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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