Optimal. Leaf size=164 \[ -\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}-\frac {\coth ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {\text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 e}+\frac {\text {Li}_2\left (1-\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e} \]
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Rubi [A] time = 0.03, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5923} \[ -\frac {\text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e}-\frac {\coth ^{-1}(c x) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {\text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 e}+\frac {\coth ^{-1}(c x) \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac {\log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e} \]
Antiderivative was successfully verified.
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Rule 5923
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(c x)^2}{d+e x} \, dx &=-\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {\coth ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {\coth ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{e}-\frac {\coth ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 e}-\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}\\ \end {align*}
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Mathematica [C] time = 7.80, size = 565, normalized size = 3.45 \[ \frac {\frac {24 (e-c d) (c d+e) \left (2 c d \sqrt {1-\frac {e^2}{c^2 d^2}} \coth ^{-1}(c x)^3 e^{-\tanh ^{-1}\left (\frac {e}{c d}\right )}+3 e \coth ^{-1}(c x)^2 \log \left (\frac {d+e x}{x \sqrt {1-\frac {1}{c^2 x^2}}}\right )-3 i \pi e \log \left (\frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}\right ) \coth ^{-1}(c x)-6 e \coth ^{-1}(c x) \text {Li}_2\left (-e^{\coth ^{-1}(c x)+\tanh ^{-1}\left (\frac {e}{c d}\right )}\right )-6 e \coth ^{-1}(c x) \text {Li}_2\left (e^{\coth ^{-1}(c x)+\tanh ^{-1}\left (\frac {e}{c d}\right )}\right )+6 e \text {Li}_3\left (-e^{\coth ^{-1}(c x)+\tanh ^{-1}\left (\frac {e}{c d}\right )}\right )+6 e \text {Li}_3\left (e^{\coth ^{-1}(c x)+\tanh ^{-1}\left (\frac {e}{c d}\right )}\right )-3 e \coth ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\coth ^{-1}(c x)} \left (c d \left (e^{2 \coth ^{-1}(c x)}-1\right )+e \left (e^{2 \coth ^{-1}(c x)}+1\right )\right )\right )-3 e \coth ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac {e}{c d}\right )+\coth ^{-1}(c x)}\right )-3 e \coth ^{-1}(c x)^2 \log \left (e^{\tanh ^{-1}\left (\frac {e}{c d}\right )+\coth ^{-1}(c x)}+1\right )-6 e \coth ^{-1}(c x) \tanh ^{-1}\left (\frac {e}{c d}\right ) \log \left (\frac {1}{2} i e^{-\tanh ^{-1}\left (\frac {e}{c d}\right )-\coth ^{-1}(c x)} \left (e^{2 \left (\tanh ^{-1}\left (\frac {e}{c d}\right )+\coth ^{-1}(c x)\right )}-1\right )\right )+6 e \coth ^{-1}(c x) \tanh ^{-1}\left (\frac {e}{c d}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {e}{c d}\right )+\coth ^{-1}(c x)\right )\right )-c d \coth ^{-1}(c x)^3+3 e \coth ^{-1}(c x)^3+3 i \pi e \coth ^{-1}(c x) \log \left (\frac {1}{2} \left (e^{-\coth ^{-1}(c x)}+e^{\coth ^{-1}(c x)}\right )\right )\right )}{3 c^2 d^2-3 e^2}+8 c d \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x) \text {Li}_2\left (e^{2 \coth ^{-1}(c x)}\right )+12 e \text {Li}_3\left (e^{2 \coth ^{-1}(c x)}\right )+8 e \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x)^2 \log \left (1-e^{2 \coth ^{-1}(c x)}\right )-i \pi ^3 e}{24 e^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.29, size = 926, normalized size = 5.65 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {arcoth}\left (c x\right )^{2}}{e x + d}\,{d x} \]
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 \[ \int \frac {{\mathrm {acoth}\left (c\,x\right )}^2}{d+e\,x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}^{2}{\left (c x \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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