Optimal. Leaf size=275 \[ \frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d}+\frac {\text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d}-\frac {\text {Li}_3\left (\frac {2}{1-c x}-1\right )}{2 d}-\frac {\text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 d}-\frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)}{d}+\frac {\text {Li}_2\left (\frac {2}{1-c x}-1\right ) \tanh ^{-1}(c x)}{d}-\frac {\text {Li}_2\left (1-\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)}{d}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)^2}{d}+\frac {\log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)^2}{d} \]
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Rubi [A] time = 0.34, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5940, 5914, 6052, 5948, 6058, 6610, 5922} \[ \frac {\text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 d}+\frac {\tanh ^{-1}(c x) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d}-\frac {\text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d}-\frac {\text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d}-\frac {\tanh ^{-1}(c x) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right )}{d}-\frac {\tanh ^{-1}(c x) \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)^2}{d}+\frac {\log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)^2}{d} \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5922
Rule 5940
Rule 5948
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(c x)^2}{x (d+e x)} \, dx &=\int \left (\frac {\tanh ^{-1}(c x)^2}{d x}-\frac {e \tanh ^{-1}(c x)^2}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\tanh ^{-1}(c x)^2}{x} \, dx}{d}-\frac {e \int \frac {\tanh ^{-1}(c x)^2}{d+e x} \, dx}{d}\\ &=\frac {2 \tanh ^{-1}(c x)^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d}+\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d}-\frac {(4 c) \int \frac {\tanh ^{-1}(c x) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=\frac {2 \tanh ^{-1}(c x)^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d}+\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d}+\frac {(2 c) \int \frac {\tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d}-\frac {(2 c) \int \frac {\tanh ^{-1}(c x) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=\frac {2 \tanh ^{-1}(c x)^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d}+\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d}+\frac {c \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d}-\frac {c \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=\frac {2 \tanh ^{-1}(c x)^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{d}-\frac {\tanh ^{-1}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d}-\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d}+\frac {\tanh ^{-1}(c x) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}+\frac {\text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d}-\frac {\text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d}-\frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d}+\frac {\text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 9.37, size = 521, normalized size = 1.89 \[ \frac {16 e \sqrt {1-\frac {c^2 d^2}{e^2}} \tanh ^{-1}(c x)^3 e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )}+24 c d \tanh ^{-1}(c x)^2 \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )+12 i \pi c d \log \left (1-c^2 x^2\right ) \tanh ^{-1}(c x)-48 c d \tanh ^{-1}(c x) \text {Li}_2\left (-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-48 c d \tanh ^{-1}(c x) \text {Li}_2\left (e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+48 c d \text {Li}_3\left (-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )+48 c d \text {Li}_3\left (e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-24 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}\right )-24 c d \tanh ^{-1}(c x)^2 \log \left (e^{\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)}+1\right )-24 c d \tanh ^{-1}(c x)^2 \log \left (e^{-\tanh ^{-1}(c x)} \left (c d \left (e^{2 \tanh ^{-1}(c x)}+1\right )+e \left (e^{2 \tanh ^{-1}(c x)}-1\right )\right )\right )-48 c d \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (\frac {1}{2} i e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )-\tanh ^{-1}(c x)} \left (e^{2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}-1\right )\right )+48 c d \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+24 c d \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )-12 c d \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )+16 c d \tanh ^{-1}(c x)^3+24 c d \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+c d \log (16777216) \tanh ^{-1}(c x)^2+24 i \pi c d \tanh ^{-1}(c x) \log \left (e^{-\tanh ^{-1}(c x)}+e^{\tanh ^{-1}(c x)}\right )-24 i \pi c d \log (2) \tanh ^{-1}(c x)+i \pi ^3 c d-16 e \tanh ^{-1}(c x)^3}{24 c d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (c x\right )^{2}}{e x^{2} + d x}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (c x\right )^{2}}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 1507, normalized size = 5.48 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {artanh}\left (c x\right )^{2}}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (c\,x\right )}^2}{x\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (c x \right )}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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