Optimal. Leaf size=295 \[ -\frac {b \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (\frac {2}{1-c x}-1\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 d^2}+\frac {b^2}{2 d^2 (c x+1)}-\frac {b^2 \tanh ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.64, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5940, 5914, 6052, 5948, 6058, 6610, 5928, 5926, 627, 44, 207, 5918, 6056} \[ -\frac {b \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^2}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {b^2}{2 d^2 (c x+1)}-\frac {b^2 \tanh ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 627
Rule 5914
Rule 5918
Rule 5926
Rule 5928
Rule 5940
Rule 5948
Rule 6052
Rule 6056
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x (d+c d x)^2} \, dx &=\int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)^2}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx}{d^2}-\frac {c \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{d^2}-\frac {c \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{d^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {(2 b c) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^2}-\frac {(2 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}-\frac {(4 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^2}+\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d^2}+\frac {(2 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}-\frac {(2 b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}+\frac {\left (b^2 c\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {\left (b^2 c\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^2}+\frac {\left (b^2 c\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (b^2 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {\left (b^2 c\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{d^2}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {\left (b^2 c\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=\frac {b^2}{2 d^2 (1+c x)}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {\left (b^2 c\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {b^2}{2 d^2 (1+c x)}-\frac {b^2 \tanh ^{-1}(c x)}{2 d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}\\ \end {align*}
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Mathematica [C] time = 0.86, size = 254, normalized size = 0.86 \[ \frac {\frac {24 a^2}{c x+1}+24 a^2 \log (c x)-24 a^2 \log (c x+1)+12 a b \left (-2 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+b^2 \left (24 \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )-12 \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )-16 \tanh ^{-1}(c x)^3+24 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-12 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )-12 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-6 \sinh \left (2 \tanh ^{-1}(c x)\right )+12 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+12 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+6 \cosh \left (2 \tanh ^{-1}(c x)\right )+i \pi ^3\right )}{24 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{c^{2} d^{2} x^{3} + 2 \, c d^{2} x^{2} + d^{2} x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 1566, normalized size = 5.31 \[ \text {result too large to display} \]
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 \[ a^{2} {\left (\frac {1}{c d^{2} x + d^{2}} - \frac {\log \left (c x + 1\right )}{d^{2}} + \frac {\log \relax (x)}{d^{2}}\right )} + \frac {{\left (b^{2} - {\left (b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{4 \, {\left (c d^{2} x + d^{2}\right )}} + \int \frac {{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c x - a b\right )} \log \left (c x + 1\right ) - 2 \, {\left (b^{2} c^{2} x^{2} - 2 \, a b + {\left (2 \, a b c + b^{2} c\right )} x - {\left (b^{2} c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c^{3} d^{2} x^{4} + c^{2} d^{2} x^{3} - c d^{2} x^{2} - d^{2} x\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x\,{\left (d+c\,d\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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