Optimal. Leaf size=84 \[ \frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 c d} \]
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Rubi [A] time = 0.15, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5918, 5948, 6056, 6610} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c d}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 c d}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c d} \]
Antiderivative was successfully verified.
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Rule 5918
Rule 5948
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+c d x} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {(2 b) \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}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c d}-\frac {b^2 \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c d}+\frac {b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 102, normalized size = 1.21 \[ \frac {2 a^2 \log (c x+1)+2 b \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)\right )-4 a b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+b^2 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )-2 b^2 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{2 c d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{c d x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c d x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 822, normalized size = 9.79 \[ \frac {a^{2} \ln \left (c x +1\right )}{c d}+\frac {b^{2} \arctanh \left (c x \right )^{2} \ln \left (c x +1\right )}{c d}-\frac {2 b^{2} \arctanh \left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )}{c d}+\frac {2 b^{2} \arctanh \left (c x \right )^{3}}{3 c d}-\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \arctanh \left (c x \right )^{2}}{2 c d}-\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}\right )^{3} \arctanh \left (c x \right )^{2}}{2 c d}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}\right ) \arctanh \left (c x \right )^{2}}{2 c d}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}\right )^{2} \arctanh \left (c x \right )^{2}}{2 c d}-\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3} \arctanh \left (c x \right )^{2}}{2 c d}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}\right )^{2} \arctanh \left (c x \right )^{2}}{2 c d}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2} \arctanh \left (c x \right )^{2}}{c d}-\frac {b^{2} \arctanh \left (c x \right )^{2} \ln \relax (2)}{c d}-\frac {b^{2} \arctanh \left (c x \right ) \polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{c d}+\frac {b^{2} \polylog \left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2 c d}+\frac {2 a b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{c d}-\frac {a b \ln \left (c x +1\right )^{2}}{2 c d}+\frac {a b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{c d}-\frac {a b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{c d}-\frac {a b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{c d} \]
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 \[ \frac {b^{2} \log \left (c x + 1\right ) \log \left (-c x + 1\right )^{2}}{4 \, c d} + \frac {a^{2} \log \left (c d x + d\right )}{c d} - \int -\frac {{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c x - a b\right )} \log \left (c x + 1\right ) - 4 \, {\left (b^{2} c x \log \left (c x + 1\right ) + a b c x - a b\right )} \log \left (-c x + 1\right )}{4 \, {\left (c^{2} d x^{2} - d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c x + 1}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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