Optimal. Leaf size=111 \[ \frac {3 b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {3 b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{c x+1}\right )}{4 c} \]
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Rubi [A] time = 0.22, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5918, 5948, 6056, 6060, 6610} \[ \frac {3 b^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {3 b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2}{c x+1}\right )}{4 c}-\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c} \]
Antiderivative was successfully verified.
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Rule 5918
Rule 5948
Rule 6056
Rule 6060
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{1+c x} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{c}+(3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{c}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c}-\left (3 b^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{c}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c}-\frac {1}{2} \left (3 b^3\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+c x}\right )}{c}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 c}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 c}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{1+c x}\right )}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 152, normalized size = 1.37 \[ \frac {4 a^3 \log (c x+1)-12 a^2 b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+6 b^2 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)\right )-12 a b^2 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+6 b \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+3 b^3 \text {Li}_4\left (-e^{-2 \tanh ^{-1}(c x)}\right )-4 b^3 \tanh ^{-1}(c x)^3 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x\right ) + a^{3}}{c x + 1}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{c x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.46, size = 1491, normalized size = 13.43 \[ \text {result too large to display} \]
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 \[ -\frac {b^{3} \log \left (c x + 1\right ) \log \left (-c x + 1\right )^{3}}{8 \, c} + \frac {a^{3} \log \left (c x + 1\right )}{c} + \int \frac {{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{3} + 6 \, {\left (a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )^{2} + 6 \, {\left (b^{3} c x \log \left (c x + 1\right ) + a b^{2} c x - a b^{2}\right )} \log \left (-c x + 1\right )^{2} + 12 \, {\left (a^{2} b c x - a^{2} b\right )} \log \left (c x + 1\right ) - 3 \, {\left (4 \, a^{2} b c x - 4 \, a^{2} b + {\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{c\,x+1} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{c x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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