Optimal. Leaf size=208 \[ -\frac {3}{2} b^2 \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {3}{2} b^2 \text {Li}_3\left (\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {3}{2} b \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{2} b \text {Li}_2\left (\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {3}{4} b^3 \text {Li}_4\left (1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{4} b^3 \text {Li}_4\left (\frac {2}{1-\frac {c}{x}}-1\right ) \]
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Rubi [A] time = 0.51, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6095, 5914, 6052, 5948, 6058, 6062, 6610} \[ -\frac {3}{2} b^2 \text {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {3}{2} b^2 \text {PolyLog}\left (3,\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {3}{2} b \text {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{2} b \text {PolyLog}\left (2,\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{4} b^3 \text {PolyLog}\left (4,1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{4} b^3 \text {PolyLog}\left (4,\frac {2}{1-\frac {c}{x}}-1\right )-2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6062
Rule 6095
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{x} \, dx,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+(6 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )-(3 b c) \operatorname {Subst}\left (\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,x,\frac {1}{x}\right )+(3 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-\frac {c}{x}}\right )-\left (3 b^2 c\right ) \operatorname {Subst}\left (\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,x,\frac {1}{x}\right )+\left (3 b^2 c\right ) \operatorname {Subst}\left (\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,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_3\left (-1+\frac {2}{1-\frac {c}{x}}\right )+\frac {1}{2} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \text {Li}_2\left (-1+\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{2} b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_3\left (-1+\frac {2}{1-\frac {c}{x}}\right )+\frac {3}{4} b^3 \text {Li}_4\left (1-\frac {2}{1-\frac {c}{x}}\right )-\frac {3}{4} b^3 \text {Li}_4\left (-1+\frac {2}{1-\frac {c}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 171, normalized size = 0.82 \[ \frac {3}{4} b \left (2 \text {Li}_2\left (\frac {c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2-2 \text {Li}_2\left (\frac {c+x}{x-c}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2+b \left (-2 \text {Li}_3\left (\frac {c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )+2 \text {Li}_3\left (\frac {c+x}{x-c}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )+b \left (\text {Li}_4\left (\frac {c+x}{c-x}\right )-\text {Li}_4\left (\frac {c+x}{x-c}\right )\right )\right )\right )-2 \tanh ^{-1}\left (\frac {c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (\frac {c}{x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (\frac {c}{x}\right ) + a^{3}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 1631, normalized size = 7.84 \[ \text {result too large to display} \]
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 \[ a^{3} \log \relax (x) + \int \frac {b^{3} {\left (\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )\right )}^{3}}{8 \, x} + \frac {3 \, a b^{2} {\left (\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )\right )}^{2}}{4 \, x} + \frac {3 \, a^{2} b {\left (\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )\right )}}{2 \, x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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