Optimal. Leaf size=133 \[ b \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-b \text {Li}_2\left (\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{2} b^2 \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {1}{2} b^2 \text {Li}_3\left (\frac {2}{1-\frac {c}{x}}-1\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6095, 5914, 6052, 5948, 6058, 6610} \[ b \text {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-b \text {PolyLog}\left (2,\frac {2}{1-\frac {c}{x}}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{2} b^2 \text {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,\frac {2}{1-\frac {c}{x}}-1\right )-2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6095
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+(4 b c) \operatorname {Subst}\left (\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,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )-(2 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )+(2 b c) \operatorname {Subst}\left (\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,x,\frac {1}{x}\right )\\ &=-2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )-b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-\frac {c}{x}}\right )-\left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\frac {1}{x}\right )+\left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {\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 )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {c}{x}}\right )+b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )-b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-\frac {c}{x}}\right )-\frac {1}{2} b^2 \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right )+\frac {1}{2} b^2 \text {Li}_3\left (-1+\frac {2}{1-\frac {c}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 114, normalized size = 0.86 \[ \frac {1}{2} b \left (2 \text {Li}_2\left (\frac {c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )-2 \text {Li}_2\left (\frac {c+x}{x-c}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )+b \left (\text {Li}_3\left (\frac {c+x}{x-c}\right )-\text {Li}_3\left (\frac {c+x}{c-x}\right )\right )\right )-2 \tanh ^{-1}\left (\frac {c+x}{c-x}\right ) \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2 \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + 2 \, a b \operatorname {artanh}\left (\frac {c}{x}\right ) + a^{2}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 780, normalized size = 5.86 \[ -a^{2} \ln \left (\frac {c}{x}\right )-b^{2} \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )^{2}+b^{2} \arctanh \left (\frac {c}{x}\right ) \polylog \left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-\frac {b^{2} \polylog \left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}+b^{2} \arctanh \left (\frac {c}{x}\right )^{2} \ln \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )-b^{2} \arctanh \left (\frac {c}{x}\right )^{2} \ln \left (1-\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 b^{2} \arctanh \left (\frac {c}{x}\right ) \polylog \left (2, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 b^{2} \polylog \left (3, \frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-b^{2} \arctanh \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-2 b^{2} \arctanh \left (\frac {c}{x}\right ) \polylog \left (2, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )+2 b^{2} \polylog \left (3, -\frac {1+\frac {c}{x}}{\sqrt {1-\frac {c^{2}}{x^{2}}}}\right )-\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right )^{3} \arctanh \left (\frac {c}{x}\right )^{2}}{2}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right )^{2} \arctanh \left (\frac {c}{x}\right )^{2}}{2}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right )^{2} \arctanh \left (\frac {c}{x}\right )^{2}}{2}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}-1\right )}{1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}}\right ) \arctanh \left (\frac {c}{x}\right )^{2}}{2}-2 a b \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )+a b \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )+a b \dilog \left (\frac {c}{x}\right )+a b \dilog \left (1+\frac {c}{x}\right ) \]
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 \[ a^{2} \log \relax (x) + \int \frac {b^{2} {\left (\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )\right )}^{2}}{4 \, x} + \frac {a b {\left (\log \left (\frac {c}{x} + 1\right ) - \log \left (-\frac {c}{x} + 1\right )\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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