Optimal. Leaf size=117 \[ 2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6033, 6199,
6095, 6205, 6745} \begin {gather*} -b \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 \text {Li}_3\left (\frac {2}{1-c x}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6033
Rule 6095
Rule 6199
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-(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\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+(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-(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\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\left (b^2 c\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 120, normalized size = 1.03 \begin {gather*} 2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (\frac {1+c x}{-1+c x}\right )+\frac {1}{2} b \left (2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {1+c x}{1-c x}\right )-2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {1+c x}{-1+c x}\right )+b \left (-\text {PolyLog}\left (3,\frac {1+c x}{1-c x}\right )+\text {PolyLog}\left (3,\frac {1+c x}{-1+c x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.78, size = 701, normalized size = 5.99
method | result | size |
derivativedivides | \(a^{2} \ln \left (c x \right )+b^{2} \ln \left (c x \right ) \arctanh \left (c x \right )^{2}-b^{2} \arctanh \left (c x \right ) \polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {b^{2} \polylog \left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}-b^{2} \arctanh \left (c x \right )^{2} \ln \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )+b^{2} \arctanh \left (c x \right )^{2} \ln \left (1-\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+2 b^{2} \arctanh \left (c x \right ) \polylog \left (2, \frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-2 b^{2} \polylog \left (3, \frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+b^{2} \arctanh \left (c x \right )^{2} \ln \left (1+\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+2 b^{2} \arctanh \left (c x \right ) \polylog \left (2, -\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-2 b^{2} \polylog \left (3, -\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-\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 (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{2} \arctanh \left (c x \right )^{2}}{2}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )\right ) \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 (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right ) \arctanh \left (c x \right )^{2}}{2}+\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{3} \arctanh \left (c x \right )^{2}}{2}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{2} \arctanh \left (c x \right )^{2}}{2}-a b \ln \left (c x \right ) \ln \left (c x +1\right )+2 a b \ln \left (c x \right ) \arctanh \left (c x \right )-a b \dilog \left (c x \right )-a b \dilog \left (c x +1\right )\) | \(701\) |
default | \(a^{2} \ln \left (c x \right )+b^{2} \ln \left (c x \right ) \arctanh \left (c x \right )^{2}-b^{2} \arctanh \left (c x \right ) \polylog \left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {b^{2} \polylog \left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}-b^{2} \arctanh \left (c x \right )^{2} \ln \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )+b^{2} \arctanh \left (c x \right )^{2} \ln \left (1-\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+2 b^{2} \arctanh \left (c x \right ) \polylog \left (2, \frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-2 b^{2} \polylog \left (3, \frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+b^{2} \arctanh \left (c x \right )^{2} \ln \left (1+\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+2 b^{2} \arctanh \left (c x \right ) \polylog \left (2, -\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-2 b^{2} \polylog \left (3, -\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )-\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 (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{2} \arctanh \left (c x \right )^{2}}{2}+\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )\right ) \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 (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right ) \arctanh \left (c x \right )^{2}}{2}+\frac {i b^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{3} \arctanh \left (c x \right )^{2}}{2}-\frac {i b^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}-1\right )}{1+\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}}\right )^{2} \arctanh \left (c x \right )^{2}}{2}-a b \ln \left (c x \right ) \ln \left (c x +1\right )+2 a b \ln \left (c x \right ) \arctanh \left (c x \right )-a b \dilog \left (c x \right )-a b \dilog \left (c x +1\right )\) | \(701\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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