Optimal. Leaf size=67 \[ -\frac {\text {Li}_3\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {\text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {\log \left (2-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1593, 5932, 5948, 6058, 6610} \[ -\frac {\text {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 c}+\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{c}+\frac {\log \left (2-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1593
Rule 5932
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{c x-a c x^2} \, dx &=\int \frac {\tanh ^{-1}(a x)^2}{x (c-a c x)} \, dx\\ &=\frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {a \int \frac {\text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{c}-\frac {\text {Li}_3\left (-1+\frac {2}{1-a x}\right )}{2 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 59, normalized size = 0.88 \[ \frac {\tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )}{c}-\frac {\text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )}{2 c}+\frac {\tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )}{c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{2}}{a c x^{2} - c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{2}}{a c x^{2} - c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.52, size = 717, normalized size = 10.70 \[ \frac {\arctanh \left (a x \right )^{2} \ln \left (a x \right )}{c}-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{c}-\frac {\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{c}+\frac {\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {2 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \pi }{2 c}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \pi }{2 c}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3} \pi }{2 c}+\frac {i \arctanh \left (a x \right )^{2} \pi }{c}+\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3} \pi }{c}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \pi }{2 c}-\frac {i \arctanh \left (a x \right )^{2} \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \pi }{c}+\frac {\arctanh \left (a x \right )^{2} \ln \relax (2)}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (-a x + 1\right )^{3}}{12 \, c} + \frac {1}{4} \, \int -\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a c x^{2} - c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{c\,x-a\,c\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a x^{2} - x}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________