Optimal. Leaf size=412 \[ \frac {b e \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b e \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b e \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{d^2}-\frac {2 e \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {e \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \text {Li}_3\left (\frac {2}{1-c x}-1\right )}{2 d^2}+\frac {b^2 e \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d^2}-\frac {b^2 c \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.60, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5940, 5916, 5988, 5932, 2447, 5914, 6052, 5948, 6058, 6610, 5922} \[ \frac {b e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b e \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {b e \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}-\frac {b^2 e \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^2}+\frac {b^2 e \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b^2 e \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 d^2}-\frac {b^2 c \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d}-\frac {2 e \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {e \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2447
Rule 5914
Rule 5916
Rule 5922
Rule 5932
Rule 5940
Rule 5948
Rule 5988
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 (d+e x)} \, dx &=\int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x^2}-\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx}{d}-\frac {e \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+e x} \, dx}{d^2}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 e \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}+\frac {(2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac {(4 b c e) \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}{d^2}\\ &=\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 e \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}+\frac {(2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx}{d}-\frac {(2 b c e) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}+\frac {(2 b c e) \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}{d^2}\\ &=\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 e \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}-\frac {\left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}-\frac {\left (b^2 c e\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}+\frac {\left (b^2 c e\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac {2 e \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}+\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d}-\frac {b e \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b^2 e \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 12.75, size = 1010, normalized size = 2.45 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{e x^{3} + d x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.77, size = 26776, normalized size = 64.99 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {1}{d x}\right )} - \frac {b^{2} \log \left (-c x + 1\right )^{2}}{4 \, d x} - \int -\frac {{\left (b^{2} c d x - b^{2} d\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c d x - a b d\right )} \log \left (c x + 1\right ) + 2 \, {\left (b^{2} c e x^{2} + 2 \, a b d - {\left (2 \, a b c d - b^{2} c d\right )} x - {\left (b^{2} c d x - b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c d e x^{4} - d^{2} x^{2} + {\left (c d^{2} - d e\right )} x^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{x^{2} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________