Optimal. Leaf size=394 \[ -\frac {4 b \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (c x+1)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (c x+1)}-\frac {20 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^5 d^2}+\frac {4 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2}-\frac {2 a b x}{c^4 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{c^5 d^2}-\frac {b^2}{2 c^5 d^2 (c x+1)}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2} \]
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Rubi [A] time = 0.84, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 19, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 321, 206, 5928, 5926, 627, 44, 207, 6056, 6610} \[ -\frac {4 b \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2}-\frac {10 b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {2 b^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{c^5 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {2 a b x}{c^4 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (c x+1)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (c x+1)}-\frac {20 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^5 d^2}+\frac {4 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (c x+1)}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 207
Rule 260
Rule 321
Rule 627
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5926
Rule 5928
Rule 5940
Rule 5948
Rule 5980
Rule 5984
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^2} \, dx &=\int \left (\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {2 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 d^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2 (1+c x)^2}-\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c^4 d^2}+\frac {3 \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^4 d^2}-\frac {4 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c^4 d^2}-\frac {2 \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^3 d^2}+\frac {\int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^2 d^2}\\ &=\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}+\frac {(2 b) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}-\frac {(8 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {(6 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^3 d^2}+\frac {(2 b) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^2 d^2}-\frac {(2 b) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c d^2}\\ &=\frac {3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^4 d^2}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^4 d^2}-\frac {(2 b) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^4 d^2}+\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {(6 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^4 d^2}+\frac {\left (4 b^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}+\frac {(2 b) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^3 d^2}-\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^3 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^4 d^2}+\frac {b^2 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^4 d^2}-\frac {\left (2 b^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^4 d^2}+\frac {\left (6 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^2}-\frac {b^2 \int \frac {x^2}{1-c^2 x^2} \, dx}{3 c^2 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^5 d^2}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{3 c^4 d^2}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^4 d^2}+\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{c^4 d^2}+\frac {\left (2 b^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c^3 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^5 d^2}+\frac {b^2 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{2 c^4 d^2}\\ &=-\frac {2 a b x}{c^4 d^2}+\frac {b^2 x}{3 c^4 d^2}-\frac {b^2}{2 c^5 d^2 (1+c x)}+\frac {b^2 \tanh ^{-1}(c x)}{6 c^5 d^2}-\frac {2 b^2 x \tanh ^{-1}(c x)}{c^4 d^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3 d^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^2 (1+c x)}+\frac {29 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^5 d^2}+\frac {3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^2}+\frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^2 d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^2 (1+c x)}-\frac {20 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^5 d^2}+\frac {4 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{c^5 d^2}-\frac {10 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^5 d^2}-\frac {4 b \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}-\frac {2 b^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{c^5 d^2}\\ \end {align*}
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Mathematica [A] time = 1.71, size = 425, normalized size = 1.08 \[ \frac {4 a^2 c^3 x^3-12 a^2 c^2 x^2+36 a^2 c x-\frac {12 a^2}{c x+1}-48 a^2 \log (c x+1)+2 a b \left (2 c^2 x^2+20 \log \left (1-c^2 x^2\right )+2 \tanh ^{-1}(c x) \left (2 c^3 x^3-6 c^2 x^2+18 c x+24 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )-3 \cosh \left (2 \tanh ^{-1}(c x)\right )+6\right )-24 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )-12 c x+3 \sinh \left (2 \tanh ^{-1}(c x)\right )-3 \cosh \left (2 \tanh ^{-1}(c x)\right )-2\right )+b^2 \left (4 c^3 x^3 \tanh ^{-1}(c x)^2-12 \log \left (1-c^2 x^2\right )-12 c^2 x^2 \tanh ^{-1}(c x)^2+4 c^2 x^2 \tanh ^{-1}(c x)-8 \left (6 \tanh ^{-1}(c x)-5\right ) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )-24 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )+4 c x+36 c x \tanh ^{-1}(c x)^2-24 c x \tanh ^{-1}(c x)-28 \tanh ^{-1}(c x)^2-4 \tanh ^{-1}(c x)+48 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-80 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+6 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+6 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )-3 \cosh \left (2 \tanh ^{-1}(c x)\right )\right )}{12 c^5 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{4} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {artanh}\left (c x\right ) + a^{2} x^{4}}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c d x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.31, size = 1467, normalized size = 3.72 \[ \text {result too large to display} \]
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 \[ -\frac {1}{3} \, a^{2} {\left (\frac {3}{c^{6} d^{2} x + c^{5} d^{2}} - \frac {c^{2} x^{3} - 3 \, c x^{2} + 9 \, x}{c^{4} d^{2}} + \frac {12 \, \log \left (c x + 1\right )}{c^{5} d^{2}}\right )} + \frac {{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{3} x^{3} + 6 \, b^{2} c^{2} x^{2} + 9 \, b^{2} c x - 3 \, b^{2} - 12 \, {\left (b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{12 \, {\left (c^{6} d^{2} x + c^{5} d^{2}\right )}} - \int -\frac {3 \, {\left (b^{2} c^{5} x^{5} - b^{2} c^{4} x^{4}\right )} \log \left (c x + 1\right )^{2} + 12 \, {\left (a b c^{5} x^{5} - a b c^{4} x^{4}\right )} \log \left (c x + 1\right ) - 2 \, {\left (4 \, b^{2} c^{3} x^{3} + 15 \, b^{2} c^{2} x^{2} + {\left (6 \, a b c^{5} + b^{2} c^{5}\right )} x^{5} - {\left (6 \, a b c^{4} + b^{2} c^{4}\right )} x^{4} + 6 \, b^{2} c x - 3 \, b^{2} + 3 \, {\left (b^{2} c^{5} x^{5} - b^{2} c^{4} x^{4} - 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2} c x - 4 \, b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, {\left (c^{7} d^{2} x^{3} + c^{6} d^{2} x^{2} - c^{5} d^{2} x - c^{4} d^{2}\right )}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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