Optimal. Leaf size=396 \[ -b c^3 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^3 d^3 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c^3 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (\frac {2}{1-c x}-1\right )+3 b^2 c^3 d^3 \log (x)+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b^2 c^2 d^3}{3 x}-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right ) \]
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Rubi [A] time = 0.93, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 17, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5940, 5916, 5982, 325, 206, 5988, 5932, 2447, 266, 36, 29, 31, 5948, 5914, 6052, 6058, 6610} \[ -b c^3 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^3 d^3 \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {10}{3} b^2 c^3 d^3 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^3 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )-\frac {b^2 c^2 d^3}{3 x}+3 b^2 c^3 d^3 \log (x)+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 206
Rule 266
Rule 325
Rule 2447
Rule 5914
Rule 5916
Rule 5932
Rule 5940
Rule 5948
Rule 5982
Rule 5988
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=\int \left (\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4}+\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx+\left (3 c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 c^2 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (c^3 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b c^4 d^3\right ) \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\\ &=3 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (3 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{3} \left (2 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (2 b c^4 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c^4 d^3\right ) \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+\left (3 b c^4 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (3 b^2 c^3 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^2 c^4 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c^4 d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b^2 c^4 d^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-3 b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (3 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 c^4 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^4 d^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (3 b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (3 b^2 c^5 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )\\ \end {align*}
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Mathematica [C] time = 0.73, size = 569, normalized size = 1.44 \[ \frac {d^3 \left (24 a^2 c^3 x^3 \log (x)-72 a^2 c^2 x^2-36 a^2 c x-8 a^2-24 a b c^3 x^3 \text {Li}_2(-c x)+24 a b c^3 x^3 \text {Li}_2(c x)+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (c x+1)-72 a b c^2 x^2-144 a b c^2 x^2 \tanh ^{-1}(c x)-80 a b c^3 x^3 \log \left (1-c^2 x^2\right )-8 a b c x-72 a b c x \tanh ^{-1}(c x)-16 a b \tanh ^{-1}(c x)+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )-80 b^2 c^3 x^3 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )+12 b^2 c^3 x^3 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 c^3 x^3 \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )+i \pi ^3 b^2 c^3 x^3-16 b^2 c^3 x^3 \tanh ^{-1}(c x)^3+116 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+8 b^2 c^3 x^3 \tanh ^{-1}(c x)+160 b^2 c^3 x^3 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-8 b^2 c^2 x^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)+72 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-36 b^2 c x \tanh ^{-1}(c x)^2-8 b^2 c x \tanh ^{-1}(c x)-8 b^2 \tanh ^{-1}(c x)^2\right )}{24 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} + {\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname {artanh}\left (c x\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.28, size = 1337, normalized size = 3.38 \[ \text {result too large to display} \]
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 \[ a^{2} c^{3} d^{3} \log \relax (x) - 3 \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} a b c^{2} d^{3} + \frac {3}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c d^{3} - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b d^{3} - \frac {3 \, a^{2} c^{2} d^{3}}{x} - \frac {3 \, a^{2} c d^{3}}{2 \, x^{2}} - \frac {a^{2} d^{3}}{3 \, x^{3}} - \frac {{\left (18 \, b^{2} c^{2} d^{3} x^{2} + 9 \, b^{2} c d^{3} x + 2 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2}}{24 \, x^{3}} - \int -\frac {3 \, {\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 12 \, {\left (a b c^{4} d^{3} x^{4} - a b c^{3} d^{3} x^{3}\right )} \log \left (c x + 1\right ) - {\left (12 \, a b c^{4} d^{3} x^{4} - 9 \, b^{2} c^{2} d^{3} x^{2} - 2 \, b^{2} c d^{3} x - 6 \, {\left (2 \, a b c^{3} d^{3} + 3 \, b^{2} c^{3} d^{3}\right )} x^{3} + 6 \, {\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, {\left (c x^{5} - x^{4}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^4} \,d x \]
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 \[ d^{3} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int \frac {a^{2} c^{3}}{x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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