Optimal. Leaf size=479 \[ \frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{60} b c^6 d^3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{c x+1}\right )+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {37 b^2 c^5 d^3}{30 x}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right ) \]
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Rubi [A] time = 0.51, antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {43, 5938, 5916, 266, 44, 325, 206, 36, 29, 31, 5912, 5918, 2402, 2315} \[ -\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,-c x)+\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,c x)+\frac {37}{40} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}+\frac {28}{15} a b c^6 d^3 \log (x)-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}+\frac {37}{20} b c^6 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{60} b c^6 d^3 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac {37 b^2 c^5 d^3}{30 x}+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 44
Rule 206
Rule 266
Rule 325
Rule 2315
Rule 2402
Rule 5912
Rule 5916
Rule 5918
Rule 5938
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^7} \, dx &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {11 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^4}-\frac {14 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac {11 c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^2}-\frac {14 c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x}+\frac {37 c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{40 (-1+c x)}+\frac {c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{120 (1+c x)}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^6} \, dx+\frac {1}{5} \left (6 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx+\frac {1}{6} \left (11 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac {1}{15} \left (28 b c^4 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac {1}{6} \left (11 b c^5 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{15} \left (28 b c^6 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx-\frac {1}{60} \left (b c^7 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx-\frac {1}{20} \left (37 b c^7 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c x} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {1}{15} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^5 \left (1-c^2 x^2\right )} \, dx+\frac {1}{10} \left (3 b^2 c^3 d^3\right ) \int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx+\frac {1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{15} \left (14 b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{20} \left (37 b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^3 d^3}{10 x^3}-\frac {14 b^2 c^5 d^3}{15 x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {1}{30} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{10} \left (3 b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^6 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} \left (37 b^2 c^6 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )+\frac {1}{15} \left (14 b^2 c^7 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^3 d^3}{10 x^3}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {14}{15} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{30} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {c^2}{x^2}+\frac {c^4}{x}-\frac {c^6}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{10} \left (3 b^2 c^7 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{12} \left (11 b^2 c^8 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}{60 x^4}-\frac {b^2 c^3 d^3}{10 x^3}-\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {37 b^2 c^5 d^3}{30 x}+\frac {37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac {11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac {14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+\frac {1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 1.39, size = 402, normalized size = 0.84 \[ -\frac {d^3 \left (60 a^2 c^3 x^3+135 a^2 c^2 x^2+108 a^2 c x+30 a^2-336 a b c^6 x^6 \log (c x)+165 a b c^6 x^6 \log (1-c x)-165 a b c^6 x^6 \log (c x+1)+330 a b c^5 x^5+168 a b c^4 x^4+110 a b c^3 x^3+54 a b c^2 x^2+168 a b c^6 x^6 \log \left (1-c^2 x^2\right )+2 b \tanh ^{-1}(c x) \left (3 a \left (20 c^3 x^3+45 c^2 x^2+36 c x+10\right )-168 b c^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+b c x \left (-111 c^5 x^5+165 c^4 x^4+84 c^3 x^3+55 c^2 x^2+27 c x+6\right )\right )+12 a b c x+168 b^2 c^6 x^6 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-64 b^2 c^6 x^6+222 b^2 c^5 x^5+61 b^2 c^4 x^4+18 b^2 c^3 x^3+3 b^2 c^2 x^2-452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+3 b^2 \left (-111 c^6 x^6+20 c^3 x^3+45 c^2 x^2+36 c x+10\right ) \tanh ^{-1}(c x)^2\right )}{180 x^6} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} + {\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname {artanh}\left (c x\right )}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 736, normalized size = 1.54 \[ -\frac {2 c^{3} d^{3} a b \arctanh \left (c x \right )}{3 x^{3}}-\frac {3 c^{2} d^{3} a b \arctanh \left (c x \right )}{2 x^{4}}-\frac {6 c \,d^{3} a b \arctanh \left (c x \right )}{5 x^{5}}-\frac {3 c^{2} d^{3} a^{2}}{4 x^{4}}-\frac {3 c \,d^{3} a^{2}}{5 x^{5}}-\frac {14 c^{6} d^{3} b^{2} \dilog \left (c x \right )}{15}-\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2}}{6 x^{6}}+\frac {14 c^{6} d^{3} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{15}-\frac {14 c^{6} d^{3} b^{2} \dilog \left (c x +1\right )}{15}+\frac {c^{6} d^{3} b^{2} \ln \left (c x +1\right )^{2}}{240}+\frac {113 c^{6} d^{3} b^{2} \ln \left (c x \right )}{45}-\frac {37 c^{6} d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80}-\frac {c^{3} d^{3} a^{2}}{3 x^{3}}-\frac {337 c^{6} d^{3} b^{2} \ln \left (c x -1\right )}{180}-\frac {23 c^{6} d^{3} b^{2} \ln \left (c x +1\right )}{36}-\frac {b^{2} c^{2} d^{3}}{60 x^{4}}-\frac {b^{2} c^{3} d^{3}}{10 x^{3}}-\frac {61 b^{2} c^{4} d^{3}}{180 x^{2}}-\frac {37 b^{2} c^{5} d^{3}}{30 x}+\frac {37 c^{6} d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{40}-\frac {d^{3} a^{2}}{6 x^{6}}+\frac {28 c^{6} d^{3} a b \ln \left (c x \right )}{15}-\frac {37 c^{6} d^{3} a b \ln \left (c x -1\right )}{20}-\frac {c^{6} d^{3} a b \ln \left (c x +1\right )}{60}-\frac {c^{3} d^{3} b^{2} \arctanh \left (c x \right )^{2}}{3 x^{3}}-\frac {14 c^{4} d^{3} b^{2} \arctanh \left (c x \right )}{15 x^{2}}-\frac {c^{6} d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {c \,d^{3} a b}{15 x^{5}}-\frac {11 c^{5} d^{3} a b}{6 x}-\frac {14 c^{4} d^{3} a b}{15 x^{2}}-\frac {14 c^{6} d^{3} b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{15}-\frac {c^{6} d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{120}+\frac {c^{6} d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{120}-\frac {11 c^{3} d^{3} a b}{18 x^{3}}-\frac {3 c^{2} d^{3} a b}{10 x^{4}}-\frac {37 c^{6} d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{20}+\frac {28 c^{6} d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x \right )}{15}-\frac {3 c^{2} d^{3} b^{2} \arctanh \left (c x \right )^{2}}{4 x^{4}}-\frac {11 c^{3} d^{3} b^{2} \arctanh \left (c x \right )}{18 x^{3}}-\frac {3 c \,d^{3} b^{2} \arctanh \left (c x \right )^{2}}{5 x^{5}}-\frac {3 c^{2} d^{3} b^{2} \arctanh \left (c x \right )}{10 x^{4}}-\frac {d^{3} a b \arctanh \left (c x \right )}{3 x^{6}}-\frac {c \,d^{3} b^{2} \arctanh \left (c x \right )}{15 x^{5}}-\frac {11 c^{5} d^{3} b^{2} \arctanh \left (c x \right )}{6 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 961, normalized size = 2.01 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^7} \,d x \]
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 \[ d^{3} \left (\int \frac {a^{2}}{x^{7}}\, dx + \int \frac {3 a^{2} c}{x^{6}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{5}}\, dx + \int \frac {a^{2} c^{3}}{x^{4}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{7}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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