Optimal. Leaf size=513 \[ \frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{60} i b c^6 d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {i b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right ) \]
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Rubi [A] time = 0.52, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {43, 4874, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391, 4854, 2402, 2315} \[ -\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {28}{15} i a b c^6 d^3 \log (x)-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}+\frac {37}{20} i b c^6 d^3 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{60} i b c^6 d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}+\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {i b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right )+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 44
Rule 203
Rule 266
Rule 325
Rule 2315
Rule 2391
Rule 2402
Rule 4848
Rule 4852
Rule 4854
Rule 4874
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^7} \, dx &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {11 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^4}+\frac {14 i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^3}-\frac {11 c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^2}-\frac {14 i c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x}+\frac {i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{120 (-i+c x)}+\frac {37 i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{40 (i+c x)}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^6} \, dx+\frac {1}{5} \left (6 i b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^5} \, dx-\frac {1}{6} \left (11 b c^3 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^4} \, dx-\frac {1}{15} \left (28 i b c^4 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+\frac {1}{6} \left (11 b c^5 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx+\frac {1}{15} \left (28 i b c^6 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx-\frac {1}{60} \left (i b c^7 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx-\frac {1}{20} \left (37 i b c^7 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i+c x} \, dx\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+\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 i 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 i b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1+i c x)}{x} \, 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 (i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {1}{20} \left (37 i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {14 i b^2 c^5 d^3}{15 x}-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i 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 i 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+i 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-i c x}\right )+\frac {1}{15} \left (14 i b^2 c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {14}{15} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+i 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 i 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 {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac {3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac {11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac {14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac {11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i 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(-i c x)+\frac {14}{15} b^2 c^6 d^3 \text {Li}_2(i c x)+\frac {37}{40} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )\\ \end {align*}
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Mathematica [A] time = 1.61, size = 401, normalized size = 0.78 \[ \frac {d^3 \left (60 i a^2 c^3 x^3+135 a^2 c^2 x^2-108 i a^2 c x-30 a^2+336 i a b c^6 x^6 \log (c x)-330 a b c^5 x^5+168 i a b c^4 x^4+110 a b c^3 x^3-54 i a b c^2 x^2-168 i a b c^6 x^6 \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (-3 a \left (55 c^6 x^6-20 i c^3 x^3-45 c^2 x^2+36 i c x+10\right )+168 i b c^6 x^6 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+b c x \left (111 i c^5 x^5-165 c^4 x^4+84 i c^3 x^3+55 c^2 x^2-27 i c x-6\right )\right )-12 a b c x+168 b^2 c^6 x^6 \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+64 b^2 c^6 x^6+222 i b^2 c^5 x^5+61 b^2 c^4 x^4-18 i b^2 c^3 x^3-3 b^2 c^2 x^2+3 b^2 (c x-i)^4 \left (c^2 x^2+4 i c x-10\right ) \tan ^{-1}(c x)^2+452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {c^2 x^2+1}}\right )\right )}{180 x^6} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ \frac {240 \, x^{6} {\rm integral}\left (\frac {-60 i \, a^{2} c^{5} d^{3} x^{5} - 180 \, a^{2} c^{4} d^{3} x^{4} + 120 i \, a^{2} c^{3} d^{3} x^{3} - 120 \, a^{2} c^{2} d^{3} x^{2} + 180 i \, a^{2} c d^{3} x + 60 \, a^{2} d^{3} + {\left (60 \, a b c^{5} d^{3} x^{5} - 20 \, {\left (9 i \, a b - b^{2}\right )} c^{4} d^{3} x^{4} - {\left (120 \, a b + 45 i \, b^{2}\right )} c^{3} d^{3} x^{3} - 12 \, {\left (10 i \, a b + 3 \, b^{2}\right )} c^{2} d^{3} x^{2} - {\left (180 \, a b - 10 i \, b^{2}\right )} c d^{3} x + 60 i \, a b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{60 \, {\left (c^{2} x^{9} + x^{7}\right )}}, x\right ) + {\left (-20 i \, b^{2} c^{3} d^{3} x^{3} - 45 \, b^{2} c^{2} d^{3} x^{2} + 36 i \, b^{2} c d^{3} x + 10 \, b^{2} d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2}}{240 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 853, normalized size = 1.66 \[ -\frac {d^{3} a^{2}}{6 x^{6}}+\frac {14 i c^{4} d^{3} a b}{15 x^{2}}+\frac {28 i c^{6} d^{3} a b \ln \left (c x \right )}{15}-\frac {14 i c^{6} d^{3} a b \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {14 i c^{6} d^{3} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {28 i c^{6} d^{3} b^{2} \arctan \left (c x \right ) \ln \left (c x \right )}{15}-\frac {3 i c^{2} d^{3} a b}{10 x^{4}}+\frac {14 i c^{4} d^{3} b^{2} \arctan \left (c x \right )}{15 x^{2}}+\frac {i c^{3} d^{3} b^{2} \arctan \left (c x \right )^{2}}{3 x^{3}}-\frac {3 i c \,d^{3} b^{2} \arctan \left (c x \right )^{2}}{5 x^{5}}-\frac {3 i c^{2} d^{3} b^{2} \arctan \left (c x \right )}{10 x^{4}}+\frac {3 c^{2} d^{3} b^{2} \arctan \left (c x \right )^{2}}{4 x^{4}}+\frac {11 c^{3} d^{3} b^{2} \arctan \left (c x \right )}{18 x^{3}}-\frac {11 c^{5} d^{3} b^{2} \arctan \left (c x \right )}{6 x}-\frac {c \,d^{3} b^{2} \arctan \left (c x \right )}{15 x^{5}}-\frac {c \,d^{3} a b}{15 x^{5}}-\frac {d^{3} a b \arctan \left (c x \right )}{3 x^{6}}-\frac {11 c^{5} d^{3} a b}{6 x}+\frac {11 c^{3} d^{3} a b}{18 x^{3}}-\frac {6 i c \,d^{3} a b \arctan \left (c x \right )}{5 x^{5}}+\frac {2 i c^{3} d^{3} a b \arctan \left (c x \right )}{3 x^{3}}-\frac {113 b^{2} c^{6} d^{3} \ln \left (c^{2} x^{2}+1\right )}{90}+\frac {14 c^{6} d^{3} b^{2} \dilog \left (-i c x +1\right )}{15}-\frac {7 c^{6} d^{3} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 c^{6} d^{3} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {7 c^{6} d^{3} b^{2} \ln \left (c x +i\right )^{2}}{30}-\frac {7 c^{6} d^{3} b^{2} \ln \left (c x -i\right )^{2}}{30}-\frac {14 c^{6} d^{3} b^{2} \dilog \left (i c x +1\right )}{15}+\frac {i c^{3} d^{3} a^{2}}{3 x^{3}}-\frac {3 i c \,d^{3} a^{2}}{5 x^{5}}+\frac {7 c^{6} d^{3} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {7 c^{6} d^{3} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {14 c^{6} d^{3} b^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 c^{6} d^{3} b^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}-\frac {7 c^{6} d^{3} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 c^{6} d^{3} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i b^{2} c^{3} d^{3}}{10 x^{3}}+\frac {37 i b^{2} c^{5} d^{3}}{30 x}+\frac {37 i b^{2} c^{6} d^{3} \arctan \left (c x \right )}{30}+\frac {113 c^{6} d^{3} b^{2} \ln \left (c x \right )}{45}-\frac {11 c^{6} d^{3} b^{2} \arctan \left (c x \right )^{2}}{12}+\frac {3 c^{2} d^{3} a^{2}}{4 x^{4}}-\frac {d^{3} b^{2} \arctan \left (c x \right )^{2}}{6 x^{6}}-\frac {11 c^{6} d^{3} a b \arctan \left (c x \right )}{6}+\frac {3 c^{2} d^{3} a b \arctan \left (c x \right )}{2 x^{4}}-\frac {b^{2} c^{2} d^{3}}{60 x^{4}}+\frac {61 b^{2} c^{4} d^{3}}{180 x^{2}} \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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