Optimal. Leaf size=402 \[ -\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {11 i a b d^3 x}{6 c^2}+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {14 i b^2 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{15 c^3}-\frac {37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac {113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}-\frac {1}{60} i b^2 c d^3 x^4+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3 \]
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Rubi [A] time = 1.20, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302} \[ -\frac {14 i b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3}-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {11 i a b d^3 x}{6 c^2}-\frac {37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac {37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac {1}{60} i b^2 c d^3 x^4+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 203
Rule 260
Rule 266
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4876
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int x^2 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-3 c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 i c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (3 c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^3\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{5} \left (6 b c^3 d^3\right ) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (i b c^4 d^3\right ) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} \left (3 i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac {1}{2} \left (3 i b d^3\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {\left (2 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (2 b d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (6 b c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (6 b c d^3\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (i b c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (i b c^2 d^3\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {1}{2} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac {1}{3} \left (i b d^3\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (3 i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac {\left (3 i b d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (2 b d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}-\frac {\left (6 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac {\left (6 b d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{2} \left (i b^2 c d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{10} \left (3 b^2 c^2 d^3\right ) \int \frac {x^4}{1+c^2 x^2} \, dx-\frac {1}{15} \left (i b^2 c^3 d^3\right ) \int \frac {x^5}{1+c^2 x^2} \, dx\\ &=\frac {3 i a b d^3 x}{2 c^2}+\frac {b^2 d^3 x}{3 c^2}-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {101 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {1}{5} \left (3 b^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^2}-\frac {\left (i b d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^2}-\frac {\left (6 b d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^2}+\frac {\left (3 i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{2 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{9} \left (i b^2 c d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {1}{4} \left (i b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (3 b^2 c^2 d^3\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{30} \left (i b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}-\frac {1}{10} b^2 d^3 x^3-\frac {b^2 d^3 \tan ^{-1}(c x)}{3 c^3}+\frac {3 i b^2 d^3 x \tan ^{-1}(c x)}{2 c^2}-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {\left (2 i b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}+\frac {\left (i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{3 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^2}-\frac {\left (3 b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^2}+\frac {\left (6 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (3 i b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c}+\frac {1}{18} \left (i b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {1}{4} \left (i b^2 c d^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{30} \left (i b^2 c^3 d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {17 i b^2 d^3 x^2}{60 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac {11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {31 i b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^3}-\frac {i b^2 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (6 i b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^3}-\frac {\left (i b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{18} \left (i b^2 c d^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {11 i a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 i b^2 d^3 x^2}{180 c}-\frac {1}{10} b^2 d^3 x^3-\frac {1}{60} i b^2 c d^3 x^4-\frac {37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac {11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac {14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac {14 i b^2 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{15 c^3}\\ \end {align*}
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Mathematica [A] time = 1.51, size = 369, normalized size = 0.92 \[ \frac {d^3 \left (-30 i a^2 c^6 x^6-108 a^2 c^5 x^5+135 i a^2 c^4 x^4+60 a^2 c^3 x^3+12 i a b c^5 x^5+54 a b c^4 x^4-110 i a b c^3 x^3-168 a b c^2 x^2+168 a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (3 a \left (-10 i c^6 x^6-36 c^5 x^5+45 i c^4 x^4+20 c^3 x^3-55 i\right )+b \left (6 i c^5 x^5+27 c^4 x^4-55 i c^3 x^3-84 c^2 x^2+165 i c x-111\right )-168 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+330 i a b c x-162 a b-3 i b^2 c^4 x^4-18 b^2 c^3 x^3+61 i b^2 c^2 x^2-226 i b^2 \log \left (c^2 x^2+1\right )+3 b^2 (c x-i)^4 \left (-10 i c^2 x^2+4 c x+i\right ) \tan ^{-1}(c x)^2+168 i b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+222 b^2 c x+64 i b^2\right )}{180 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ \frac {1}{240} \, {\left (10 i \, b^{2} c^{3} d^{3} x^{6} + 36 \, b^{2} c^{2} d^{3} x^{5} - 45 i \, b^{2} c d^{3} x^{4} - 20 \, b^{2} d^{3} x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\rm integral}\left (\frac {-60 i \, a^{2} c^{5} d^{3} x^{7} - 180 \, a^{2} c^{4} d^{3} x^{6} + 120 i \, a^{2} c^{3} d^{3} x^{5} - 120 \, a^{2} c^{2} d^{3} x^{4} + 180 i \, a^{2} c d^{3} x^{3} + 60 \, a^{2} d^{3} x^{2} + {\left (60 \, a b c^{5} d^{3} x^{7} - 10 \, {\left (18 i \, a b + b^{2}\right )} c^{4} d^{3} x^{6} - {\left (120 \, a b - 36 i \, b^{2}\right )} c^{3} d^{3} x^{5} - 15 \, {\left (8 i \, a b - 3 \, b^{2}\right )} c^{2} d^{3} x^{4} - {\left (180 \, a b + 20 i \, b^{2}\right )} c d^{3} x^{3} + 60 i \, a b d^{3} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{60 \, {\left (c^{2} x^{2} + 1\right )}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 712, normalized size = 1.77 \[ -\frac {6 c^{2} d^{3} a b \arctan \left (c x \right ) x^{5}}{5}+\frac {d^{3} a^{2} x^{3}}{3}+\frac {i c^{2} d^{3} a b \,x^{5}}{15}+\frac {3 i c \,d^{3} b^{2} \arctan \left (c x \right )^{2} x^{4}}{4}-\frac {i c^{3} d^{3} b^{2} \arctan \left (c x \right )^{2} x^{6}}{6}-\frac {11 i d^{3} a b \arctan \left (c x \right )}{6 c^{3}}+\frac {i c^{2} d^{3} b^{2} \arctan \left (c x \right ) x^{5}}{15}-\frac {7 i d^{3} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+\frac {7 i d^{3} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}+\frac {7 i d^{3} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {7 i d^{3} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}+\frac {37 b^{2} d^{3} x}{30 c^{2}}-\frac {37 b^{2} d^{3} \arctan \left (c x \right )}{30 c^{3}}-\frac {11 i d^{3} b^{2} \arctan \left (c x \right )^{2}}{12 c^{3}}-\frac {7 i d^{3} b^{2} \ln \left (c x -i\right )^{2}}{30 c^{3}}+\frac {7 i d^{3} b^{2} \ln \left (c x +i\right )^{2}}{30 c^{3}}-\frac {11 i d^{3} a b \,x^{3}}{18}-\frac {11 i d^{3} b^{2} \arctan \left (c x \right ) x^{3}}{18}-\frac {i c^{3} d^{3} a^{2} x^{6}}{6}+\frac {7 i d^{3} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}-\frac {7 i d^{3} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}+\frac {3 i c \,d^{3} a^{2} x^{4}}{4}-\frac {i b^{2} c \,d^{3} x^{4}}{60}-\frac {113 i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{90 c^{3}}+\frac {61 i b^{2} d^{3} x^{2}}{180 c}+\frac {14 d^{3} a b \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {14 d^{3} b^{2} \arctan \left (c x \right ) x^{2}}{15 c}-\frac {3 c^{2} d^{3} b^{2} \arctan \left (c x \right )^{2} x^{5}}{5}+\frac {3 c \,d^{3} b^{2} \arctan \left (c x \right ) x^{4}}{10}+\frac {2 d^{3} a b \arctan \left (c x \right ) x^{3}}{3}+\frac {14 d^{3} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+\frac {3 c \,d^{3} a b \,x^{4}}{10}-\frac {14 d^{3} a b \,x^{2}}{15 c}-\frac {3 c^{2} d^{3} a^{2} x^{5}}{5}+\frac {d^{3} b^{2} \arctan \left (c x \right )^{2} x^{3}}{3}+\frac {11 i a b \,d^{3} x}{6 c^{2}}+\frac {11 i b^{2} d^{3} x \arctan \left (c x \right )}{6 c^{2}}-\frac {b^{2} d^{3} x^{3}}{10}-\frac {i c^{3} d^{3} a b \arctan \left (c x \right ) x^{6}}{3}+\frac {3 i c \,d^{3} a b \arctan \left (c x \right ) x^{4}}{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 \[ -\frac {1}{6} i \, a^{2} c^{3} d^{3} x^{6} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {3}{4} i \, a^{2} c d^{3} x^{4} - \frac {1}{45} i \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b c^{3} d^{3} - \frac {3}{10} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} a b c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {1}{2} i \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b c d^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b d^{3} - \frac {1}{960} \, {\left (40 i \, b^{2} c^{3} d^{3} x^{6} + 144 \, b^{2} c^{2} d^{3} x^{5} - 180 i \, b^{2} c d^{3} x^{4} - 80 \, b^{2} d^{3} x^{3}\right )} \arctan \left (c x\right )^{2} + \frac {1}{960} \, {\left (40 \, b^{2} c^{3} d^{3} x^{6} - 144 i \, b^{2} c^{2} d^{3} x^{5} - 180 \, b^{2} c d^{3} x^{4} + 80 i \, b^{2} d^{3} x^{3}\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) - \frac {1}{960} \, {\left (-10 i \, b^{2} c^{3} d^{3} x^{6} - 36 \, b^{2} c^{2} d^{3} x^{5} + 45 i \, b^{2} c d^{3} x^{4} + 20 \, b^{2} d^{3} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - i \, \int \frac {180 \, {\left (b^{2} c^{5} d^{3} x^{7} - 2 \, b^{2} c^{3} d^{3} x^{5} - 3 \, b^{2} c d^{3} x^{3}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (b^{2} c^{5} d^{3} x^{7} - 2 \, b^{2} c^{3} d^{3} x^{5} - 3 \, b^{2} c d^{3} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 2 \, {\left (46 \, b^{2} c^{4} d^{3} x^{6} - 65 \, b^{2} c^{2} d^{3} x^{4}\right )} \arctan \left (c x\right ) + {\left (10 \, b^{2} c^{5} d^{3} x^{7} - 81 \, b^{2} c^{3} d^{3} x^{5} + 20 \, b^{2} c d^{3} x^{3} - 60 \, {\left (3 \, b^{2} c^{4} d^{3} x^{6} + 2 \, b^{2} c^{2} d^{3} x^{4} - b^{2} d^{3} x^{2}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{240 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} - \int \frac {180 \, {\left (3 \, b^{2} c^{4} d^{3} x^{6} + 2 \, b^{2} c^{2} d^{3} x^{4} - b^{2} d^{3} x^{2}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (3 \, b^{2} c^{4} d^{3} x^{6} + 2 \, b^{2} c^{2} d^{3} x^{4} - b^{2} d^{3} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, {\left (10 \, b^{2} c^{5} d^{3} x^{7} - 81 \, b^{2} c^{3} d^{3} x^{5} + 20 \, b^{2} c d^{3} x^{3}\right )} \arctan \left (c x\right ) + {\left (46 \, b^{2} c^{4} d^{3} x^{6} - 65 \, b^{2} c^{2} d^{3} x^{4} + 60 \, {\left (b^{2} c^{5} d^{3} x^{7} - 2 \, b^{2} c^{3} d^{3} x^{5} - 3 \, b^{2} c d^{3} x^{3}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{240 \, {\left (c^{2} x^{2} + 1\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 \[ \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]
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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