Optimal. Leaf size=333 \[ -\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i a b d^2 x}{c^2}+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {8 i b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{15 c^3}-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3 \]
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Rubi [A] time = 0.86, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 36, 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} \[ -\frac {8 i b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{15 c^3}-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i a b d^2 x}{c^2}-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {2 i b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3 \]
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)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+2 i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (2 i c d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (c^2 d^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\left (i b c^2 d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{5} \left (2 b c^3 d^2\right ) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\left (i b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (i b d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (2 b c d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} \left (b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {\left (i b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2}-\frac {\left (i b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2}-\frac {\left (2 b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}-\frac {\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{3} \left (i b^2 c d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{10} \left (b^2 c^2 d^2\right ) \int \frac {x^4}{1+c^2 x^2} \, dx\\ &=\frac {i a b d^2 x}{c^2}+\frac {b^2 d^2 x}{3 c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {1}{5} \left (b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^2}+\frac {\left (i b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{6} \left (i b^2 c d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (b^2 c^2 d^2\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 {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}-\frac {1}{30} b^2 d^2 x^3-\frac {b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \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^2\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 (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (i b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c}+\frac {1}{6} \left (i b^2 c d^2\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 {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (2 i b^2 d^2\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 {i a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {i b^2 d^2 x^2}{6 c}-\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac {i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac {8 b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {1}{3} i b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {31 i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{2} i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{15 c^3}-\frac {2 i b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {8 i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{15 c^3}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 306, normalized size = 0.92 \[ -\frac {d^2 \left (6 a^2 c^5 x^5-15 i a^2 c^4 x^4-10 a^2 c^3 x^3-3 a b c^4 x^4+10 i a b c^3 x^3+16 a b c^2 x^2-16 a b \log \left (c^2 x^2+1\right )+b \tan ^{-1}(c x) \left (2 a \left (6 c^5 x^5-15 i c^4 x^4-10 c^3 x^3+15 i\right )+b \left (-3 c^4 x^4+10 i c^3 x^3+16 c^2 x^2-30 i c x+19\right )+32 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-30 i a b c x+9 a b+b^2 c^3 x^3-5 i b^2 c^2 x^2+20 i b^2 \log \left (c^2 x^2+1\right )+b^2 (c x-i)^3 \left (6 c^2 x^2+3 i c x-1\right ) \tan ^{-1}(c x)^2-16 i b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-19 b^2 c x-5 i b^2\right )}{30 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ \frac {1}{120} \, {\left (6 \, b^{2} c^{2} d^{2} x^{5} - 15 i \, b^{2} c d^{2} x^{4} - 10 \, b^{2} d^{2} x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\rm integral}\left (-\frac {30 \, a^{2} c^{4} d^{2} x^{6} - 60 i \, a^{2} c^{3} d^{2} x^{5} - 60 i \, a^{2} c d^{2} x^{3} - 30 \, a^{2} d^{2} x^{2} - {\left (-30 i \, a b c^{4} d^{2} x^{6} - {\left (60 \, a b - 6 i \, b^{2}\right )} c^{3} d^{2} x^{5} + 15 \, b^{2} c^{2} d^{2} x^{4} - {\left (60 \, a b + 10 i \, b^{2}\right )} c d^{2} x^{3} + 30 i \, a b d^{2} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{30 \, {\left (c^{2} x^{2} + 1\right )}}, x\right ) \]
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 = 0.11, size = 612, normalized size = 1.84 \[ \frac {d^{2} a^{2} x^{3}}{3}+\frac {i b^{2} d^{2} x \arctan \left (c x \right )}{c^{2}}+\frac {19 b^{2} d^{2} x}{30 c^{2}}+\frac {i c \,d^{2} b^{2} \arctan \left (c x \right )^{2} x^{4}}{2}+\frac {4 i d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}-\frac {i d^{2} a b \arctan \left (c x \right )}{c^{3}}+\frac {i a b \,d^{2} x}{c^{2}}-\frac {4 i d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}-\frac {4 i d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+\frac {4 i d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}+i c \,d^{2} a b \arctan \left (c x \right ) x^{4}-\frac {2 i d^{2} b^{2} \ln \left (c x -i\right )^{2}}{15 c^{3}}-\frac {i d^{2} a b \,x^{3}}{3}-\frac {i d^{2} b^{2} \arctan \left (c x \right )^{2}}{2 c^{3}}+\frac {2 i d^{2} b^{2} \ln \left (c x +i\right )^{2}}{15 c^{3}}-\frac {4 i d^{2} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{15 c^{3}}-\frac {i d^{2} b^{2} \arctan \left (c x \right ) x^{3}}{3}+\frac {i c \,d^{2} a^{2} x^{4}}{2}-\frac {2 i b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}+\frac {4 i d^{2} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{15 c^{3}}+\frac {i b^{2} d^{2} x^{2}}{6 c}+\frac {8 d^{2} a b \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} a b \,x^{2}}{15 c}+\frac {c \,d^{2} a b \,x^{4}}{10}+\frac {2 d^{2} a b \arctan \left (c x \right ) x^{3}}{3}+\frac {8 d^{2} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {8 d^{2} b^{2} \arctan \left (c x \right ) x^{2}}{15 c}-\frac {c^{2} d^{2} b^{2} \arctan \left (c x \right )^{2} x^{5}}{5}+\frac {c \,d^{2} b^{2} \arctan \left (c x \right ) x^{4}}{10}-\frac {c^{2} d^{2} a^{2} x^{5}}{5}-\frac {b^{2} d^{2} x^{3}}{30}-\frac {19 b^{2} d^{2} \arctan \left (c x \right )}{30 c^{3}}+\frac {d^{2} b^{2} \arctan \left (c x \right )^{2} x^{3}}{3}-\frac {2 c^{2} d^{2} a b \arctan \left (c x \right ) x^{5}}{5} \]
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}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {1}{2} i \, a^{2} c d^{2} x^{4} - \frac {1}{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^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{3} 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^{2} + \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^{2} - \frac {1}{480} \, {\left (24 \, b^{2} c^{2} d^{2} x^{5} - 60 i \, b^{2} c d^{2} x^{4} - 40 \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} - \frac {1}{480} \, {\left (24 i \, b^{2} c^{2} d^{2} x^{5} + 60 \, b^{2} c d^{2} x^{4} - 40 i \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) + \frac {1}{480} \, {\left (6 \, b^{2} c^{2} d^{2} x^{5} - 15 i \, b^{2} c d^{2} x^{4} - 10 \, b^{2} d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - \int \frac {180 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (21 \, b^{2} c^{3} d^{2} x^{5} - 10 \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right ) + 2 \, {\left (6 \, b^{2} c^{4} d^{2} x^{6} - 25 \, b^{2} c^{2} d^{2} x^{4} - 60 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} 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} + i \, \int \frac {180 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} + 15 \, {\left (b^{2} c^{3} d^{2} x^{5} + b^{2} c d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, {\left (6 \, b^{2} c^{4} d^{2} x^{6} - 25 \, b^{2} c^{2} d^{2} x^{4}\right )} \arctan \left (c x\right ) + {\left (21 \, b^{2} c^{3} d^{2} x^{5} - 10 \, b^{2} c d^{2} x^{3} + 30 \, {\left (b^{2} c^{4} d^{2} x^{6} - b^{2} d^{2} x^{2}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{120 \, {\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 \[ \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \]
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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