Optimal. Leaf size=256 \[ \frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (-c x+i)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (-c x+i)^2}+\frac {6 i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {3 a x}{c^4 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c^5 d^3}-\frac {15 i b}{8 c^5 d^3 (-c x+i)}-\frac {b}{8 c^5 d^3 (-c x+i)^2}+\frac {19 i b \tan ^{-1}(c x)}{8 c^5 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {3 b \log \left (c^2 x^2+1\right )}{2 c^5 d^3} \]
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Rubi [A] time = 0.28, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4876, 4846, 260, 4852, 321, 203, 4862, 627, 44, 4854, 2402, 2315} \[ -\frac {3 b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (-c x+i)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (-c x+i)^2}+\frac {6 i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3}-\frac {3 a x}{c^4 d^3}+\frac {3 b \log \left (c^2 x^2+1\right )}{2 c^5 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {15 i b}{8 c^5 d^3 (-c x+i)}-\frac {b}{8 c^5 d^3 (-c x+i)^2}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {19 i b \tan ^{-1}(c x)}{8 c^5 d^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 260
Rule 321
Rule 627
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4862
Rule 4876
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{(d+i c d x)^3} \, dx &=\int \left (-\frac {3 \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3 (-i+c x)^3}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3 (-i+c x)^2}-\frac {6 i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3 (-i+c x)}\right ) \, dx\\ &=\frac {i \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{c^4 d^3}-\frac {(6 i) \int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx}{c^4 d^3}-\frac {3 \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^4 d^3}+\frac {4 \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^4 d^3}+\frac {i \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3 d^3}\\ &=-\frac {3 a x}{c^4 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (i-c x)^2}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (i-c x)}+\frac {6 i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {(i b) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{2 c^4 d^3}-\frac {(6 i b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3}-\frac {(3 b) \int \tan ^{-1}(c x) \, dx}{c^4 d^3}+\frac {(4 b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^4 d^3}-\frac {(i b) \int \frac {x^2}{1+c^2 x^2} \, dx}{2 c^2 d^3}\\ &=-\frac {3 a x}{c^4 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (i-c x)^2}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (i-c x)}+\frac {6 i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {(6 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^5 d^3}+\frac {(i b) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{2 c^4 d^3}+\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^4 d^3}+\frac {(4 b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^4 d^3}+\frac {(3 b) \int \frac {x}{1+c^2 x^2} \, dx}{c^3 d^3}\\ &=-\frac {3 a x}{c^4 d^3}-\frac {i b x}{2 c^4 d^3}+\frac {i b \tan ^{-1}(c x)}{2 c^5 d^3}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (i-c x)^2}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (i-c x)}+\frac {6 i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 b \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {(i b) \int \left (-\frac {i}{2 (-i+c x)^3}+\frac {1}{4 (-i+c x)^2}-\frac {1}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{2 c^4 d^3}+\frac {(4 b) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}\\ &=-\frac {3 a x}{c^4 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (i-c x)^2}-\frac {15 i b}{8 c^5 d^3 (i-c x)}+\frac {i b \tan ^{-1}(c x)}{2 c^5 d^3}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (i-c x)^2}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (i-c x)}+\frac {6 i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 b \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^4 d^3}+\frac {(2 i b) \int \frac {1}{1+c^2 x^2} \, dx}{c^4 d^3}\\ &=-\frac {3 a x}{c^4 d^3}-\frac {i b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (i-c x)^2}-\frac {15 i b}{8 c^5 d^3 (i-c x)}+\frac {19 i b \tan ^{-1}(c x)}{8 c^5 d^3}-\frac {3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac {i x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^3 d^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 c^5 d^3 (i-c x)^2}+\frac {4 \left (a+b \tan ^{-1}(c x)\right )}{c^5 d^3 (i-c x)}+\frac {6 i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 b \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^5 d^3}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 235, normalized size = 0.92 \[ \frac {16 i a c^2 x^2-96 i a \log \left (c^2 x^2+1\right )-96 a c x-\frac {128 a}{c x-i}-\frac {16 i a}{(c x-i)^2}+192 a \tan ^{-1}(c x)+b \left (48 \log \left (c^2 x^2+1\right )+4 i \tan ^{-1}(c x) \left (4 c^2 x^2+24 i c x+48 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+14 i \sin \left (2 \tan ^{-1}(c x)\right )-i \sin \left (4 \tan ^{-1}(c x)\right )-14 \cos \left (2 \tan ^{-1}(c x)\right )+\cos \left (4 \tan ^{-1}(c x)\right )+4\right )+96 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-16 i c x+192 \tan ^{-1}(c x)^2+28 i \sin \left (2 \tan ^{-1}(c x)\right )-i \sin \left (4 \tan ^{-1}(c x)\right )-28 \cos \left (2 \tan ^{-1}(c x)\right )+\cos \left (4 \tan ^{-1}(c x)\right )\right )}{32 c^5 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x^{4} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 i \, a x^{4}}{2 \, c^{3} d^{3} x^{3} - 6 i \, c^{2} d^{3} x^{2} - 6 \, c d^{3} x + 2 i \, d^{3}}, x\right ) \]
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 [A] time = 0.07, size = 423, normalized size = 1.65 \[ -\frac {3 a x}{c^{4} d^{3}}-\frac {3 i a \ln \left (c^{2} x^{2}+1\right )}{c^{5} d^{3}}-\frac {4 a}{c^{5} d^{3} \left (c x -i\right )}-\frac {i b \arctan \left (c x \right )}{2 c^{5} d^{3} \left (c x -i\right )^{2}}+\frac {6 a \arctan \left (c x \right )}{c^{5} d^{3}}+\frac {i a \,x^{2}}{2 c^{3} d^{3}}-\frac {3 b x \arctan \left (c x \right )}{c^{4} d^{3}}+\frac {5 i b \arctan \left (\frac {c x}{2}\right )}{32 c^{5} d^{3}}-\frac {4 b \arctan \left (c x \right )}{c^{5} d^{3} \left (c x -i\right )}+\frac {15 i b}{8 c^{5} d^{3} \left (c x -i\right )}-\frac {5 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 c^{5} d^{3}}-\frac {b}{2 c^{5} d^{3}}-\frac {5 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 c^{5} d^{3}}+\frac {5 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 c^{5} d^{3}}-\frac {i b x}{2 c^{4} d^{3}}-\frac {i a}{2 c^{5} d^{3} \left (c x -i\right )^{2}}+\frac {i b \arctan \left (c x \right ) x^{2}}{2 c^{3} d^{3}}-\frac {6 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{5} d^{3}}-\frac {b}{8 c^{5} d^{3} \left (c x -i\right )^{2}}+\frac {43 b \ln \left (c^{2} x^{2}+1\right )}{32 c^{5} d^{3}}+\frac {43 i b \arctan \left (c x \right )}{16 c^{5} d^{3}}+\frac {3 b \ln \left (c x -i\right )^{2}}{2 c^{5} d^{3}}-\frac {3 b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{5} d^{3}}-\frac {3 b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 356, normalized size = 1.39 \[ \frac {8 i \, a c^{4} x^{4} - 8 \, {\left (4 \, a + i \, b\right )} c^{3} x^{3} + {\left (b {\left (5 i \, \arctan \left (1, c x\right ) - 16\right )} + 88 i \, a\right )} c^{2} x^{2} + {\left (b {\left (10 \, \arctan \left (1, c x\right ) + 38 i\right )} - 16 \, a\right )} c x + {\left (24 \, b c^{2} x^{2} - 48 i \, b c x - 24 \, b\right )} \arctan \left (c x\right )^{2} + {\left (6 \, b c^{2} x^{2} - 12 i \, b c x - 6 \, b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + {\left (-24 i \, b c^{2} x^{2} - 48 \, b c x + 24 i \, b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + b {\left (-5 i \, \arctan \left (1, c x\right ) + 28\right )} + {\left (8 i \, b c^{4} x^{4} - 32 \, b c^{3} x^{3} + {\left (96 \, a + 131 i \, b\right )} c^{2} x^{2} + {\left (-192 i \, a + 70 \, b\right )} c x - 96 \, a + 13 i \, b\right )} \arctan \left (c x\right ) - {\left (48 \, b c^{2} x^{2} - 96 i \, b c x - 48 \, b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) + {\left ({\left (-48 i \, a + 24 \, b\right )} c^{2} x^{2} - 48 \, {\left (2 \, a + i \, b\right )} c x - {\left (12 \, b c^{2} x^{2} - 24 i \, b c x - 12 \, b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + 48 i \, a - 24 \, b\right )} \log \left (c^{2} x^{2} + 1\right ) + 56 i \, a}{16 \, c^{7} d^{3} x^{2} - 32 i \, c^{6} d^{3} x - 16 \, c^{5} d^{3}} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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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