Optimal. Leaf size=88 \[ \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}+\frac {3 b}{8 c^2 d^3 (-c x+i)}-\frac {i b}{8 c^2 d^3 (-c x+i)^2}+\frac {b \tan ^{-1}(c x)}{8 c^2 d^3} \]
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Rubi [A] time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {37, 4872, 12, 88, 203} \[ \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}+\frac {3 b}{8 c^2 d^3 (-c x+i)}-\frac {i b}{8 c^2 d^3 (-c x+i)^2}+\frac {b \tan ^{-1}(c x)}{8 c^2 d^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 88
Rule 203
Rule 4872
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{(d+i c d x)^3} \, dx &=\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}-(b c) \int \frac {x^2}{2 d^3 (i-c x)^3 (i+c x)} \, dx\\ &=\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}-\frac {(b c) \int \frac {x^2}{(i-c x)^3 (i+c x)} \, dx}{2 d^3}\\ &=\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}-\frac {(b c) \int \left (-\frac {i}{2 c^2 (-i+c x)^3}-\frac {3}{4 c^2 (-i+c x)^2}-\frac {1}{4 c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^3}\\ &=-\frac {i b}{8 c^2 d^3 (i-c x)^2}+\frac {3 b}{8 c^2 d^3 (i-c x)}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{8 c d^3}\\ &=-\frac {i b}{8 c^2 d^3 (i-c x)^2}+\frac {3 b}{8 c^2 d^3 (i-c x)}+\frac {b \tan ^{-1}(c x)}{8 c^2 d^3}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 d^3 (1+i c x)^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 63, normalized size = 0.72 \[ \frac {a (-4-8 i c x)-b \left (3 c^2 x^2+2 i c x+1\right ) \tan ^{-1}(c x)+b (-3 c x+2 i)}{8 c^2 d^3 (c x-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 83, normalized size = 0.94 \[ \frac {{\left (-16 i \, a - 6 \, b\right )} c x + {\left (-3 i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 8 \, a + 4 i \, b}{16 \, {\left (c^{4} d^{3} x^{2} - 2 i \, c^{3} d^{3} x - c^{2} d^{3}\right )}} \]
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 = 0.06, size = 128, normalized size = 1.45 \[ -\frac {i a}{c^{2} d^{3} \left (c x -i\right )}+\frac {a}{2 c^{2} d^{3} \left (c x -i\right )^{2}}-\frac {i b \arctan \left (c x \right )}{c^{2} d^{3} \left (c x -i\right )}+\frac {b \arctan \left (c x \right )}{2 c^{2} d^{3} \left (c x -i\right )^{2}}-\frac {3 b \arctan \left (c x \right )}{8 c^{2} d^{3}}-\frac {i b}{8 c^{2} d^{3} \left (c x -i\right )^{2}}-\frac {3 b}{8 c^{2} d^{3} \left (c x -i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 71, normalized size = 0.81 \[ -\frac {{\left (8 i \, a + 3 \, b\right )} c x + {\left (3 \, b c^{2} x^{2} + 2 i \, b c x + b\right )} \arctan \left (c x\right ) + 4 \, a - 2 i \, b}{8 \, c^{4} d^{3} x^{2} - 16 i \, c^{3} d^{3} x - 8 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.22, size = 194, normalized size = 2.20 \[ \frac {b \left (\frac {3 i \log {\left (x - \frac {i}{c} \right )}}{16} - \frac {3 i \log {\left (x + \frac {i}{c} \right )}}{16}\right )}{c^{2} d^{3}} + \frac {\left (- 2 i b c x - b\right ) \log {\left (i c x + 1 \right )}}{4 i c^{4} d^{3} x^{2} + 8 c^{3} d^{3} x - 4 i c^{2} d^{3}} + \frac {\left (- 2 i b c x - b\right ) \log {\left (- i c x + 1 \right )}}{- 4 i c^{4} d^{3} x^{2} - 8 c^{3} d^{3} x + 4 i c^{2} d^{3}} + \frac {4 a - 2 i b + x \left (8 i a c + 3 b c\right )}{- 8 c^{4} d^{3} x^{2} + 16 i c^{3} d^{3} x + 8 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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