Optimal. Leaf size=69 \[ \frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}+\frac {i b}{2 c d^2 (-c x+i)}-\frac {i b \tan ^{-1}(c x)}{2 c d^2} \]
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Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4862, 627, 44, 203} \[ \frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}+\frac {i b}{2 c d^2 (-c x+i)}-\frac {i b \tan ^{-1}(c x)}{2 c d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 4862
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{(d+i c d x)^2} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}-\frac {(i b) \int \frac {1}{(d+i c d x) \left (1+c^2 x^2\right )} \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}-\frac {(i b) \int \frac {1}{\left (\frac {1}{d}-\frac {i c x}{d}\right ) (d+i c d x)^2} \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}-\frac {(i b) \int \left (-\frac {1}{2 d (-i+c x)^2}+\frac {1}{2 d \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=\frac {i b}{2 c d^2 (i-c x)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}-\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {i b}{2 c d^2 (i-c x)}-\frac {i b \tan ^{-1}(c x)}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.61 \[ \frac {2 a+(b-i b c x) \tan ^{-1}(c x)-i b}{2 c d^2 (c x-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 50, normalized size = 0.72 \[ \frac {{\left (b c x + i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 4 \, a - 2 i \, b}{4 \, c^{2} d^{2} x - 4 i \, c d^{2}} \]
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 = 0.05, size = 76, normalized size = 1.10 \[ \frac {i a}{c \,d^{2} \left (i c x +1\right )}+\frac {i b \arctan \left (c x \right )}{c \,d^{2} \left (i c x +1\right )}-\frac {i b \arctan \left (c x \right )}{2 c \,d^{2}}-\frac {i b}{2 c \,d^{2} \left (c x -i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.94, size = 116, normalized size = 1.68 \[ \frac {i b \log {\left (- i c x + 1 \right )}}{2 c^{2} d^{2} x - 2 i c d^{2}} - \frac {i b \log {\left (i c x + 1 \right )}}{2 c^{2} d^{2} x - 2 i c d^{2}} - \frac {b \left (\frac {\log {\left (b x - \frac {i b}{c} \right )}}{4} - \frac {\log {\left (b x + \frac {i b}{c} \right )}}{4}\right )}{c d^{2}} - \frac {- 2 a + i b}{2 c^{2} d^{2} x - 2 i c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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