Optimal. Leaf size=59 \[ \frac {1}{4 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {i x}{4 a^2}-\frac {1}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3526, 3479, 8} \[ \frac {1}{4 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {i x}{4 a^2}-\frac {1}{4 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rule 3526
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {1}{4 d (a+i a \tan (c+d x))^2}-\frac {i \int \frac {1}{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=-\frac {1}{4 d (a+i a \tan (c+d x))^2}+\frac {1}{4 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {i \int 1 \, dx}{4 a^2}\\ &=-\frac {i x}{4 a^2}-\frac {1}{4 d (a+i a \tan (c+d x))^2}+\frac {1}{4 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 66, normalized size = 1.12 \[ \frac {\sec ^2(c+d x) ((1+4 i d x) \cos (2 (c+d x))-(4 d x+i) \sin (2 (c+d x)))}{16 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 32, normalized size = 0.54 \[ \frac {{\left (-4 i \, d x e^{\left (4 i \, d x + 4 i \, c\right )} - 1\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 70, normalized size = 1.19 \[ -\frac {\frac {\log \left (\tan \left (2 \, d x + 2 \, c\right ) - i\right )}{a^{2}} - \frac {\log \left (-i \, \tan \left (2 \, d x + 2 \, c\right ) + 1\right )}{a^{2}} - \frac {\tan \left (2 \, d x + 2 \, c\right ) + i}{a^{2} {\left (\tan \left (2 \, d x + 2 \, c\right ) - i\right )}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 77, normalized size = 1.31 \[ \frac {\ln \left (\tan \left (d x +c \right )+i\right )}{8 d \,a^{2}}-\frac {i}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )}+\frac {1}{4 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {\ln \left (\tan \left (d x +c \right )-i\right )}{8 d \,a^{2}} \]
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 [B] time = 3.97, size = 46, normalized size = 0.78 \[ -\frac {x\,1{}\mathrm {i}}{4\,a^2}+\frac {\mathrm {tan}\left (c+d\,x\right )}{4\,a^2\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 75, normalized size = 1.27 \[ \begin {cases} - \frac {e^{- 4 i c} e^{- 4 i d x}}{16 a^{2} d} & \text {for}\: 16 a^{2} d e^{4 i c} \neq 0 \\x \left (\frac {\left (- i e^{4 i c} + i\right ) e^{- 4 i c}}{4 a^{2}} + \frac {i}{4 a^{2}}\right ) & \text {otherwise} \end {cases} - \frac {i x}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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