Optimal. Leaf size=25 \[ -\frac {i a}{2 f (c-i c \tan (e+f x))^2} \]
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Rubi [A] time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ -\frac {i a}{2 f (c-i c \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^2} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^3} \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {i a}{2 f (c-i c \tan (e+f x))^2}\\ \end {align*}
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Mathematica [B] time = 0.73, size = 51, normalized size = 2.04 \[ \frac {a (3 \cos (e+f x)-i \sin (e+f x)) (\sin (3 (e+f x))-i \cos (3 (e+f x)))}{8 c^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 1.32 \[ \frac {-i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 2 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.71, size = 65, normalized size = 2.60 \[ -\frac {2 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{c^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 22, normalized size = 0.88 \[ \frac {i a}{2 f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}} \]
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 = 4.61, size = 21, normalized size = 0.84 \[ \frac {a\,1{}\mathrm {i}}{2\,c^2\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 90, normalized size = 3.60 \[ \begin {cases} \frac {- 4 i a c^{2} f e^{4 i e} e^{4 i f x} - 8 i a c^{2} f e^{2 i e} e^{2 i f x}}{32 c^{4} f^{2}} & \text {for}\: 32 c^{4} f^{2} \neq 0 \\\frac {x \left (a e^{4 i e} + a e^{2 i e}\right )}{2 c^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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