Optimal. Leaf size=25 \[ -\frac {i a}{4 f (c-i c \tan (e+f x))^4} \]
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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} \[ -\frac {i a}{4 f (c-i c \tan (e+f x))^4} \]
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))^4} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^5} \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{(c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {i a}{4 f (c-i c \tan (e+f x))^4}\\ \end {align*}
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Mathematica [B] time = 0.61, size = 74, normalized size = 2.96 \[ \frac {a (-i (2 \sin (e+f x)+3 \sin (3 (e+f x)))+10 \cos (e+f x)+5 \cos (3 (e+f x))) (\sin (5 (e+f x))-i \cos (5 (e+f x)))}{64 c^4 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 57, normalized size = 2.28 \[ \frac {-i \, a e^{\left (8 i \, f x + 8 i \, e\right )} - 4 i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, c^{4} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.62, size = 125, normalized size = 5.00 \[ -\frac {2 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 3 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 7 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 8 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 7 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 22, normalized size = 0.88 \[ -\frac {i a}{4 f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}} \]
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 = 4.65, size = 21, normalized size = 0.84 \[ -\frac {a\,1{}\mathrm {i}}{4\,c^4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 168, normalized size = 6.72 \[ \begin {cases} \frac {- 8192 i a c^{12} f^{3} e^{8 i e} e^{8 i f x} - 32768 i a c^{12} f^{3} e^{6 i e} e^{6 i f x} - 49152 i a c^{12} f^{3} e^{4 i e} e^{4 i f x} - 32768 i a c^{12} f^{3} e^{2 i e} e^{2 i f x}}{524288 c^{16} f^{4}} & \text {for}\: 524288 c^{16} f^{4} \neq 0 \\\frac {x \left (a e^{8 i e} + 3 a e^{6 i e} + 3 a e^{4 i e} + a e^{2 i e}\right )}{8 c^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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