Optimal. Leaf size=25 \[ \frac {i a (c-i c \tan (e+f x))^4}{4 f} \]
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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} \[ \frac {i a (c-i c \tan (e+f x))^4}{4 f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx &=(a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^3 \, dx\\ &=\frac {(i a) \operatorname {Subst}\left (\int (c+x)^3 \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac {i a (c-i c \tan (e+f x))^4}{4 f}\\ \end {align*}
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Mathematica [B] time = 1.56, size = 85, normalized size = 3.40 \[ \frac {a c^4 \sec (e) \sec ^4(e+f x) (2 \sin (e+2 f x)-2 \sin (3 e+2 f x)+\sin (3 e+4 f x)-2 i \cos (e+2 f x)-2 i \cos (3 e+2 f x)-3 \sin (e)-3 i \cos (e))}{4 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 57, normalized size = 2.28 \[ \frac {4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 61, normalized size = 2.44 \[ \frac {4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 48, normalized size = 1.92 \[ \frac {a \,c^{4} \left (\tan \left (f x +e \right )+\frac {i \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\left (\tan ^{3}\left (f x +e \right )\right )-\frac {3 i \left (\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 60, normalized size = 2.40 \[ \frac {i \, a c^{4} \tan \left (f x + e\right )^{4} - 4 \, a c^{4} \tan \left (f x + e\right )^{3} - 6 i \, a c^{4} \tan \left (f x + e\right )^{2} + 4 \, a c^{4} \tan \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 78, normalized size = 3.12 \[ -\frac {a\,c^4\,\sin \left (e+f\,x\right )\,\left (-4\,{\cos \left (e+f\,x\right )}^3+{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )\,6{}\mathrm {i}+4\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^2-{\sin \left (e+f\,x\right )}^3\,1{}\mathrm {i}\right )}{4\,f\,{\cos \left (e+f\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 90, normalized size = 3.60 \[ \frac {4 a c^{4}}{- i f e^{8 i e} e^{8 i f x} - 4 i f e^{6 i e} e^{6 i f x} - 6 i f e^{4 i e} e^{4 i f x} - 4 i f e^{2 i e} e^{2 i f x} - i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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