Optimal. Leaf size=59 \[ \frac {(A+i B) c}{3 a^3 f (i-\tan (e+f x))^3}-\frac {B c}{2 a^3 f (i-\tan (e+f x))^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3669, 45}
\begin {gather*} \frac {c (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac {B c}{2 a^3 f (-\tan (e+f x)+i)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3669
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {A+i B}{a^4 (-i+x)^4}+\frac {B}{a^4 (-i+x)^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(A+i B) c}{3 a^3 f (i-\tan (e+f x))^3}-\frac {B c}{2 a^3 f (i-\tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 81, normalized size = 1.37 \begin {gather*} \frac {c \sec ^2(e+f x) (3 i A+2 (2 i A+B) \cos (2 (e+f x))-2 (A-2 i B) \sin (2 (e+f x))) (i+\tan (e+f x))}{24 a^3 f (-i+\tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 43, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {c \left (-\frac {i B +A}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,a^{3}}\) | \(43\) |
default | \(\frac {c \left (-\frac {i B +A}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,a^{3}}\) | \(43\) |
risch | \(\frac {c \,{\mathrm e}^{-2 i \left (f x +e \right )} B}{8 a^{3} f}+\frac {i c \,{\mathrm e}^{-2 i \left (f x +e \right )} A}{8 a^{3} f}+\frac {i A c \,{\mathrm e}^{-4 i \left (f x +e \right )}}{8 a^{3} f}-\frac {c \,{\mathrm e}^{-6 i \left (f x +e \right )} B}{24 a^{3} f}+\frac {i c \,{\mathrm e}^{-6 i \left (f x +e \right )} A}{24 a^{3} f}\) | \(100\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.19, size = 63, normalized size = 1.07 \begin {gather*} -\frac {{\left (3 \, {\left (-i \, A - B\right )} c e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, A c e^{\left (2 i \, f x + 2 i \, e\right )} - {\left (i \, A - B\right )} c\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 206 vs. \(2 (44) = 88\).
time = 0.27, size = 206, normalized size = 3.49 \begin {gather*} \begin {cases} \frac {\left (192 i A a^{6} c f^{2} e^{8 i e} e^{- 4 i f x} + \left (64 i A a^{6} c f^{2} e^{6 i e} - 64 B a^{6} c f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (192 i A a^{6} c f^{2} e^{10 i e} + 192 B a^{6} c f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{1536 a^{9} f^{3}} & \text {for}\: a^{9} f^{3} e^{12 i e} \neq 0 \\\frac {x \left (A c e^{4 i e} + 2 A c e^{2 i e} + A c - i B c e^{4 i e} + i B c\right ) e^{- 6 i e}}{4 a^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 149 vs. \(2 (47) = 94\).
time = 0.77, size = 149, normalized size = 2.53 \begin {gather*} -\frac {2 \, {\left (3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 i \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 i \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 i \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.84, size = 62, normalized size = 1.05 \begin {gather*} \frac {\frac {c\,\left (B+A\,2{}\mathrm {i}\right )}{6}+\frac {B\,c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{2}}{a^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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