Optimal. Leaf size=82 \[ -\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}+\frac {a^3 c^4 \tan (e+f x)}{f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan ^5(e+f x)}{5 f} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3567,
3852} \begin {gather*} \frac {a^3 c^4 \tan ^5(e+f x)}{5 f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan (e+f x)}{f}-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3603
Rule 3852
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4 \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) (c-i c \tan (e+f x)) \, dx\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}+\left (a^3 c^4\right ) \int \sec ^6(e+f x) \, dx\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}-\frac {\left (a^3 c^4\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}+\frac {a^3 c^4 \tan (e+f x)}{f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A]
time = 1.04, size = 63, normalized size = 0.77 \begin {gather*} \frac {a^3 c^4 \sec (e) \sec ^6(e+f x) (-10 i \cos (e)-10 \sin (e)+15 \sin (e+2 f x)+6 \sin (3 e+4 f x)+\sin (5 e+6 f x))}{60 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 75, normalized size = 0.91
method | result | size |
risch | \(\frac {16 i a^{3} c^{4} \left (15 \,{\mathrm e}^{4 i \left (f x +e \right )}+6 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}\) | \(50\) |
derivativedivides | \(-\frac {i a^{3} c^{4} \left (\frac {\left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{2}+\frac {i \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {2 i \left (\tan ^{3}\left (f x +e \right )\right )}{3}+i \tan \left (f x +e \right )\right )}{f}\) | \(75\) |
default | \(-\frac {i a^{3} c^{4} \left (\frac {\left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{2}+\frac {i \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {2 i \left (\tan ^{3}\left (f x +e \right )\right )}{3}+i \tan \left (f x +e \right )\right )}{f}\) | \(75\) |
norman | \(\frac {a^{3} c^{4} \tan \left (f x +e \right )}{f}+\frac {2 a^{3} c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {a^{3} c^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}-\frac {i a^{3} c^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {i a^{3} c^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{2 f}-\frac {i a^{3} c^{4} \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 106, normalized size = 1.29 \begin {gather*} -\frac {5 i \, a^{3} c^{4} \tan \left (f x + e\right )^{6} - 6 \, a^{3} c^{4} \tan \left (f x + e\right )^{5} + 15 i \, a^{3} c^{4} \tan \left (f x + e\right )^{4} - 20 \, a^{3} c^{4} \tan \left (f x + e\right )^{3} + 15 i \, a^{3} c^{4} \tan \left (f x + e\right )^{2} - 30 \, a^{3} c^{4} \tan \left (f x + e\right )}{30 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 128, normalized size = 1.56 \begin {gather*} -\frac {16 \, {\left (-15 i \, a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 i \, a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{3} c^{4}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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. 175 vs. \(2 (73) = 146\).
time = 0.42, size = 175, normalized size = 2.13 \begin {gather*} \frac {240 i a^{3} c^{4} e^{4 i e} e^{4 i f x} + 96 i a^{3} c^{4} e^{2 i e} e^{2 i f x} + 16 i a^{3} c^{4}}{15 f e^{12 i e} e^{12 i f x} + 90 f e^{10 i e} e^{10 i f x} + 225 f e^{8 i e} e^{8 i f x} + 300 f e^{6 i e} e^{6 i f x} + 225 f e^{4 i e} e^{4 i f x} + 90 f e^{2 i e} e^{2 i f x} + 15 f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 128, normalized size = 1.56 \begin {gather*} -\frac {16 \, {\left (-15 i \, a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 i \, a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{3} c^{4}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.55, size = 117, normalized size = 1.43 \begin {gather*} \frac {a^3\,c^4\,\sin \left (e+f\,x\right )\,\left (30\,{\cos \left (e+f\,x\right )}^5-{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )\,15{}\mathrm {i}+20\,{\cos \left (e+f\,x\right )}^3\,{\sin \left (e+f\,x\right )}^2-{\cos \left (e+f\,x\right )}^2\,{\sin \left (e+f\,x\right )}^3\,15{}\mathrm {i}+6\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^4-{\sin \left (e+f\,x\right )}^5\,5{}\mathrm {i}\right )}{30\,f\,{\cos \left (e+f\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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