Optimal. Leaf size=223 \[ \frac {35 x}{128 a^3 c^4}-\frac {i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac {i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac {i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac {5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac {5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac {15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )} \]
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Rubi [A]
time = 0.15, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3603, 3568, 46,
212} \begin {gather*} -\frac {5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac {15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}+\frac {35 x}{128 a^3 c^4}-\frac {5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac {i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac {i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac {i}{64 a^3 f (c-i c \tan (e+f x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx &=\frac {\int \frac {\cos ^6(e+f x)}{c-i c \tan (e+f x)} \, dx}{a^3 c^3}\\ &=\frac {\left (i c^4\right ) \text {Subst}\left (\int \frac {1}{(c-x)^4 (c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=\frac {\left (i c^4\right ) \text {Subst}\left (\int \left (\frac {1}{32 c^5 (c-x)^4}+\frac {5}{64 c^6 (c-x)^3}+\frac {15}{128 c^7 (c-x)^2}+\frac {1}{16 c^4 (c+x)^5}+\frac {1}{8 c^5 (c+x)^4}+\frac {5}{32 c^6 (c+x)^3}+\frac {5}{32 c^7 (c+x)^2}+\frac {35}{128 c^7 \left (c^2-x^2\right )}\right ) \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=-\frac {i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac {i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac {i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac {5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac {5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac {15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}+\frac {(35 i) \text {Subst}\left (\int \frac {1}{c^2-x^2} \, dx,x,-i c \tan (e+f x)\right )}{128 a^3 c^3 f}\\ &=\frac {35 x}{128 a^3 c^4}-\frac {i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac {i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac {i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac {5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac {5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac {15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.43, size = 133, normalized size = 0.60 \begin {gather*} \frac {(\cos (e+f x)+i \sin (e+f x)) (420 (-i+2 f x) \cos (e+f x)+126 i \cos (3 (e+f x))+14 i \cos (5 (e+f x))+i \cos (7 (e+f x))+420 \sin (e+f x)-840 i f x \sin (e+f x)+378 \sin (3 (e+f x))+70 \sin (5 (e+f x))+7 \sin (7 (e+f x)))}{3072 a^3 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 132, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {35 i \ln \left (\tan \left (f x +e \right )-i\right )}{256}-\frac {5 i}{128 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{96 \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {15}{128 \left (\tan \left (f x +e \right )-i\right )}+\frac {5 i}{64 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{64 \left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {35 i \ln \left (\tan \left (f x +e \right )+i\right )}{256}-\frac {1}{24 \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {5}{32 \left (\tan \left (f x +e \right )+i\right )}}{f \,a^{3} c^{4}}\) | \(132\) |
default | \(\frac {-\frac {35 i \ln \left (\tan \left (f x +e \right )-i\right )}{256}-\frac {5 i}{128 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{96 \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {15}{128 \left (\tan \left (f x +e \right )-i\right )}+\frac {5 i}{64 \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{64 \left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {35 i \ln \left (\tan \left (f x +e \right )+i\right )}{256}-\frac {1}{24 \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {5}{32 \left (\tan \left (f x +e \right )+i\right )}}{f \,a^{3} c^{4}}\) | \(132\) |
risch | \(\frac {35 x}{128 a^{3} c^{4}}-\frac {i {\mathrm e}^{8 i \left (f x +e \right )}}{1024 a^{3} c^{4} f}-\frac {i \cos \left (6 f x +6 e \right )}{128 a^{3} c^{4} f}+\frac {\sin \left (6 f x +6 e \right )}{96 a^{3} c^{4} f}-\frac {7 i \cos \left (4 f x +4 e \right )}{256 a^{3} c^{4} f}+\frac {7 \sin \left (4 f x +4 e \right )}{128 a^{3} c^{4} f}-\frac {7 i \cos \left (2 f x +2 e \right )}{128 a^{3} c^{4} f}+\frac {7 \sin \left (2 f x +2 e \right )}{32 a^{3} c^{4} f}\) | \(155\) |
norman | \(\frac {\frac {35 x}{128 a c}-\frac {i}{8 a c f}+\frac {93 \tan \left (f x +e \right )}{128 a c f}+\frac {511 \left (\tan ^{3}\left (f x +e \right )\right )}{384 a c f}+\frac {385 \left (\tan ^{5}\left (f x +e \right )\right )}{384 a c f}+\frac {35 \left (\tan ^{7}\left (f x +e \right )\right )}{128 a c f}+\frac {35 x \left (\tan ^{2}\left (f x +e \right )\right )}{32 a c}+\frac {105 x \left (\tan ^{4}\left (f x +e \right )\right )}{64 a c}+\frac {35 x \left (\tan ^{6}\left (f x +e \right )\right )}{32 a c}+\frac {35 x \left (\tan ^{8}\left (f x +e \right )\right )}{128 a c}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{2} c^{3}}\) | \(184\) |
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 = 1.84, size = 109, normalized size = 0.49 \begin {gather*} \frac {{\left (840 \, f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (14 i \, f x + 14 i \, e\right )} - 28 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 126 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 420 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 252 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 42 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{3072 \, a^{3} c^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 333, normalized size = 1.49 \begin {gather*} \begin {cases} \frac {\left (- 10133099161583616 i a^{18} c^{24} f^{6} e^{20 i e} e^{8 i f x} - 94575592174780416 i a^{18} c^{24} f^{6} e^{18 i e} e^{6 i f x} - 425590164786511872 i a^{18} c^{24} f^{6} e^{16 i e} e^{4 i f x} - 1418633882621706240 i a^{18} c^{24} f^{6} e^{14 i e} e^{2 i f x} + 851180329573023744 i a^{18} c^{24} f^{6} e^{10 i e} e^{- 2 i f x} + 141863388262170624 i a^{18} c^{24} f^{6} e^{8 i e} e^{- 4 i f x} + 13510798882111488 i a^{18} c^{24} f^{6} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{10376293541461622784 a^{21} c^{28} f^{7}} & \text {for}\: a^{21} c^{28} f^{7} e^{12 i e} \neq 0 \\x \left (\frac {\left (e^{14 i e} + 7 e^{12 i e} + 21 e^{10 i e} + 35 e^{8 i e} + 35 e^{6 i e} + 21 e^{4 i e} + 7 e^{2 i e} + 1\right ) e^{- 6 i e}}{128 a^{3} c^{4}} - \frac {35}{128 a^{3} c^{4}}\right ) & \text {otherwise} \end {cases} + \frac {35 x}{128 a^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 160, normalized size = 0.72 \begin {gather*} -\frac {\frac {420 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4}} - \frac {420 i \, \log \left (i \, \tan \left (f x + e\right ) - 1\right )}{a^{3} c^{4}} - \frac {2 \, {\left (385 \, \tan \left (f x + e\right )^{3} - 1335 i \, \tan \left (f x + e\right )^{2} - 1575 \, \tan \left (f x + e\right ) + 641 i\right )}}{a^{3} c^{4} {\left (i \, \tan \left (f x + e\right ) + 1\right )}^{3}} + \frac {875 i \, \tan \left (f x + e\right )^{4} - 3980 \, \tan \left (f x + e\right )^{3} - 6930 i \, \tan \left (f x + e\right )^{2} + 5548 \, \tan \left (f x + e\right ) + 1771 i}{a^{3} c^{4} {\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{3072 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.10, size = 109, normalized size = 0.49 \begin {gather*} \frac {35\,x}{128\,a^3\,c^4}-\frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^6\,35{}\mathrm {i}}{128}-\frac {35\,{\mathrm {tan}\left (e+f\,x\right )}^5}{128}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,35{}\mathrm {i}}{48}-\frac {35\,{\mathrm {tan}\left (e+f\,x\right )}^3}{48}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,77{}\mathrm {i}}{128}-\frac {77\,\mathrm {tan}\left (e+f\,x\right )}{128}+\frac {1}{8}{}\mathrm {i}}{a^3\,c^4\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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