Optimal. Leaf size=209 \[ \frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 0.26, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3479, 8, 3730} \[ \frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rule 3730
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+i a \tan (e+f x))^3} \, dx &=\frac {x (c+d x)}{8 a^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-d \int \left (\frac {x}{8 a^3}+\frac {i}{6 f (a+i a \tan (e+f x))^3}+\frac {i}{8 a f (a+i a \tan (e+f x))^2}+\frac {i}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\right ) \, dx\\ &=-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(i d) \int \frac {1}{a^3+i a^3 \tan (e+f x)} \, dx}{8 f}-\frac {(i d) \int \frac {1}{(a+i a \tan (e+f x))^3} \, dx}{6 f}-\frac {(i d) \int \frac {1}{(a+i a \tan (e+f x))^2} \, dx}{8 a f}\\ &=-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {d}{32 a f^2 (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {d}{16 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(i d) \int 1 \, dx}{16 a^3 f}-\frac {(i d) \int \frac {1}{a+i a \tan (e+f x)} \, dx}{16 a^2 f}-\frac {(i d) \int \frac {1}{(a+i a \tan (e+f x))^2} \, dx}{12 a f}\\ &=-\frac {i d x}{16 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {3 d}{32 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(i d) \int 1 \, dx}{32 a^3 f}-\frac {(i d) \int \frac {1}{a+i a \tan (e+f x)} \, dx}{24 a^2 f}\\ &=-\frac {3 i d x}{32 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(i d) \int 1 \, dx}{48 a^3 f}\\ &=-\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac {i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac {i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 205, normalized size = 0.98 \[ \frac {i \sec ^3(e+f x) \left (4 \left (6 c f (6 f x+i)+d \left (18 f^2 x^2+6 i f x+1\right )\right ) \cos (3 (e+f x))+27 (12 i c f+d (5+12 i f x)) \cos (e+f x)+144 i c f^2 x \sin (3 (e+f x))-108 c f \sin (e+f x)+24 c f \sin (3 (e+f x))+72 i d f^2 x^2 \sin (3 (e+f x))-108 d f x \sin (e+f x)+24 d f x \sin (3 (e+f x))+81 i d \sin (e+f x)-4 i d \sin (3 (e+f x))\right )}{1152 a^3 f^2 (\tan (e+f x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 103, normalized size = 0.49 \[ \frac {{\left (24 i \, d f x + 24 i \, c f + 72 \, {\left (d f^{2} x^{2} + 2 \, c f^{2} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (216 i \, d f x + 216 i \, c f + 108 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (108 i \, d f x + 108 i \, c f + 27 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.13, size = 151, normalized size = 0.72 \[ \frac {{\left (72 \, d f^{2} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 144 \, c f^{2} x e^{\left (6 i \, f x + 6 i \, e\right )} + 216 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 108 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, d f x + 216 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 108 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f + 108 \, d e^{\left (4 i \, f x + 4 i \, e\right )} + 27 \, d e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.85, size = 527, normalized size = 2.52 \[ \frac {-\frac {4 i d \left (-\frac {\left (f x +e \right ) \left (\cos ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{36}+\frac {5 f x}{96}+\frac {5 e}{96}\right )}{f}+\frac {2 i c \left (\cos ^{6}\left (f x +e \right )\right )}{3}-\frac {2 i d e \left (\cos ^{6}\left (f x +e \right )\right )}{3 f}+\frac {4 d \left (\left (f x +e \right ) \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {5 \left (f x +e \right )^{2}}{32}+\frac {\left (\cos ^{6}\left (f x +e \right )\right )}{36}+\frac {5 \left (\cos ^{4}\left (f x +e \right )\right )}{96}+\frac {5 \left (\cos ^{2}\left (f x +e \right )\right )}{32}\right )}{f}+4 c \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {4 d e \left (\frac {\left (\cos ^{5}\left (f x +e \right )+\frac {5 \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \cos \left (f x +e \right )}{8}\right ) \sin \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}+\frac {i d \left (-\frac {\left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{16}+\frac {3 f x}{32}+\frac {3 e}{32}\right )}{f}-\frac {i c \left (\cos ^{4}\left (f x +e \right )\right )}{4}+\frac {i d e \left (\cos ^{4}\left (f x +e \right )\right )}{4 f}-\frac {3 d \left (\left (f x +e \right ) \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {3 \left (f x +e \right )^{2}}{16}+\frac {\left (\cos ^{4}\left (f x +e \right )\right )}{16}+\frac {3 \left (\cos ^{2}\left (f x +e \right )\right )}{16}\right )}{f}-3 c \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {3 d e \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}}{a^{3} f} \]
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 = 3.03, size = 146, normalized size = 0.70 \[ \frac {d\,x^2}{16\,a^3}-{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (-12\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{128\,a^3\,f^2}-\frac {d\,x\,3{}\mathrm {i}}{32\,a^3\,f}\right )-{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (-6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{288\,a^3\,f^2}-\frac {d\,x\,1{}\mathrm {i}}{48\,a^3\,f}\right )-{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-6\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^3\,f^2}-\frac {d\,x\,3{}\mathrm {i}}{16\,a^3\,f}\right )+\frac {c\,x}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 320, normalized size = 1.53 \[ \begin {cases} - \frac {\left (\left (- 24576 i a^{6} c f^{5} e^{6 i e} - 24576 i a^{6} d f^{5} x e^{6 i e} - 4096 a^{6} d f^{4} e^{6 i e}\right ) e^{- 6 i f x} + \left (- 110592 i a^{6} c f^{5} e^{8 i e} - 110592 i a^{6} d f^{5} x e^{8 i e} - 27648 a^{6} d f^{4} e^{8 i e}\right ) e^{- 4 i f x} + \left (- 221184 i a^{6} c f^{5} e^{10 i e} - 221184 i a^{6} d f^{5} x e^{10 i e} - 110592 a^{6} d f^{4} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{1179648 a^{9} f^{6}} & \text {for}\: 1179648 a^{9} f^{6} e^{12 i e} \neq 0 \\\frac {x^{2} \left (3 d e^{4 i e} + 3 d e^{2 i e} + d\right ) e^{- 6 i e}}{16 a^{3}} + \frac {x \left (3 c e^{4 i e} + 3 c e^{2 i e} + c\right ) e^{- 6 i e}}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c x}{8 a^{3}} + \frac {d x^{2}}{16 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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