Optimal. Leaf size=122 \[ \frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac{3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f} \]
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Rubi [A] time = 0.23504, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3494, 3493} \[ \frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{7 f}+\frac{3 i a (a+i a \tan (e+f x))^{5/3} (d \sec (e+f x))^{2/3}}{7 f} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{8/3} \, dx &=\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac{1}{7} (12 a) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3} \, dx\\ &=\frac{9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}+\frac{1}{7} \left (18 a^2\right ) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx\\ &=\frac{54 i a^3 (d \sec (e+f x))^{2/3}}{7 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{9 i a^2 (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{7 f}+\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{5/3}}{7 f}\\ \end{align*}
Mathematica [A] time = 0.595814, size = 100, normalized size = 0.82 \[ \frac{3 a^2 (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{5/3} (\sin (e-f x)+i \cos (e-f x)) (5 i \sin (2 (e+f x))+23 \cos (2 (e+f x))+21)}{14 d f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{2}{3}}} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{{\frac{8}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.20946, size = 543, normalized size = 4.45 \begin{align*} \frac{42 \,{\left (-i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (\frac{4}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 2^{\frac{1}{3}} a^{2} \sin \left (\frac{4}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} \sqrt{\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1} a^{\frac{2}{3}} d^{\frac{2}{3}} -{\left (-12 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (\frac{7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 12 \cdot 2^{\frac{1}{3}} a^{2} \sin \left (\frac{7}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) +{\left (-84 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} - 84 i \cdot 2^{\frac{1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 168 i \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 84 i \cdot 2^{\frac{1}{3}} a^{2}\right )} \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 84 \,{\left (2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 2^{\frac{1}{3}} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \cdot 2^{\frac{1}{3}} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2^{\frac{1}{3}} a^{2}\right )} \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} a^{\frac{2}{3}} d^{\frac{2}{3}}}{7 \,{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac{7}{6}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06202, size = 305, normalized size = 2.5 \begin{align*} \frac{2 \cdot 2^{\frac{1}{3}}{\left (42 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 63 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 27 i \, a^{2}\right )} \left (\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}} e^{\left (2 i \, f x + 2 i \, e\right )}}{7 \,{\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{8}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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