Optimal. Leaf size=81 \[ \frac{9 i a^2 (d \sec (e+f x))^{2/3}}{2 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{3 i a (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{4 f} \]
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Rubi [A] time = 0.154354, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3494, 3493} \[ \frac{9 i a^2 (d \sec (e+f x))^{2/3}}{2 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{3 i a (a+i a \tan (e+f x))^{2/3} (d \sec (e+f x))^{2/3}}{4 f} \]
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))^{5/3} \, dx &=\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{4 f}+\frac{1}{2} (3 a) \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx\\ &=\frac{9 i a^2 (d \sec (e+f x))^{2/3}}{2 f \sqrt [3]{a+i a \tan (e+f x)}}+\frac{3 i a (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3}}{4 f}\\ \end{align*}
Mathematica [A] time = 0.514008, size = 70, normalized size = 0.86 \[ -\frac{3 a d (\cos (e)-i \sin (e)) (\tan (e+f x)-7 i) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{2/3}}{4 f \sqrt [3]{d \sec (e+f x)}} \]
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{5}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.00544, size = 429, normalized size = 5.3 \begin{align*} \frac{3 \,{\left (-i \cdot 2^{\frac{1}{3}} a \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 \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 ({\left (-12 i \cdot 2^{\frac{1}{3}} a \cos \left (2 \, f x + 2 \, e\right )^{2} - 12 i \cdot 2^{\frac{1}{3}} a \sin \left (2 \, f x + 2 \, e\right )^{2} - 24 i \cdot 2^{\frac{1}{3}} a \cos \left (2 \, f x + 2 \, e\right ) - 12 i \cdot 2^{\frac{1}{3}} a\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 ) - 12 \,{\left (2^{\frac{1}{3}} a \cos \left (2 \, f x + 2 \, e\right )^{2} + 2^{\frac{1}{3}} a \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \cdot 2^{\frac{1}{3}} a \cos \left (2 \, f x + 2 \, e\right ) + 2^{\frac{1}{3}} a\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}}}{2 \,{\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 = 2.0753, size = 167, normalized size = 2.06 \begin{align*} \frac{2^{\frac{1}{3}}{\left (12 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} + 9 i \, a\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}}}{2 \, f} \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{5}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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