Optimal. Leaf size=71 \[ \frac {12 i 2^{5/6} a^2 (d \sec (e+f x))^{5/3} \, _2F_1\left (-\frac {11}{6},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f (1+i \tan (e+f x))^{5/6}} \]
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Rubi [A] time = 0.18, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3505, 3523, 70, 69} \[ \frac {12 i 2^{5/6} a^2 (d \sec (e+f x))^{5/3} \text {Hypergeometric2F1}\left (-\frac {11}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f (1+i \tan (e+f x))^{5/6}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))^2 \, dx &=\frac {(d \sec (e+f x))^{5/3} \int (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{17/6} \, dx}{(a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}}\\ &=\frac {\left (a^2 (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{11/6}}{\sqrt [6]{a-i a x}} \, dx,x,\tan (e+f x)\right )}{f (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}}\\ &=\frac {\left (2\ 2^{5/6} a^3 (d \sec (e+f x))^{5/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{11/6}}{\sqrt [6]{a-i a x}} \, dx,x,\tan (e+f x)\right )}{f (a-i a \tan (e+f x))^{5/6} \left (\frac {a+i a \tan (e+f x)}{a}\right )^{5/6}}\\ &=\frac {12 i 2^{5/6} a^2 \, _2F_1\left (-\frac {11}{6},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{5/3}}{5 f (1+i \tan (e+f x))^{5/6}}\\ \end {align*}
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Mathematica [B] time = 2.82, size = 267, normalized size = 3.76 \[ \frac {(a+i a \tan (e+f x))^2 (d \sec (e+f x))^{5/3} \left (\frac {3}{4} \csc (e) (\cos (2 e)-i \sin (2 e)) \sec ^{\frac {8}{3}}(e+f x) (64 i \sin (2 e+f x)+75 \cos (2 e+f x)+55 \cos (2 e+3 f x)-64 i \sin (f x)+90 \cos (f x))-\frac {33 i 2^{2/3} \left (5 \sqrt [3]{1+e^{2 i (e+f x)}}-\left (-1+e^{2 i e}\right ) e^{2 i f x} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};-e^{2 i (e+f x)}\right )\right )}{\left (-1+e^{2 i e}\right ) \sqrt [3]{\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt [3]{1+e^{2 i (e+f x)}}}\right )}{80 f \sec ^{\frac {11}{3}}(e+f x) (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ \frac {2^{\frac {2}{3}} {\left (-165 i \, a^{2} d e^{\left (5 i \, f x + 5 i \, e\right )} - 78 i \, a^{2} d e^{\left (3 i \, f x + 3 i \, e\right )} - 33 i \, a^{2} d e^{\left (i \, f x + i \, e\right )}\right )} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} e^{\left (\frac {2}{3} i \, f x + \frac {2}{3} i \, e\right )} + 80 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} {\rm integral}\left (\frac {11 i \cdot 2^{\frac {2}{3}} a^{2} d \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} e^{\left (\frac {2}{3} i \, f x + \frac {2}{3} i \, e\right )}}{16 \, f}, x\right )}{80 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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