Optimal. Leaf size=71 \[ -\frac {3 i (1+i \tan (e+f x))^{5/6} \, _2F_1\left (-\frac {5}{6},\frac {17}{6};\frac {1}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}} \]
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Rubi [A] time = 0.21, 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} \[ -\frac {3 i (1+i \tan (e+f x))^{5/6} \text {Hypergeometric2F1}\left (-\frac {5}{6},\frac {17}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {1}{(d \sec (e+f x))^{5/3} (a+i a \tan (e+f x))} \, dx &=\frac {\left ((a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}\right ) \int \frac {1}{(a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{11/6}} \, dx}{(d \sec (e+f x))^{5/3}}\\ &=\frac {\left (a^2 (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-i a x)^{11/6} (a+i a x)^{17/6}} \, dx,x,\tan (e+f x)\right )}{f (d \sec (e+f x))^{5/3}}\\ &=\frac {\left ((a-i a \tan (e+f x))^{5/6} \left (\frac {a+i a \tan (e+f x)}{a}\right )^{5/6}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{17/6} (a-i a x)^{11/6}} \, dx,x,\tan (e+f x)\right )}{4\ 2^{5/6} f (d \sec (e+f x))^{5/3}}\\ &=-\frac {3 i \, _2F_1\left (-\frac {5}{6},\frac {17}{6};\frac {1}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{5/6}}{10\ 2^{5/6} a f (d \sec (e+f x))^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 119, normalized size = 1.68 \[ -\frac {3 \sec ^2(e+f x) \left (\frac {128 e^{2 i (e+f x)} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-e^{2 i (e+f x)}\right )}{\left (1+e^{2 i (e+f x)}\right )^{2/3}}+16 i \sin (2 (e+f x))+6 \cos (2 (e+f x))-26\right )}{220 a f (\tan (e+f x)-i) (d \sec (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ \frac {{\left (440 \, a d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} {\rm integral}\left (-\frac {16 i \cdot 2^{\frac {1}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} e^{\left (-\frac {2}{3} i \, f x - \frac {2}{3} i \, e\right )}}{55 \, a d^{2} f}, x\right ) + 2^{\frac {1}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} {\left (-33 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 45 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 93 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 15 i\right )} e^{\left (\frac {1}{3} i \, f x + \frac {1}{3} i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{440 \, a d^{2} f} \]
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 \[ \int \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +i a \tan \left (f x +e \right )\right )}\, dx \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}} \tan {\left (e + f x \right )} - i \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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