Optimal. Leaf size=69 \[ -\frac {3 i \sqrt [6]{2} a (1+i \tan (e+f x))^{5/6} \, _2F_1\left (-\frac {5}{6},\frac {5}{6};\frac {1}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f (d \sec (e+f x))^{5/3}} \]
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Rubi [A] time = 0.15, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3505, 3523, 70, 69} \[ -\frac {3 i \sqrt [6]{2} a (1+i \tan (e+f x))^{5/6} \text {Hypergeometric2F1}\left (-\frac {5}{6},\frac {5}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f (d \sec (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx &=\frac {\left ((a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}\right ) \int \frac {\sqrt [6]{a+i a \tan (e+f x)}}{(a-i a \tan (e+f x))^{5/6}} \, 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)^{5/6}} \, dx,x,\tan (e+f x)\right )}{f (d \sec (e+f x))^{5/3}}\\ &=\frac {\left (a^2 (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 )^{5/6} (a-i a x)^{11/6}} \, dx,x,\tan (e+f x)\right )}{2^{5/6} f (d \sec (e+f x))^{5/3}}\\ &=-\frac {3 i \sqrt [6]{2} a \, _2F_1\left (-\frac {5}{6},\frac {5}{6};\frac {1}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{5/6}}{5 f (d \sec (e+f x))^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 106, normalized size = 1.54 \[ -\frac {3 i a e^{i (e+f x)} \left (4 \sqrt [3]{1+e^{2 i (e+f x)}} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-e^{2 i (e+f x)}\right )+e^{2 i (e+f x)}+1\right )}{5 d f \left (1+e^{2 i (e+f x)}\right ) (d \sec (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ \frac {10 \, d^{2} f {\rm integral}\left (-\frac {2 i \cdot 2^{\frac {1}{3}} a \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} e^{\left (-\frac {2}{3} i \, f x - \frac {2}{3} i \, e\right )}}{5 \, d^{2} f}, x\right ) + 2^{\frac {1}{3}} {\left (-3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, a\right )} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {1}{3} i \, f x + \frac {1}{3} i \, e\right )}}{10 \, d^{2} f} \]
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 \[ \int \frac {i \, a \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {a +i a \tan \left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {i \, a \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}} \,d x \]
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 \[ i a \left (\int \left (- \frac {i}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\right )\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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