Optimal. Leaf size=83 \[ \frac {3 i \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{5/3} \sqrt [6]{1+i \tan (e+f x)}}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))} \]
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Rubi [A]
time = 0.14, antiderivative size = 83, 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 = {3586, 3604, 72,
71} \begin {gather*} \frac {3 i \sqrt [6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{5/3}}{a+i a \tan (e+f x)} \, dx &=\frac {(d \sec (e+f x))^{5/3} \int \frac {(a-i a \tan (e+f x))^{5/6}}{\sqrt [6]{a+i a \tan (e+f x)}} \, 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 ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{a-i a x} (a+i a x)^{7/6}} \, 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 (a (d \sec (e+f x))^{5/3} \sqrt [6]{\frac {a+i a \tan (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{7/6} \sqrt [6]{a-i a x}} \, dx,x,\tan (e+f x)\right )}{2 \sqrt [6]{2} f (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))}\\ &=\frac {3 i \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {11}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{5/3} \sqrt [6]{1+i \tan (e+f x)}}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 84, normalized size = 1.01 \begin {gather*} \frac {6 d e^{i (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-e^{2 i (e+f x)}\right ) (d \sec (e+f x))^{2/3}}{a \sqrt [3]{1+e^{2 i (e+f x)}} f (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.47, size = 0, normalized size = 0.00 \[\int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}{a +i a \tan \left (f x +e \right )}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \frac {i \int \frac {\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}}{a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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