Optimal. Leaf size=67 \[ -\frac {3 i 2^{5/6} a \, _2F_1\left (-\frac {1}{6},\frac {1}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A]
time = 0.13, antiderivative size = 67, 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 = {3586, 3604, 72,
71} \begin {gather*} -\frac {3 i 2^{5/6} a \sqrt [6]{1+i \tan (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [3]{d \sec (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 {a+i a \tan (e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx &=\frac {\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \int \frac {(a+i a \tan (e+f x))^{5/6}}{\sqrt [6]{a-i a \tan (e+f x)}} \, dx}{\sqrt [3]{d \sec (e+f x)}}\\ &=\frac {\left (a^2 \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{(a-i a x)^{7/6} \sqrt [6]{a+i a x}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac {\left (a^2 \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{\frac {a+i a \tan (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{\frac {1}{2}+\frac {i x}{2}} (a-i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{\sqrt [6]{2} f \sqrt [3]{d \sec (e+f x)}}\\ &=-\frac {3 i 2^{5/6} a \, _2F_1\left (-\frac {1}{6},\frac {1}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{f \sqrt [3]{d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 98, normalized size = 1.46 \begin {gather*} -\frac {3 i 2^{2/3} a e^{2 i (e+f x)} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};-e^{2 i (e+f x)}\right )}{5 \sqrt [3]{\frac {d e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt [3]{1+e^{2 i (e+f x)}} f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {a +i a \tan \left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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*} i a \left (\int \left (- \frac {i}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\right )\, dx + \int \frac {\tan {\left (e + f x \right )}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx\right ) \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 {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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