Optimal. Leaf size=71 \[ -\frac {3 i \, _2F_1\left (-\frac {1}{6},\frac {19}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{4 \sqrt [6]{2} a^2 f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A]
time = 0.14, 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 = {3586, 3604, 72,
71} \begin {gather*} -\frac {3 i \sqrt [6]{1+i \tan (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {19}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{4 \sqrt [6]{2} a^2 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 {1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2} \, dx &=\frac {\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \int \frac {1}{\sqrt [6]{a-i a \tan (e+f x)} (a+i a \tan (e+f x))^{13/6}} \, 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} (a+i a x)^{19/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac {\left (\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}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{19/6} (a-i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{8 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}}\\ &=-\frac {3 i \, _2F_1\left (-\frac {1}{6},\frac {19}{6};\frac {5}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{4 \sqrt [6]{2} a^2 f \sqrt [3]{d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.58, size = 141, normalized size = 1.99 \begin {gather*} \frac {(d \sec (e+f x))^{2/3} \left (16 e^{3 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};-e^{2 i (e+f x)}\right )-10 \left (7 \cos (e+f x)+5 \cos (3 (e+f x))+18 i \cos ^2(e+f x) \sin (e+f x)\right )\right ) (-3 i \cos (2 (e+f x))-3 \sin (2 (e+f x)))}{260 a^2 d f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-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 {\int \frac {1}{\sqrt [3]{d \sec {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i \sqrt [3]{d \sec {\left (e + f x \right )}} \tan {\left (e + f x \right )} - \sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx}{a^{2}} \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 {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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