Optimal. Leaf size=83 \[ \frac {3 i \sqrt [3]{1+i \tan (c+d x)} \sqrt [3]{e \sec (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {4}{3};\frac {7}{6};\frac {1}{2} (1-i \tan (c+d x))\right )}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.18, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3505, 3523, 70, 69} \[ \frac {3 i \sqrt [3]{1+i \tan (c+d x)} \sqrt [3]{e \sec (c+d x)} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {4}{3},\frac {7}{6},\frac {1}{2} (1-i \tan (c+d x))\right )}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\sqrt [3]{e \sec (c+d x)} \int \frac {\sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{\sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a^2 \sqrt [3]{e \sec (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-i a x)^{5/6} (a+i a x)^{4/3}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a \sqrt [3]{e \sec (c+d x)} \sqrt [3]{\frac {a+i a \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{4/3} (a-i a x)^{5/6}} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [3]{2} d \sqrt [6]{a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {3 i \, _2F_1\left (\frac {1}{6},\frac {4}{3};\frac {7}{6};\frac {1}{2} (1-i \tan (c+d x))\right ) \sqrt [3]{e \sec (c+d x)} \sqrt [3]{1+i \tan (c+d x)}}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 95, normalized size = 1.14 \[ \frac {3 \left (8 i-\frac {2 i e^{2 i (c+d x)} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {5}{3};-e^{2 i (c+d x)}\right )}{\sqrt [6]{1+e^{2 i (c+d x)}}}\right ) \sqrt [3]{e \sec (c+d x)}}{16 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ \frac {{\left (4 \, a d e^{\left (i \, d x + i \, c\right )} {\rm integral}\left (-\frac {i \cdot 2^{\frac {5}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {1}{3} i \, d x + \frac {1}{3} i \, c\right )}}{4 \, a d}, x\right ) + 2^{\frac {5}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {1}{3} i \, d x + \frac {1}{3} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a d} \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {1}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{1/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {\sqrt [3]{e \sec {\left (c + d x \right )}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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