Optimal. Leaf size=84 \[ \frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {2}{3}}(c+d x) F_1\left (\frac {2}{3};\frac {5}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 84, 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 = {3564, 130, 511, 510} \[ \frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {2}{3}}(c+d x) F_1\left (\frac {2}{3};\frac {5}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3564
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-\frac {i x}{a}} (a+x)^{5/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a+i a x^3\right )^{5/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (3 a \sqrt {1+i \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+i x^3\right )^{5/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {3 F_1\left (\frac {2}{3};\frac {5}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {2}{3}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 11.78, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (79 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 12 \, e^{\left (5 i \, d x + 5 i \, c\right )} + 170 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 24 \, e^{\left (3 i \, d x + 3 i \, c\right )} + 103 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 12 \, e^{\left (i \, d x + i \, c\right )} + 12\right )} + 36 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - 4 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (27 \, e^{\left (5 i \, d x + 5 i \, c\right )} + 750 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 484 \, e^{\left (3 i \, d x + 3 i \, c\right )} + 40 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 457 \, e^{\left (i \, d x + i \, c\right )} - 710\right )}}{108 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - 6 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 12 \, a^{2} d e^{\left (i \, d x + i \, c\right )} + 8 \, a^{2} d\right )}}, x\right )}{36 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - 4 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
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 \[ \int \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {1}{\tan \left (d x +c \right )^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt [3]{\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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