Optimal. Leaf size=82 \[ \frac {3 a \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)} F_1\left (\frac {2}{3};-\frac {1}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 d \sqrt {1+i \tan (c+d x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 82, 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} \[ \frac {3 a \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)} F_1\left (\frac {2}{3};-\frac {1}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 d \sqrt {1+i \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3564
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{\sqrt [3]{-\frac {i x}{a}} \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 \sqrt {a+i a x^3}}{-a^2+i a^2 x^3} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (3 a^3 \sqrt {a+i a \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {1+i x^3}}{-a^2+i a^2 x^3} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {1+i \tan (c+d x)}}\\ &=\frac {3 a F_1\left (\frac {2}{3};-\frac {1}{2},1;\frac {5}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d \sqrt {1+i \tan (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 16.88, size = 0, normalized size = 0.00 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{\tan \left (d x +c \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\tan \left (d x + c\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}{\sqrt [3]{\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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