Optimal. Leaf size=82 \[ \frac{3 a \tan ^{\frac{5}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left (\frac{5}{3};-\frac{1}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{5 d \sqrt{1+i \tan (c+d x)}} \]
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Rubi [A] time = 0.146763, antiderivative size = 82, normalized size of antiderivative = 1., 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} \[ \frac{3 a \tan ^{\frac{5}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left (\frac{5}{3};-\frac{1}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{5 d \sqrt{1+i \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3564
Rule 130
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i x}{a}\right )^{2/3} \sqrt{a+x}}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \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^4 \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{5}{3};-\frac{1}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \tan ^{\frac{5}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d \sqrt{1+i \tan (c+d x)}}\\ \end{align*}
Mathematica [F] time = 13.5732, size = 0, normalized size = 0. \[ \int \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.304, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \tan \left (d x + c\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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