Optimal. Leaf size=80 \[ \frac{3 a \sqrt{1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)} F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.116587, antiderivative size = 80, 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, 430, 429} \[ \frac{3 a \sqrt{1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)} F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3564
Rule 130
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{i x}{a}\right )^{2/3} \sqrt{a+x} \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{1}{\sqrt{a+i a x^3} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d}\\ &=-\frac{\left (3 a^3 \sqrt{1+i \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+i x^3} \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 a F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt{1+i \tan (c+d x)} \sqrt [3]{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [F] time = 0.934482, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+ia\tan \left ( dx+c \right ) } \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, 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 \frac{\sqrt{i \, a \tan \left (d x + c\right ) + a}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )}}{\tan ^{\frac{2}{3}}{\left (c + d x \right )}}\, dx \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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