Optimal. Leaf size=84 \[ \frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{2}{3}}(c+d x) F_1\left (\frac{2}{3};\frac{5}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 a d \sqrt{a+i a \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.132579, antiderivative size = 84, 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} \[ \frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{2}{3}}(c+d x) F_1\left (\frac{2}{3};\frac{5}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{2 a d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3564
Rule 130
Rule 511
Rule 510
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*}
Mathematica [F] time = 10.9793, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{\tan \left ( dx+c \right ) }}} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )} e^{\left (i \, d x + i \, c\right )} + 36 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - 4 \, 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 )}\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 )} e^{\left (i \, d x + i \, c\right )}}{108 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - 6 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + 11 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} - 12 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + 8 \, a^{2} d e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{36 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - 4 \, 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 )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}} \sqrt [3]{\tan{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]