Optimal. Leaf size=111 \[ -\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.102424, antiderivative size = 111, 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 = {3496, 3486, 3771, 2641} \[ -\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3496
Rule 3486
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{3/2}} \, dx &=-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}}-\frac{\left (5 a^2\right ) \int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx}{3 e^2}\\ &=-\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}}-\frac{\left (5 a^3\right ) \int \sqrt{e \sec (c+d x)} \, dx}{3 e^2}\\ &=-\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}}-\frac{\left (5 a^3 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 e^2}\\ &=-\frac{10 i a^3 \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 d e^2}-\frac{4 i a (a+i a \tan (c+d x))^2}{3 d (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.654838, size = 123, normalized size = 1.11 \[ -\frac{2 a^3 \sec ^2(c+d x) (\cos (c+4 d x)+i \sin (c+4 d x)) \left (3 \sin (c+d x)+7 i \cos (c+d x)+5 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)-i \sin (c+d x))\right )}{3 d (\cos (d x)+i \sin (d x))^3 (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.274, size = 175, normalized size = 1.6 \begin{align*} -{\frac{2\,{a}^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( 5\,i\cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +5\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +4\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +3\,i \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \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{3 \, d e^{2}{\rm integral}\left (\frac{5 i \, \sqrt{2} a^{3} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \, d e^{2}}, x\right ) + \sqrt{2}{\left (-4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 10 i \, a^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{3 \, d e^{2}} \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*} a^{3} \left (\int \frac{1}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int - \frac{3 \tan ^{2}{\left (c + d x \right )}}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{3 i \tan{\left (c + d x \right )}}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int - \frac{i \tan ^{3}{\left (c + d x \right )}}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]