Optimal. Leaf size=66 \[ \frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0756069, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2682, 2642, 2641} \[ \frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2682
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx &=\frac{2 e \sqrt{e \cos (c+d x)}}{a d}+\frac{e^2 \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{a}\\ &=\frac{2 e \sqrt{e \cos (c+d x)}}{a d}+\frac{\left (e^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{a \sqrt{e \cos (c+d x)}}\\ &=\frac{2 e \sqrt{e \cos (c+d x)}}{a d}+\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0747973, size = 66, normalized size = 1. \[ -\frac{2 \sqrt [4]{2} (e \cos (c+d x))^{5/2} \, _2F_1\left (\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{5 a d e (\sin (c+d x)+1)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.345, size = 110, normalized size = 1.7 \begin{align*} -2\,{\frac{{e}^{2} \left ( \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-\sin \left ( 1/2\,dx+c/2 \right ) \right ) }{a\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}d}} \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 (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{a \sin \left (d x + c\right ) + a}\,{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*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )} e \cos \left (d x + c\right )}{a \sin \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]