Optimal. Leaf size=118 \[ -\frac{2 \left (a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.138044, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2691, 2669, 2640, 2639} \[ -\frac{2 \left (a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2691
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}}-\frac{2 \int \left (\frac{a^2}{2}+b^2+\frac{3}{2} a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)} \, dx}{e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac{\left (a^2+2 b^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac{\left (\left (a^2+2 b^2\right ) \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}}-\frac{2 \left (a^2+2 b^2\right ) E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}\\ \end{align*}
Mathematica [A] time = 0.23882, size = 75, normalized size = 0.64 \[ \frac{-2 \left (a^2+b^2\right ) \cos (c+d x)+2 \left (a^2+2 b^2\right ) \sqrt{\sin (c+d x)} E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )-4 a b}{d e \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.618, size = 277, normalized size = 2.4 \begin{align*}{\frac{1}{ed\cos \left ( dx+c \right ) } \left ( \left ( -2\,{a}^{2}-2\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,ab\cos \left ( dx+c \right ) +2\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){a}^{2}+4\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2}-\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ){a}^{2}-2\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2} \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \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 (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \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*}{\rm integral}\left (-\frac{{\left (b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{e \sin \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2} - e^{2}}, 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 (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \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]