Optimal. Leaf size=114 \[ \frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d \sqrt{e \sin (c+d x)}}+\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129492, antiderivative size = 114, 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 = {2692, 2669, 2642, 2641} \[ \frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d \sqrt{e \sin (c+d x)}}+\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2692
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx &=\frac{2 b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2}{3} \int \frac{\frac{3 a^2}{2}+b^2+\frac{5}{2} a b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 d e}+\frac{1}{3} \left (3 a^2+2 b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 d e}+\frac{\left (\left (3 a^2+2 b^2\right ) \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 \left (3 a^2+2 b^2\right ) F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d \sqrt{e \sin (c+d x)}}+\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 d e}\\ \end{align*}
Mathematica [A] time = 0.379236, size = 79, normalized size = 0.69 \[ \frac{2 b \sin (c+d x) (6 a+b \cos (c+d x))-2 \left (3 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )}{3 d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.77, size = 170, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,d\cos \left ( dx+c \right ) } \left ( 3\,\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 ){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}-2\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-12\,ab\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \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}}{\sqrt{e \sin \left (d x + c\right )}}\,{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 \sin \left (d x + c\right )}, x\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{\left (a + b \cos{\left (c + d x \right )}\right )^{2}}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx \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}}{\sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]