Optimal. Leaf size=66 \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e} \]
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Rubi [A] time = 0.0509949, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2669, 2642, 2641} \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx &=\frac{2 b \sqrt{e \sin (c+d x)}}{d e}+a \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{2 b \sqrt{e \sin (c+d x)}}{d e}+\frac{\left (a \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{\sqrt{e \sin (c+d x)}}\\ &=\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e}\\ \end{align*}
Mathematica [A] time = 0.201985, size = 54, normalized size = 0.82 \[ \frac{2 \left (b \sin (c+d x)-a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.162, size = 92, normalized size = 1.4 \begin{align*} -{\frac{1}{d\cos \left ( dx+c \right ) } \left ( a\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 ) -2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) b \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \cos \left (d x + c\right ) + a}{\sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \cos \left (d x + c\right ) + a\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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \cos{\left (c + d x \right )}}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \cos \left (d x + c\right ) + a}{\sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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