Optimal. Leaf size=95 \[ -\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}} \]
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Rubi [A] time = 0.207713, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3872, 2839, 2564, 30, 2567, 2640, 2639} \[ -\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2567
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sqrt{e \sin (c+d x)}}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sqrt{e \sin (c+d x)}}{-a-a \cos (c+d x)} \, dx\\ &=\frac{e^2 \int \frac{\cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a}-\frac{e^2 \int \frac{\cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a}\\ &=\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{2 \int \sqrt{e \sin (c+d x)} \, dx}{a}+\frac{e \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,e \sin (c+d x)\right )}{a d}\\ &=-\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{\left (2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{a \sqrt{\sin (c+d x)}}\\ &=-\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.587984, size = 249, normalized size = 2.62 \[ \frac{2 \left (12 e^{2 i c} \sqrt{1-e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},e^{2 i (c+d x)}\right )+4 e^{2 i (c+d x)} \sqrt{1-e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 i (c+d x)}\right )+6 e^{i (c+d x)}-9 e^{2 i (c+d x)}+3 e^{2 i (2 c+d x)}+6 e^{i (3 c+d x)}-9 e^{2 i c}+3\right ) \sqrt{e \sin (c+d x)}}{3 a \left (1+i e^{i c}\right ) \left (e^{i c}+i\right ) d \left (-1+e^{i (c+d x)}\right ) \left (1+e^{i (c+d x)}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.421, size = 149, normalized size = 1.6 \begin{align*} -2\,{\frac{e \left ( 2\,\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) - \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) \right ) }{a\cos \left ( dx+c \right ) \sqrt{e\sin \left ( dx+c \right ) }d}} \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{\sqrt{e \sin \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}\,{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{\sqrt{e \sin \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}, 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*} \frac{\int \frac{\sqrt{e \sin{\left (c + d x \right )}}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \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{\sqrt{e \sin \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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