Optimal. Leaf size=103 \[ \frac{2 a \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}} \]
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Rubi [A] time = 0.151885, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3872, 2838, 2564, 329, 212, 206, 203, 2642, 2641} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2838
Rule 2564
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+a \sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx &=-\int \frac{(-a-a \cos (c+d x)) \sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=a \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx+a \int \frac{\sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{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 a) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^4}{e^2}} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e}\\ &=\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{a \operatorname{Subst}\left (\int \frac{1}{e-x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{e+x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}\\ &=\frac{a \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d \sqrt{e}}+\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)}}\\ \end{align*}
Mathematica [A] time = 3.20554, size = 201, normalized size = 1.95 \[ \frac{4 a \cos \left (\frac{1}{2} (c+d x)\right ) \left (4 \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{1}{4} (c+d x)\right )}}\right ),-1\right )+\sqrt{2} \left (-\Pi \left (-1-\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{1}{4} (c+d x)\right )}}\right )\right |-1\right )+\Pi \left (1-\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{1}{4} (c+d x)\right )}}\right )\right |-1\right )+\Pi \left (-1+\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{1}{4} (c+d x)\right )}}\right )\right |-1\right )-\Pi \left (1+\sqrt{2};\left .-\sin ^{-1}\left (\frac{1}{\sqrt{\tan \left (\frac{1}{4} (c+d x)\right )}}\right )\right |-1\right )\right )\right )}{d \sqrt{\tan \left (\frac{1}{4} (c+d x)\right )} \sqrt{1-\cot ^2\left (\frac{1}{4} (c+d x)\right )} \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.998, size = 122, normalized size = 1.2 \begin{align*}{\frac{a}{d}{\it Artanh} \left ({\sqrt{e\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){\frac{1}{\sqrt{e}}}}+{\frac{a}{d}\arctan \left ({\sqrt{e\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){\frac{1}{\sqrt{e}}}}-{\frac{a}{d\cos \left ( dx+c \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},{\frac{\sqrt{2}}{2}} \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{a \sec \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 (a \sec \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*} a \left (\int \frac{1}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx + \int \frac{\sec{\left (c + d x \right )}}{\sqrt{e \sin{\left (c + d x \right )}}}\, dx\right ) \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{a \sec \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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