Optimal. Leaf size=104 \[ -\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}} \]
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Rubi [A] time = 0.150244, antiderivative size = 104, 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, 298, 203, 206, 2640, 2639} \[ -\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2838
Rule 2564
Rule 329
Rule 298
Rule 203
Rule 206
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \sqrt{e \sin (c+d x)} \, dx &=-\int (-a-a \cos (c+d x)) \sec (c+d x) \sqrt{e \sin (c+d x)} \, dx\\ &=a \int \sqrt{e \sin (c+d x)} \, dx+a \int \sec (c+d x) \sqrt{e \sin (c+d x)} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac{\left (a \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{\sqrt{\sin (c+d x)}}\\ &=\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{e^2}} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d e}\\ &=\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{e-x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}-\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{e+x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{d}\\ &=-\frac{a \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d}+\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.108921, size = 69, normalized size = 0.66 \[ \frac{a \sqrt{e \sin (c+d x)} \left (-2 E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )-\tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )+\tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )\right )}{d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.14, size = 198, normalized size = 1.9 \begin{align*}{\frac{a}{d}{\it Artanh} \left ({\sqrt{e\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ) \sqrt{e}}-{\frac{a}{d}\arctan \left ({\sqrt{e\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ) \sqrt{e}}-2\,{\frac{ae\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 ) }{d\cos \left ( dx+c \right ) \sqrt{e\sin \left ( dx+c \right ) }}}+{\frac{ae}{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{\left (a \sec \left (d x + c\right ) + a\right )} \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 ({\left (a \sec \left (d x + c\right ) + a\right )} \sqrt{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 \sqrt{e \sin{\left (c + d x \right )}}\, dx + \int \sqrt{e \sin{\left (c + d x \right )}} \sec{\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{\left (a \sec \left (d x + c\right ) + a\right )} \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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