Optimal. Leaf size=169 \[ -\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.205837, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{-\frac{1}{2} a (c-d)+\frac{1}{2} a (c-d) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{c^2-d^2}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}+\frac{a \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{d}-\frac{a \int \sqrt{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\left (a \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{d (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (a \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{2 a E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{d (c+d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{d f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 6.39223, size = 938, normalized size = 5.55 \[ a \left (\frac{\sqrt{c+d \sin (e+f x)} \left (\frac{2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac{2 \csc (e) \sec (e)}{d (c+d) f}\right ) (\sin (e+f x)+1)}{\left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}-\frac{\sec (e) \left (-\frac{F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right )},-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}\right ) (\sin (e+f x)+1)}{(c+d) f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}+\frac{2 F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right )},-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d (c+d) f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2 \sqrt{\tan ^2(e)+1}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.955, size = 246, normalized size = 1.5 \begin{align*} 2\,{\frac{a}{{d}^{2} \left ( c+d \right ) \cos \left ( fx+e \right ) \sqrt{c+d\sin \left ( fx+e \right ) }f} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){c}^{2}-\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){d}^{2}+{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}-{d}^{2} \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 \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{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 \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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