Optimal. Leaf size=156 \[ -\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{4 \sqrt{d} f} \]
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Rubi [A] time = 0.290306, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2770, 2775, 205} \[ -\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{4 \sqrt{d} f} \]
Antiderivative was successfully verified.
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Rule 2770
Rule 2775
Rule 205
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}+\frac{1}{4} (3 (c+d)) \int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx\\ &=-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}+\frac{1}{8} \left (3 (c+d)^2\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx\\ &=-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{\left (3 a (c+d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{4 f}\\ &=-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{4 \sqrt{d} f}-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 2.07325, size = 365, normalized size = 2.34 \[ \frac{\sqrt{a (\sin (e+f x)+1)} \left (-2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x)) (5 c+2 d \sin (e+f x)+3 d)-\frac{3 i (c+d)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-i \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\log \left (\frac{e^{-i e} \left (2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+2 \sqrt [4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right )}{\sqrt{d}}\right )-\log \left (\frac{2 f e^{\frac{1}{2} i (e-2 f x)} \left (i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt [4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right )}{\sqrt{d}}\right )\right ) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d}}\right )}{8 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a\sin \left ( fx+e \right ) } \left ( c+d\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \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 \sqrt{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 [B] time = 5.21785, size = 2600, normalized size = 16.67 \begin{align*} \text{result too large to display} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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