Optimal. Leaf size=75 \[ \frac{6 c^2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b \sqrt{\sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b} \]
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Rubi [A] time = 0.0316081, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2640, 2639} \[ \frac{6 c^2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b \sqrt{\sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (c \sin (a+b x))^{5/2} \, dx &=-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b}+\frac{1}{5} \left (3 c^2\right ) \int \sqrt{c \sin (a+b x)} \, dx\\ &=-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b}+\frac{\left (3 c^2 \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{5 \sqrt{\sin (a+b x)}}\\ &=\frac{6 c^2 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{5 b \sqrt{\sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.104112, size = 66, normalized size = 0.88 \[ -\frac{(c \sin (a+b x))^{5/2} \left (\sqrt{\sin (a+b x)} \sin (2 (a+b x))+6 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{5 b \sin ^{\frac{5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 152, normalized size = 2. \begin{align*} -{\frac{{c}^{3}}{5\,b\cos \left ( bx+a \right ) } \left ( 6\,\sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2}\sqrt{\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2}\sqrt{\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}+2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \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 (c \sin \left (b x + a\right )\right )^{\frac{5}{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 (-{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt{c \sin \left (b x + a\right )}, 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 \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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