Optimal. Leaf size=103 \[ -\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b \sqrt{c \sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]
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Rubi [A] time = 0.0523121, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2642, 2641} \[ -\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{21 b \sqrt{c \sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (c \sin (a+b x))^{7/2} \, dx &=-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac{1}{7} \left (5 c^2\right ) \int (c \sin (a+b x))^{3/2} \, dx\\ &=-\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac{1}{21} \left (5 c^4\right ) \int \frac{1}{\sqrt{c \sin (a+b x)}} \, dx\\ &=-\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}+\frac{\left (5 c^4 \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{21 \sqrt{c \sin (a+b x)}}\\ &=\frac{10 c^4 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{21 b \sqrt{c \sin (a+b x)}}-\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}\\ \end{align*}
Mathematica [A] time = 0.153076, size = 80, normalized size = 0.78 \[ \frac{c^3 \sqrt{c \sin (a+b x)} \left (\sqrt{\sin (a+b x)} (3 \cos (3 (a+b x))-23 \cos (a+b x))-20 F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{42 b \sqrt{\sin (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 108, normalized size = 1.1 \begin{align*} -{\frac{{c}^{4}}{21\,b\cos \left ( bx+a \right ) } \left ( -6\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}+5\,\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 ) -4\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}+10\,\sin \left ( bx+a \right ) \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{7}{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^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sqrt{c \sin \left (b x + a\right )} \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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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