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 = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2639
Rule 2640
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*}
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Mathematica [A] time = 0.10, 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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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {c \sin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 152, normalized size = 2.03 \[ -\frac {c^{3} \left (6 \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticE \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{4}\left (b x +a \right )\right )+2 \left (\sin ^{2}\left (b x +a \right )\right )\right )}{5 \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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