Optimal. Leaf size=75 \[ \frac {2 c^2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b} \]
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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, 2642, 2641} \[ \frac {2 c^2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int (c \sin (a+b x))^{3/2} \, dx &=-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}+\frac {1}{3} c^2 \int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx\\ &=-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}+\frac {\left (c^2 \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{3 \sqrt {c \sin (a+b x)}}\\ &=\frac {2 c^2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b \sqrt {c \sin (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \sin (a+b x)}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.83 \[ -\frac {2 (c \sin (a+b x))^{3/2} \left (F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )+\sqrt {\sin (a+b x)} \cos (a+b x)\right )}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \sin \left (b x + a\right )} c \sin \left (b x + a\right ), x\right ) \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 97, normalized size = 1.29 \[ -\frac {c^{2} \left (\sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{3}\left (b x +a \right )\right )+2 \sin \left (b x +a \right )\right )}{3 \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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