Optimal. Leaf size=75 \[ \frac{\sin (a+b x) \cos (a+b x) \left (c \sin ^3(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};\sin ^2(a+b x)\right )}{b (3 p+1) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0346214, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3207, 2643} \[ \frac{\sin (a+b x) \cos (a+b x) \left (c \sin ^3(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};\sin ^2(a+b x)\right )}{b (3 p+1) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2643
Rubi steps
\begin{align*} \int \left (c \sin ^3(a+b x)\right )^p \, dx &=\left (\sin ^{-3 p}(a+b x) \left (c \sin ^3(a+b x)\right )^p\right ) \int \sin ^{3 p}(a+b x) \, dx\\ &=\frac{\cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1+3 p);\frac{3 (1+p)}{2};\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^3(a+b x)\right )^p}{b (1+3 p) \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0840974, size = 67, normalized size = 0.89 \[ \frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \left (c \sin ^3(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};\sin ^2(a+b x)\right )}{3 b p+b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.829, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( bx+a \right ) \right ) ^{3} \right ) ^{p}\, 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 \left (c \sin \left (b x + a\right )^{3}\right )^{p}\,{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 (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin ^{3}{\left (a + b x \right )}\right )^{p}\, dx \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 )^{3}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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