Optimal. Leaf size=75 \[ \frac {\cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+3 p);\frac {3 (1+p)}{2};\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^3(a+b x)\right )^p}{b (1+3 p) \sqrt {\cos ^2(a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {3286, 2722}
\begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3286
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*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 0.89 \begin {gather*} \frac {\sqrt {\cos ^2(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 ) \left (c \sin ^3(a+b x)\right )^p \tan (a+b x)}{b+3 b p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (c \left (\sin ^{3}\left (b x +a \right )\right )\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.39, size = 26, normalized size = 0.35 \begin {gather*} {\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 {gather*}
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 \begin {gather*} \int \left (c \sin ^{3}{\left (a + b x \right )}\right )^{p}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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