Optimal. Leaf size=77 \[ \frac {\sin (a+b x) \cos (a+b x) \left (c \sin ^2(a+b x)\right )^p \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2 p+1);\frac {1}{2} (2 p+3);\sin ^2(a+b x)\right )}{b (2 p+1) \sqrt {\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, 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 = {3207, 2643} \[ \frac {\sin (a+b x) \cos (a+b x) \left (c \sin ^2(a+b x)\right )^p \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2 p+1);\frac {1}{2} (2 p+3);\sin ^2(a+b x)\right )}{b (2 p+1) \sqrt {\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3207
Rubi steps
\begin {align*} \int \left (c \sin ^2(a+b x)\right )^p \, dx &=\left (\sin ^{-2 p}(a+b x) \left (c \sin ^2(a+b x)\right )^p\right ) \int \sin ^{2 p}(a+b x) \, dx\\ &=\frac {\cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+2 p);\frac {1}{2} (3+2 p);\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^2(a+b x)\right )^p}{b (1+2 p) \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 0.79 \[ \frac {\sqrt {\cos ^2(a+b x)} \tan (a+b x) \left (c \sin ^2(a+b x)\right )^p \, _2F_1\left (\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\sin ^2(a+b x)\right )}{2 b p+b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-c \cos \left (b x + a\right )^{2} + c\right )}^{p}, x\right ) \]
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 \[ \int \left (c \sin \left (b x + a\right )^{2}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.84, size = 0, normalized size = 0.00 \[ \int \left (c \left (\sin ^{2}\left (b x +a \right )\right )\right )^{p}\, dx \]
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 \[ \int \left (c \sin \left (b x + a\right )^{2}\right )^{p}\,{d x} \]
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 \[ \int {\left (c\,{\sin \left (a+b\,x\right )}^2\right )}^p \,d x \]
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 \[ \int \left (c \sin ^{2}{\left (a + b x \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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