Integrand size = 12, antiderivative size = 77 \[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+4 p),\frac {1}{2} (3+4 p),\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^4(a+b x)\right )^p}{b (1+4 p) \sqrt {\cos ^2(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3286, 2722} \[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\frac {\sin (a+b x) \cos (a+b x) \left (c \sin ^4(a+b x)\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (4 p+1),\frac {1}{2} (4 p+3),\sin ^2(a+b x)\right )}{b (4 p+1) \sqrt {\cos ^2(a+b x)}} \]
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Rule 2722
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \left (\sin ^{-4 p}(a+b x) \left (c \sin ^4(a+b x)\right )^p\right ) \int \sin ^{4 p}(a+b x) \, dx \\ & = \frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+4 p),\frac {1}{2} (3+4 p),\sin ^2(a+b x)\right ) \sin (a+b x) \left (c \sin ^4(a+b x)\right )^p}{b (1+4 p) \sqrt {\cos ^2(a+b x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}+2 p,\frac {3}{2}+2 p,\sin ^2(a+b x)\right ) \left (c \sin ^4(a+b x)\right )^p \tan (a+b x)}{b+4 b p} \]
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\[\int {\left (c \left (\sin ^{4}\left (b x +a \right )\right )\right )}^{p}d x\]
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\[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{4}\right )^{p} \,d x } \]
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\[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\int \left (c \sin ^{4}{\left (a + b x \right )}\right )^{p}\, dx \]
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\[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{4}\right )^{p} \,d x } \]
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\[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\int { \left (c \sin \left (b x + a\right )^{4}\right )^{p} \,d x } \]
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Timed out. \[ \int \left (c \sin ^4(a+b x)\right )^p \, dx=\int {\left (c\,{\sin \left (a+b\,x\right )}^4\right )}^p \,d x \]
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