Optimal. Leaf size=77 \[ \frac{\sin (a+b x) \cos (a+b x) \left (c \sin ^4(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);\sin ^2(a+b x)\right )}{b (4 p+1) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0332862, antiderivative size = 77, 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} \[ \frac{\sin (a+b x) \cos (a+b x) \left (c \sin ^4(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);\sin ^2(a+b x)\right )}{b (4 p+1) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2643
Rubi steps
\begin{align*} \int \left (c \sin ^4(a+b x)\right )^p \, dx &=\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) \, _2F_1\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*}
Mathematica [A] time = 0.089348, size = 65, normalized size = 0.84 \[ \frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \left (c \sin ^4(a+b x)\right )^p \, _2F_1\left (\frac{1}{2},2 p+\frac{1}{2};2 p+\frac{3}{2};\sin ^2(a+b x)\right )}{4 b p+b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.917, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( bx+a \right ) \right ) ^{4} \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 )^{4}\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 (c \cos \left (b x + a\right )^{4} - 2 \, c \cos \left (b x + a\right )^{2} + c\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 ^{4}{\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 )^{4}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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