Optimal. Leaf size=77 \[ \frac{\sin (c+d x) \cos (c+d x) \left (b \sin ^n(c+d x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right )}{d (n p+1) \sqrt{\cos ^2(c+d x)}} \]
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Rubi [A] time = 0.0354642, 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 = {3208, 2643} \[ \frac{\sin (c+d x) \cos (c+d x) \left (b \sin ^n(c+d x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right )}{d (n p+1) \sqrt{\cos ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \left (b \sin ^n(c+d x)\right )^p \, dx &=\left (\sin ^{-n p}(c+d x) \left (b \sin ^n(c+d x)\right )^p\right ) \int \sin ^{n p}(c+d x) \, dx\\ &=\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1+n p);\frac{1}{2} (3+n p);\sin ^2(c+d x)\right ) \sin (c+d x) \left (b \sin ^n(c+d x)\right )^p}{d (1+n p) \sqrt{\cos ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0641188, size = 71, normalized size = 0.92 \[ \frac{\sqrt{\cos ^2(c+d x)} \tan (c+d x) \left (b \sin ^n(c+d x)\right )^p \, _2F_1\left (\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right )}{d (n p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.312, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \sin \left ( dx+c \right ) \right ) ^{n} \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 (b \sin \left (d x + c\right )^{n}\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 (b \sin \left (d x + c\right )^{n}\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 (b \sin ^{n}{\left (c + d 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 (b \sin \left (d x + c\right )^{n}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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