Optimal. Leaf size=77 \[ \frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(c+d x)\right ) \sin (c+d x) \left (b \sin ^n(c+d x)\right )^p}{d (1+n p) \sqrt {\cos ^2(c+d x)}} \]
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Rubi [A]
time = 0.02, 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 = {3287, 2722}
\begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3287
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*}
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Mathematica [A]
time = 0.04, size = 71, normalized size = 0.92 \begin {gather*} \frac {\sqrt {\cos ^2(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 ) \left (b \sin ^n(c+d x)\right )^p \tan (c+d x)}{d (1+n p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (b \left (\sin ^{n}\left (d x +c \right )\right )\right )^{p}\, dx\]
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 = 0.43, size = 14, normalized size = 0.18 \begin {gather*} {\rm integral}\left (\left (b \sin \left (d x + c\right )^{n}\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 (b \sin ^{n}{\left (c + d 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 (b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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