Optimal. Leaf size=163 \[ \frac {a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2826, 2748, 2643} \[ \frac {a \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2826
Rubi steps
\begin {align*} \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x)) \, dx &=\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} (a+b \sin (e+f x)) \, dx\\ &=\left (a (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx+\frac {\left (b (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{1+n p} \, dx}{d}\\ &=\frac {a \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 129, normalized size = 0.79 \[ \frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a (n p+2) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )+b (n p+1) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2}+1;\frac {n p}{2}+2;\sin ^2(e+f x)\right )\right )}{f (n p+1) (n p+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
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 \[ \int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,\left (a+b\,\sin \left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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