Optimal. Leaf size=323 \[ \frac {b \left (3 a^2 (n p+3)+b^2 (n p+2)\right ) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3) \sqrt {\cos ^2(e+f x)}}+\frac {a \left (a^2 (n p+2)+3 b^2 (n p+1)\right ) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {a b^2 (2 n p+7) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3)}-\frac {b^2 \sin (e+f x) \cos (e+f x) (a+b \sin (e+f x)) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+3)} \]
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Rubi [A] time = 0.55, antiderivative size = 303, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2826, 2793, 3023, 2748, 2643} \[ \frac {b \left (\frac {3 a^2}{n p+2}+\frac {b^2}{n p+3}\right ) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f \sqrt {\cos ^2(e+f x)}}+\frac {a \left (\frac {a^2}{n p+1}+\frac {3 b^2}{n p+2}\right ) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f \sqrt {\cos ^2(e+f x)}}-\frac {a b^2 (2 n p+7) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3)}-\frac {b^2 \sin (e+f x) \cos (e+f x) (a+b \sin (e+f x)) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+3)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2793
Rule 2826
Rule 3023
Rubi steps
\begin {align*} \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^3 \, 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))^3 \, dx\\ &=-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))}{f (3+n p)}+\frac {\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} \left (a d \left (b^2 (1+n p)+a^2 (3+n p)\right )+b d \left (b^2 (2+n p)+3 a^2 (3+n p)\right ) \sin (e+f x)+a b^2 d (7+2 n p) \sin ^2(e+f x)\right ) \, dx}{d (3+n p)}\\ &=-\frac {a b^2 (7+2 n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p)}-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))}{f (3+n p)}+\frac {\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} \left (a d^2 (3+n p) \left (3 b^2 (1+n p)+a^2 (2+n p)\right )+b d^2 (2+n p) \left (b^2 (2+n p)+3 a^2 (3+n p)\right ) \sin (e+f x)\right ) \, dx}{d^2 (2+n p) (3+n p)}\\ &=-\frac {a b^2 (7+2 n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p)}-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))}{f (3+n p)}+\frac {\left (a \left (3 b^2 (1+n p)+a^2 (2+n p)\right ) (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}{2+n p}+\frac {\left (b \left (b^2 (2+n p)+3 a^2 (3+n p)\right ) (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 (3+n p)}\\ &=-\frac {a b^2 (7+2 n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p)}+\frac {a \left (\frac {a^2}{1+n p}+\frac {3 b^2}{2+n p}\right ) \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 \sqrt {\cos ^2(e+f x)}}+\frac {b \left (\frac {3 a^2}{2+n p}+\frac {b^2}{3+n p}\right ) \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 \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))}{f (3+n p)}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 230, normalized size = 0.71 \[ \frac {\sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (\frac {b \left (3 a^2 (n p+3)+b^2 (n p+2)\right ) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2}+1;\frac {n p}{2}+2;\sin ^2(e+f x)\right )}{(n p+2) \sqrt {\cos ^2(e+f x)}}+\frac {a (n p+3) \left (a^2 (n p+2)+3 b^2 (n p+1)\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )}{(n p+1) (n p+2) \sqrt {\cos ^2(e+f x)}}-b^2 (a+b \sin (e+f x))-\frac {a b^2 (2 n p+7)}{n p+2}\right )}{f (n p+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int \left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \sin \left (f x +e \right )\right )^{3}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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