Optimal. Leaf size=299 \[ \frac {a^3 (4 n p+11) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 (4 n p+5) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {a^3 (2 n p+7) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3)}-\frac {\sin (e+f x) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+3)} \]
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Rubi [A] time = 0.50, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2826, 2763, 2968, 3023, 2748, 2643} \[ \frac {a^3 (4 n p+11) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+2);\frac {1}{2} (n p+4);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 (4 n p+5) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {a^3 (2 n p+7) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+2) (n p+3)}-\frac {\sin (e+f x) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+3)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2763
Rule 2826
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \left (c (d \sin (e+f x))^p\right )^n (a+a \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+a \sin (e+f x))^3 \, dx\\ &=-\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{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} (a+a \sin (e+f x)) \left (2 a^2 d (2+n p)+a^2 d (7+2 n p) \sin (e+f x)\right ) \, dx}{d (3+n p)}\\ &=-\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{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 (2 a^3 d (2+n p)+\left (2 a^3 d (2+n p)+a^3 d (7+2 n p)\right ) \sin (e+f x)+a^3 d (7+2 n p) \sin ^2(e+f x)\right ) \, dx}{d (3+n p)}\\ &=-\frac {a^3 (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 {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{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^3 d^2 (3+n p) (5+4 n p)+a^3 d^2 (2+n p) (11+4 n p) \sin (e+f x)\right ) \, dx}{d^2 (2+n p) (3+n p)}\\ &=-\frac {a^3 (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 {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{f (3+n p)}+\frac {\left (a^3 (5+4 n p) (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 (a^3 (11+4 n p) (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^3 (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^3 (5+4 n p) \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) (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 (11+4 n p) \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) (3+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{f (3+n p)}\\ \end {align*}
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Mathematica [A] time = 1.37, size = 297, normalized size = 0.99 \[ -\frac {a^3 \sin (e+f x) \cos (e+f x) \sqrt {\cos ^2(e+f x)} \left (\frac {1}{2} (n p+1) \sin (e+f x) \left (6 \left (n^2 p^2+7 n p+12\right ) \, _2F_1\left (\frac {1}{2},\frac {n p}{2}+1;\frac {n p}{2}+2;\sin ^2(e+f x)\right )+2 (n p+2) \sin (e+f x) \left (3 (n p+4) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+3);\frac {1}{2} (n p+5);\sin ^2(e+f x)\right )+(n p+3) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2}+2;\frac {n p}{2}+3;\sin ^2(e+f x)\right )\right )\right )+\left (n^3 p^3+9 n^2 p^2+26 n p+24\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )\right ) \left (c (d \sin (e+f x))^p\right )^n}{f (n p+1) (n p+2) (n p+3) (n p+4) (\sin (e+f x)-1) (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}, x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int \left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{3}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \]
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 \[ \int {\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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 \[ a^{3} \left (\int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}\, dx + \int 3 \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \sin {\left (e + f x \right )}\, dx + \int 3 \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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