Optimal. Leaf size=299 \[ -\frac {a^3 (7+2 n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p)}+\frac {a^3 (5+4 n p) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {a^3 (11+4 n p) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (2+n p);\frac {1}{2} (4+n p);\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) (3+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (a^3+a^3 \sin (e+f x)\right )}{f (3+n p)} \]
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Rubi [A]
time = 0.34, 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 = {2905, 2842,
3047, 3102, 2827, 2722} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 2827
Rule 2842
Rule 2905
Rule 3047
Rule 3102
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 = 0.90, size = 297, normalized size = 0.99 \begin {gather*} -\frac {a^3 \cos (e+f x) \sqrt {\cos ^2(e+f x)} \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (\left (24+26 n p+9 n^2 p^2+n^3 p^3\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right )+\frac {1}{2} (1+n p) \sin (e+f x) \left (6 \left (12+7 n p+n^2 p^2\right ) \, _2F_1\left (\frac {1}{2},1+\frac {n p}{2};2+\frac {n p}{2};\sin ^2(e+f x)\right )+2 (2+n p) \sin (e+f x) \left (3 (4+n p) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (3+n p);\frac {1}{2} (5+n p);\sin ^2(e+f x)\right )+(3+n p) \, _2F_1\left (\frac {1}{2},2+\frac {n p}{2};3+\frac {n p}{2};\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right )\right )}{f (1+n p) (2+n p) (3+n p) (4+n p) (-1+\sin (e+f x)) (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.53, 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 \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 = 0.00, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \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 = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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