Optimal. Leaf size=189 \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \]
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Rubi [A] time = 0.23, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2826, 2769, 2748, 2643} \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2769
Rule 2826
Rubi steps
\begin {align*} \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{a+a \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 \frac {(d \sin (e+f x))^{n p}}{a+a \sin (e+f x)} \, dx\\ &=-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}+\frac {\left (d 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} (a-a \sin (e+f x)) \, dx}{a^2}\\ &=-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}-\frac {\left (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}{a}+\frac {\left (d 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}{a}\\ &=\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (2+n p);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {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}{a f (1+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 157, normalized size = 0.83 \[ \frac {\sin (e+f x) \cos (e+f x) \sqrt {\cos ^2(e+f x)} \left ((n p+1) \sin (e+f x) \, _2F_1\left (\frac {3}{2},\frac {n p}{2}+1;\frac {n p}{2}+2;\sin ^2(e+f x)\right )-(n p+2) \, _2F_1\left (\frac {3}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) (n p+2) (\sin (e+f x)-1) (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{a +a \sin \left (f x +e \right )}\, dx \]
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 \[ \int \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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 \[ \frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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