Optimal. Leaf size=69 \[ \frac {\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3223, 246, 245} \[ \frac {\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 3223
Rubi steps
\begin {align*} \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b x^n\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \left (1+\frac {b x^n}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.00 \[ \frac {\sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (\frac {b \sin ^n(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b \sin ^n(e+f x)}{a}\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right ), x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.03, size = 0, normalized size = 0.00 \[ \int \cos \left (f x +e \right ) \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.91, size = 70, normalized size = 1.01 \[ \frac {\sin \left (e+f\,x\right )\,{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n},-p;\ \frac {1}{n}+1;\ -\frac {b\,{\sin \left (e+f\,x\right )}^n}{a}\right )}{f\,{\left (\frac {b\,{\sin \left (e+f\,x\right )}^n}{a}+1\right )}^p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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