Optimal. Leaf size=124 \[ \frac {(a+b (2 p+3)) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right )}{b f (2 p+3)}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3190, 388, 246, 245} \[ \frac {\left (\frac {a}{2 b p+3 b}+1\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right )}{f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 3190
Rubi steps
\begin {align*} \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (1+\frac {a}{3 b+2 b p}\right ) \operatorname {Subst}\left (\int \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (\left (1+\frac {a}{3 b+2 b p}\right ) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (1+\frac {a}{3 b+2 b p}\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{f}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 120, normalized size = 0.97 \[ -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \left (\left (a+b \sin ^2(e+f x)\right ) \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^p-(a+b (2 p+3)) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right )\right )}{b f (2 p+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \cos \left (f x + e\right )^{3}, x\right ) \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.10, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{3}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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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