Optimal. Leaf size=124 \[ -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {(a+b (3+2 p)) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{b f (3+2 p)} \]
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Rubi [A]
time = 0.07, 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 = {3269, 396, 252,
251} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 396
Rule 3269
Rubi steps
\begin {align*} \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=\frac {\text {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 ) \text {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 ) \text {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 = 0.15, size = 120, normalized size = 0.97 \begin {gather*} -\frac {\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} \left (-\left ((a+b (3+2 p)) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b \sin ^2(e+f x)}{a}\right )\right )+\left (a+b \sin ^2(e+f x)\right ) \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^p\right )}{b f (3+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.58, 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 \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 = 0.44, size = 27, normalized size = 0.22 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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