Optimal. Leaf size=93 \[ \frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)} \]
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Rubi [A] time = 0.117641, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2836, 77} \[ \frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 77
Rubi steps
\begin{align*} \int \cos ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^{1+m} \left (A+\frac{B x}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^3 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (A-B) (a+x)^{1+m}+(-A+3 B) (a+x)^{2+m}-\frac{B (a+x)^{3+m}}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^3 f}\\ &=\frac{2 (A-B) (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}-\frac{(A-3 B) (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}-\frac{B (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}\\ \end{align*}
Mathematica [A] time = 0.297884, size = 93, normalized size = 1. \[ \frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.612, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{3} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5578, size = 333, normalized size = 3.58 \begin{align*} -\frac{{\left ({\left (B m^{2} + 5 \, B m + 6 \, B\right )} \cos \left (f x + e\right )^{4} -{\left ({\left (A + B\right )} m^{2} + 4 \, A m\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left (A + B\right )} m -{\left ({\left ({\left (A + B\right )} m^{2} + 2 \,{\left (3 \, A + B\right )} m + 8 \, A\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left (A + B\right )} m + 16 \, A\right )} \sin \left (f x + e\right ) - 16 \, A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{3} + 9 \, f m^{2} + 26 \, f m + 24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27483, size = 618, normalized size = 6.65 \begin{align*} -\frac{\frac{{\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m - 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m + 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 6 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a\right )} A}{a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}} + \frac{{\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{4}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m^{2} - 3 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m^{2} + 2 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m^{2} + 5 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{4}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} m - 18 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m + 14 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m + 6 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{4}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 24 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a + 24 \,{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2}\right )} B}{{\left (a^{2} m^{3} + 9 \, a^{2} m^{2} + 26 \, a^{2} m + 24 \, a^{2}\right )} a}}{a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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