3.644 \(\int (-3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx\)

Optimal. Leaf size=45 \[ \frac{\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]

[Out]

(Cos[e + f*x]*(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))

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Rubi [A]  time = 0.0630852, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2742} \[ \frac{\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]

[Out]

(Cos[e + f*x]*(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))

Rule 2742

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f
, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int (-3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\frac{\cos (e+f x) (-3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f (1+2 m)}\\ \end{align*}

Mathematica [B]  time = 0.668599, size = 110, normalized size = 2.44 \[ \frac{2^{-m} 3^{-m-1} \sin \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (\sin (e+f x)-1)^{-m-1} \cos ^{-2 m-1}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{2 (m+1)}}{2 f m+f} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]

[Out]

(3^(-1 - m)*Cos[(2*e + Pi + 2*f*x)/4]^(-1 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(1 + m))*(-1 + Sin[e
+ f*x])^(-1 - m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4])/(2^m*(f + 2*f*m))

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Maple [F]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int \left ( -3+3\,\sin \left ( fx+e \right ) \right ) ^{-1-m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x)

[Out]

int((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (3 \, \sin \left (f x + e\right ) - 3\right )}^{-m - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^m*(3*sin(f*x + e) - 3)^(-m - 1), x)

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Fricas [A]  time = 1.5001, size = 107, normalized size = 2.38 \begin{align*} \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (3 \, \sin \left (f x + e\right ) - 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

(a*sin(f*x + e) + a)^m*(3*sin(f*x + e) - 3)^(-m - 1)*cos(f*x + e)/(2*f*m + f)

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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.

[In]

integrate((-3+3*sin(f*x+e))**(-1-m)*(a+a*sin(f*x+e))**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (3 \, \sin \left (f x + e\right ) - 3\right )}^{-m - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^m*(3*sin(f*x + e) - 3)^(-m - 1), x)