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

Optimal. Leaf size=39 \[ -\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f} \]

[Out]

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

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Rubi [A]  time = 0.0170718, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {23, 2648} \[ -\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f} \]

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)

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 2648

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

Rubi steps

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

Mathematica [B]  time = 5.03509, size = 104, normalized size = 2.67 \[ -\frac{2^{-m} 3^{-m-1} \cos \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (\sin (e+f x)+1)^{-m-1} \sin ^{-2 m-1}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^{2 (m+1)}}{f} \]

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]*(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]^(-1 - 2*m))/(2^m*f))

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Maple [F]  time = 0.254, 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 [A]  time = 1.75836, size = 51, normalized size = 1.31 \begin{align*} -\frac{2 \, a^{m}}{{\left (3^{m + 1} + \frac{3^{m + 1} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} 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="maxima")

[Out]

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

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Fricas [A]  time = 1.0135, size = 119, normalized size = 3.05 \begin{align*} -\frac{\left (\frac{1}{3} \, a\right )^{m}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{3 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \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]

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