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

Optimal. Leaf size=28 \[ -\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \]

[Out]

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

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Rubi [A]  time = 0.0196123, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {22, 2648} \[ -\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 22

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

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

Mathematica [A]  time = 0.037168, size = 45, normalized size = 1.61 \[ \frac{2\ 3^{-m-1} \sin \left (\frac{1}{2} (e+f x)\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.22, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+\sin \left ( fx+e \right ) \right ) ^{m} \left ( 3+3\,\sin \left ( fx+e \right ) \right ) ^{-1-m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.79773, size = 47, normalized size = 1.68 \begin{align*} -\frac{2}{{\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((1+sin(f*x+e))^m*(3+3*sin(f*x+e))^(-1-m),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.00792, size = 132, normalized size = 4.71 \begin{align*} -\frac{3^{-m - 1}{\left (\cos \left (f x + e\right ) + 1\right )} - 3^{-m - 1} \sin \left (f x + e\right )}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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