\(\int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx\) [622]

Optimal result

Integrand size = 27, antiderivative size = 28 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=-\frac {3^{-1-m} \cos (e+f x)}{f (1+\sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {22, 2727} \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=-\frac {3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \]

[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 2727

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*} \text {integral}& = 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] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=\frac {2\ 3^{-1-m} \sin \left (\frac {1}{2} (e+f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[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]))

Maple [F]

\[\int \left (\sin \left (f x +e \right )+1\right )^{m} \left (3+3 \sin \left (f x +e \right )\right )^{-1-m}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=-\frac {3^{-m - 1} {\left (\cos \left (f x + e\right ) + 1\right )} - 3^{-m - 1} \sin \left (f x + e\right )}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f} \]

[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)

Sympy [F]

\[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=3^{- m - 1} \int \left (\sin {\left (e + f x \right )} + 1\right )^{m} \left (\sin {\left (e + f x \right )} + 1\right )^{- m - 1}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=-\frac {2}{{\left (3^{m + 1} + \frac {3^{m + 1} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=\frac {3^{-m - 1} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3^{-m - 1}}{f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx=\frac {\frac {1}{3^{m+1}}\,\left (-\cos \left (e+f\,x\right )+\sin \left (e+f\,x\right )\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{f\,\left (\sin \left (e+f\,x\right )+1\right )} \]

[In]

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

[Out]

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