\(\int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx\) [644]

Optimal result

Integrand size = 29, antiderivative size = 45 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m}{f (1+2 m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2821} \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]

[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 2821

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

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.84 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\left (-\frac {1}{1+2 m}-\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{(1+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (-1+\sin (e+f x))^{-m} (1+\sin (e+f x))^m}{3 f} \]

[In]

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

[Out]

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

Maple [F]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (3 \, \sin \left (f x + e\right ) - 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 7.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (\frac {a\,\left (\sin \left (e+f\,x\right )+1\right )}{3}\right )}^m}{3\,f\,\left (2\,m+1\right )\,{\left (\sin \left (e+f\,x\right )-1\right )}^{m+1}} \]

[In]

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

[Out]

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