\(\int 3^{-1-m} (1+\sin (e+f x))^m \, dx\) [625]

Optimal result

Integrand size = 18, antiderivative size = 65 \[ \int 3^{-1-m} (1+\sin (e+f x))^m \, dx=-\frac {2^{\frac {1}{2}+m} 3^{-1-m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {1+\sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2730} \[ \int 3^{-1-m} (1+\sin (e+f x))^m \, dx=-\frac {2^{m+\frac {1}{2}} 3^{-m-1} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]

[In]

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

[Out]

-((2^(1/2 + m)*3^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2])/(f*Sqrt[1 +
Sin[e + f*x]]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = 3^{-1-m} \int (1+\sin (e+f x))^m \, dx \\ & = -\frac {2^{\frac {1}{2}+m} 3^{-1-m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {1+\sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82 \[ \int 3^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {2^m 3^{-1-m} B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {1}{2}+m,\frac {1}{2}\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)}{f} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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