\(\int (3+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx\) [635]

Optimal result

Integrand size = 27, antiderivative size = 80 \[ \int (3+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {2^{-1-m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,\frac {3}{2},-\frac {3-3 \sin (e+f x)}{2 (3+3 \sin (e+f x))}\right ) (1+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2867, 134} \[ \int (3+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\sqrt {-\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,\frac {\sin (e+f x)+3}{2 (\sin (e+f x)+1)}\right )}{2 \sqrt {2} f m (1-\sin (e+f x))} \]

[In]

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

[Out]

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

Rule 134

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c
*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f*x))))^n, x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rule 2867

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dis
t[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])), Subst[Int[(a + b*x)^(m - 1/2)*((c
+ d*x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] &
& EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !IntegerQ[m]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.32 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.19 \[ \int (3+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {3^m \left (-1+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (1+m,1+2 m,2 (1+m),\frac {2 i \sqrt {2} \left (-1+\sqrt {2}\right ) (\cos (e+f x)+i (1+\sin (e+f x)))}{-3+2 \sqrt {2}+i \cos (e+f x)-\sin (e+f x)}\right ) (\cos (e+f x)-i \sin (e+f x)) (1+\sin (e+f x))^m (3+\sin (e+f x))^{-1-m} (1-i \cos (e+f x)+\sin (e+f x)) \left (3+2 \sqrt {2}-i \cos (e+f x)+\sin (e+f x)\right ) \left (\frac {1+i \left (-3+2 \sqrt {2}\right ) \cos (e+f x)+\left (3-2 \sqrt {2}\right ) \sin (e+f x)}{-3+2 \sqrt {2}+i \cos (e+f x)-\sin (e+f x)}\right )^m}{2 f (1+2 m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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