\(\int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx\) [629]

Optimal result

Integrand size = 27, antiderivative size = 83 \[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {2^{1+m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,\frac {3}{2},\frac {7 (1-\sin (e+f x))}{1+\sin (e+f x)}\right ) (3-4 \sin (e+f x))^{-m} (-3+4 \sin (e+f x))^m}{f (1+\sin (e+f x))} \]

[Out]

2^(1+m)*cos(f*x+e)*hypergeom([1/2, 1+m],[3/2],7*(1-sin(f*x+e))/(1+sin(f*x+e)))*(-3+4*sin(f*x+e))^m/f/((3-4*sin
(f*x+e))^m)/(1+sin(f*x+e))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2867, 134} \[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {\sqrt {\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right )}{\sqrt {7} f m (1-\sin (e+f x))} \]

[In]

Int[(3 - 4*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]

[Out]

(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (-2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[(1 - Sin[e
+ f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])^m)/(Sqrt[7]*f*m*(3 - 4*Sin[e + f*x])^m*(1 - Sin[e + f*x]))

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 {\cos (e+f x) \text {Subst}\left (\int \frac {(3-4 x)^{-1-m} (1+x)^{-\frac {1}{2}+m}}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {2 (3-4 \sin (e+f x))}{1+\sin (e+f x)}\right ) (3-4 \sin (e+f x))^{-m} \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (1+\sin (e+f x))^m}{\sqrt {7} f m (1-\sin (e+f x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.42 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.27 \[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1+m,1+2 m,2 (1+m),-\frac {8 \sqrt {7} (\cos (e+f x)+i (1+\sin (e+f x)))}{\left (-7 i+\sqrt {7}\right ) \left (3 i+\sqrt {7}-4 \cos (e+f x)-4 i \sin (e+f x)\right )}\right ) (3-4 \sin (e+f x))^{-1-m} \left (-3 i+\sqrt {7}+4 \cos (e+f x)+4 i \sin (e+f x)\right ) \left (\frac {\left (7 i+\sqrt {7}\right ) \left (-3 i+\sqrt {7}+4 \cos (e+f x)+4 i \sin (e+f x)\right )}{\left (-7 i+\sqrt {7}\right ) \left (3 i+\sqrt {7}-4 \cos (e+f x)-4 i \sin (e+f x)\right )}\right )^m (1+\sin (e+f x))^m (i \cos (e+f x)+\sin (e+f x)) (\cos (e+f x)+i (1+\sin (e+f x)))}{\left (-7 i+\sqrt {7}\right ) f (1+2 m)} \]

[In]

Integrate[(3 - 4*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]

[Out]

-((Hypergeometric2F1[1 + m, 1 + 2*m, 2*(1 + m), (-8*Sqrt[7]*(Cos[e + f*x] + I*(1 + Sin[e + f*x])))/((-7*I + Sq
rt[7])*(3*I + Sqrt[7] - 4*Cos[e + f*x] - (4*I)*Sin[e + f*x]))]*(3 - 4*Sin[e + f*x])^(-1 - m)*(-3*I + Sqrt[7] +
4*Cos[e + f*x] + (4*I)*Sin[e + f*x])*(((7*I + Sqrt[7])*(-3*I + Sqrt[7] + 4*Cos[e + f*x] + (4*I)*Sin[e + f*x])
)/((-7*I + Sqrt[7])*(3*I + Sqrt[7] - 4*Cos[e + f*x] - (4*I)*Sin[e + f*x])))^m*(1 + Sin[e + f*x])^m*(I*Cos[e +
f*x] + Sin[e + f*x])*(Cos[e + f*x] + I*(1 + Sin[e + f*x])))/((-7*I + Sqrt[7])*f*(1 + 2*m)))

Maple [F]

\[\int \left (3-4 \sin \left (f x +e \right )\right )^{-1-m} \left (\sin \left (f x +e \right )+1\right )^{m}d x\]

[In]

int((3-4*sin(f*x+e))^(-1-m)*(sin(f*x+e)+1)^m,x)

[Out]

int((3-4*sin(f*x+e))^(-1-m)*(sin(f*x+e)+1)^m,x)

Fricas [F]

\[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int { {\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]

[In]

integrate((3-4*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((sin(f*x + e) + 1)^m*(-4*sin(f*x + e) + 3)^(-m - 1), x)

Sympy [F]

\[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int \left (3 - 4 \sin {\left (e + f x \right )}\right )^{- m - 1} \left (\sin {\left (e + f x \right )} + 1\right )^{m}\, dx \]

[In]

integrate((3-4*sin(f*x+e))**(-1-m)*(1+sin(f*x+e))**m,x)

[Out]

Integral((3 - 4*sin(e + f*x))**(-m - 1)*(sin(e + f*x) + 1)**m, x)

Maxima [F]

\[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int { {\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]

[In]

integrate((3-4*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((sin(f*x + e) + 1)^m*(-4*sin(f*x + e) + 3)^(-m - 1), x)

Giac [F]

\[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int { {\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]

[In]

integrate((3-4*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((sin(f*x + e) + 1)^m*(-4*sin(f*x + e) + 3)^(-m - 1), x)

Mupad [F(-1)]

Timed out. \[ \int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int \frac {{\left (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (3-4\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \]

[In]

int((sin(e + f*x) + 1)^m/(3 - 4*sin(e + f*x))^(m + 1),x)

[Out]

int((sin(e + f*x) + 1)^m/(3 - 4*sin(e + f*x))^(m + 1), x)