\(\int (3+3 \sin (e+f x))^m (-3+4 \sin (e+f x))^{-1-m} \, dx\) [643]

Optimal result

Integrand size = 29, antiderivative size = 76 \[ \int (3+3 \sin (e+f x))^m (-3+4 \sin (e+f x))^{-1-m} \, dx=-\frac {2^{1+m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,\frac {3}{2},\frac {7 (3-3 \sin (e+f x))}{3+3 \sin (e+f x)}\right ) (1+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m}{f} \]

[Out]

-2^(1+m)*cos(f*x+e)*hypergeom([1/2, 1+m],[3/2],7*(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 = 116, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2867, 134} \[ \int (3+3 \sin (e+f x))^m (-3+4 \sin (e+f x))^{-1-m} \, dx=-\frac {\sqrt {\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (4 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,-\frac {2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right )}{\sqrt {7} f m (1-\sin (e+f x))} \]

[In]

Int[(-3 + 4*Sin[e + f*x])^(-1 - m)*(a + a*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])]*(a + a*Sin[e + f*x])^m)/(Sqrt[7]*f*m*(1 - Sin[e + f*x])*(-3 + 4*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+4 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 {2 (3-4 \sin (e+f x))}{1+\sin (e+f x)}\right ) \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (-3+4 \sin (e+f x))^{-m} (a+a \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 = 8.35 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.58 \[ \int (3+3 \sin (e+f x))^m (-3+4 \sin (e+f x))^{-1-m} \, dx=\frac {3^m \operatorname {Hypergeometric2F1}\left (1+m,1+2 m,2 (1+m),-\frac {8 \sqrt {7} (\cos (e+f x)+i (1+\sin (e+f x)))}{\left (-7 i+\sqrt {7}\right ) \left (3 i+\sqrt {7}-4 \cos (e+f x)-4 i \sin (e+f x)\right )}\right ) (\cos (e+f x)-i \sin (e+f x)) \left (-3 i+\sqrt {7}+4 \cos (e+f x)+4 i \sin (e+f x)\right ) \left (\frac {\left (7 i+\sqrt {7}\right ) \left (-3 i+\sqrt {7}+4 \cos (e+f x)+4 i \sin (e+f x)\right )}{\left (-7 i+\sqrt {7}\right ) \left (3 i+\sqrt {7}-4 \cos (e+f x)-4 i \sin (e+f x)\right )}\right )^m (1+\sin (e+f x))^m (1-i \cos (e+f x)+\sin (e+f x)) (-3+4 \sin (e+f x))^{-1-m}}{\left (-7 i+\sqrt {7}\right ) f (1+2 m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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