\(\int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx\) [135]

Optimal result

Integrand size = 23, antiderivative size = 90 \[ \int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx=\frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\frac {3}{2},1+\sin (e+f x),\frac {1}{2} (1+\sin (e+f x))\right ) \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n}{f \sqrt {1-\sin (e+f x)}} \]

[Out]

2^(1/2+m)*AppellF1(1/2,-n,1/2-m,3/2,1+sin(f*x+e),1/2+1/2*sin(f*x+e))*cos(f*x+e)*(d*sin(f*x+e))^n/f/((-sin(f*x+
e))^n)/(1-sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2865, 2864, 138} \[ \int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx=\frac {2^{m+\frac {1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\frac {3}{2},\sin (e+f x)+1,\frac {1}{2} (\sin (e+f x)+1)\right )}{f \sqrt {1-\sin (e+f x)}} \]

[In]

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

[Out]

(2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1 + Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f
*x])^n)/(f*Sqrt[1 - Sin[e + f*x]]*(-Sin[e + f*x])^n)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 2864

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

Rule 2865

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(d/b)
^IntPart[n]*((d*Sin[e + f*x])^FracPart[n]/(b*Sin[e + f*x])^FracPart[n]), Int[(a + b*Sin[e + f*x])^m*(b*Sin[e +
f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && GtQ[a, 0] &&  !Gt
Q[d/b, 0]

Rubi steps \begin{align*} \text {integral}& = \left ((-\sin (e+f x))^{-n} (d \sin (e+f x))^n\right ) \int (1-\sin (e+f x))^m (-\sin (e+f x))^n \, dx \\ & = \frac {\left (\cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {(1-x)^n (2-x)^{-\frac {1}{2}+m}}{\sqrt {x}} \, dx,x,1+\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\frac {3}{2},1+\sin (e+f x),\frac {1}{2} (1+\sin (e+f x))\right ) \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n}{f \sqrt {1-\sin (e+f x)}} \\ \end{align*}

Mathematica [F]

\[ \int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx=\int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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