\(\int (d \sin (e+f x))^n (a-a \sin (e+f x)) (a+a \sin (e+f x))^m \, dx\) [13]

Optimal result

Integrand size = 34, antiderivative size = 114 \[ \int (d \sin (e+f x))^n (a-a \sin (e+f x)) (a+a \sin (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (1+n,-\frac {1}{2},\frac {1}{2}-m,2+n,\sin (e+f x),-\sin (e+f x)\right ) \sec (e+f x) (d \sin (e+f x))^{1+n} (1+\sin (e+f x))^{\frac {1}{2}-m} (a-a \sin (e+f x)) (a+a \sin (e+f x))^m}{d f (1+n) \sqrt {1-\sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3087, 140, 138} \[ \int (d \sin (e+f x))^n (a-a \sin (e+f x)) (a+a \sin (e+f x))^m \, dx=\frac {\sec (e+f x) (a-a \sin (e+f x)) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,-\frac {1}{2},\frac {1}{2}-m,n+2,\sin (e+f x),-\sin (e+f x)\right )}{d f (n+1) \sqrt {1-\sin (e+f x)}} \]

[In]

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

[Out]

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

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 140

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

Rule 3087

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((c_) + (d_.)*si
n[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[c + d*Sin[e + f*x]]/(f*Cos[e +
f*x])), Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b
, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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