\(\int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx\) [121]

Optimal result

Integrand size = 23, antiderivative size = 46 \[ \int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx=-\frac {2 a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2855, 67} \[ \int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx=-\frac {2 a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}} \]

[In]

Int[Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

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

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 2855

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

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 3.98 \[ \int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx=\frac {2^{1-n} e^{i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{1+n} \left (i (-1+2 n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 n),\frac {1}{4} (3-2 n),e^{2 i (e+f x)}\right )+e^{i (e+f x)} (1+2 n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (5+2 n),\frac {1}{4} (5-2 n),e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{\left (i+e^{i (e+f x)}\right ) f \left (-1+4 n^2\right )} \]

[In]

Integrate[Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(2^(1 - n)*E^(I*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hypergeom
etric2F1[1, (3 + 2*n)/4, (3 - 2*n)/4, E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(1 + 2*n)*Hypergeometric2F1[1, (5
+ 2*n)/4, (5 - 2*n)/4, E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/((I + E^(I*(e + f*x)))*f*(-1 + 4*n^2
))

Maple [F]

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

[In]

int(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x)

[Out]

int(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x)

Fricas [F]

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

[In]

integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sin(f*x + e) + a)*sin(f*x + e)^n, x)

Sympy [F]

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

[In]

integrate(sin(f*x+e)**n*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*sin(e + f*x)**n, x)

Maxima [F]

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

[In]

integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*sin(f*x + e)^n, x)

Giac [F]

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

[In]

integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*sin(f*x + e)^n, x)

Mupad [F(-1)]

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

[In]

int(sin(e + f*x)^n*(a + a*sin(e + f*x))^(1/2),x)

[Out]

int(sin(e + f*x)^n*(a + a*sin(e + f*x))^(1/2), x)