\(\int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx\) [129]

Optimal result

Integrand size = 25, antiderivative size = 66 \[ \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx=-\frac {2 a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2855, 69, 67} \[ \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx=-\frac {2 a \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}} \]

[In]

Int[(d*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]]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqr
t[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 69

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-b)*(c/d))^IntPart[m]*((b*x)^FracPart[m]/(
(-d)*(x/c))^FracPart[m]), Int[((-d)*(x/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !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 {(d 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 {\left (a^2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\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 ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.21 \[ \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx=\frac {(1-i) 2^{-n} e^{\frac {1}{2} i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{1+n} \left (i (-1+2 n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 n),\frac {1}{4} (3-2 n),e^{2 i (e+f x)}\right )+e^{i (e+f x)} (1+2 n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (5+2 n),\frac {1}{4} (5-2 n),e^{2 i (e+f x)}\right )\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt {a (1+\sin (e+f x))}}{f \left (-1+4 n^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

((1 - I)*E^((I/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^(1 + n)*(I*(-1 + 2*n)*Hyperge
ometric2F1[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))])*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])])/(2^n*f*(-1 + 4*
n^2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]^n)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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