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

Optimal result

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

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.21 \[ \int (d \sin (e+f x))^m \sqrt {a+a \sin (e+f x)} \, dx=\frac {(1-i) 2^{-m} e^{\frac {1}{2} i (e+f x)} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{1+m} \left (i (-1+2 m) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3+2 m),\frac {1}{4} (3-2 m),e^{2 i (e+f x)}\right )+e^{i (e+f x)} (1+2 m) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (5+2 m),\frac {1}{4} (5-2 m),e^{2 i (e+f x)}\right )\right ) \sin ^{-m}(e+f x) (d \sin (e+f x))^m \sqrt {a (1+\sin (e+f x))}}{f \left (-1+4 m^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3255, 77, 75}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {a \sin (e+f x)+a} (d \sin (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a \sin (e+f x)+a} (d \sin (e+f x))^mdx\)

\(\Big \downarrow \) 3255

\(\displaystyle \frac {a^2 \cos (e+f x) \int \frac {(d \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 77

\(\displaystyle \frac {a^2 \cos (e+f x) \sin ^{-m}(e+f x) (d \sin (e+f x))^m \int \frac {\sin ^m(e+f x)}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {2 a \cos (e+f x) \sin ^{-m}(e+f x) (d \sin (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 75

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] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 77

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3255

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

d**m*sqrt(a)*int(sin(e + f*x)**m*sqrt(sin(e + f*x) + 1),x)