\(\int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx\) [10]

Optimal result

Integrand size = 35, antiderivative size = 161 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {(A-B) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}}+\frac {B \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+n,2+n,\sin (e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \] Output:

-(A-B)*AppellF1(1/2,-n,1,3/2,1-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)/(a+a*sin(f*x+e))^(1/2)+B*cos(f*x+e)*hyper 
geom([1/2, 1+n],[2+n],sin(f*x+e))*(d*sin(f*x+e))^(1+n)/d/f/(1+n)/(1-sin(f* 
x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 19.34 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.55 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \sin ^n(e+f x) (d \sin (e+f x))^n \left (-\sin ^2(e+f x)\right )^{-n} \sqrt {a (1+\sin (e+f x))} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (4 (A-B) \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (-\sin (e+f x))^n \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}}-(A+B) (1+2 n) \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} \left (1-\frac {1}{1+\sin (e+f x)}\right )^n\right )}{4 a f (1+2 n) (-1+\sin (e+f x))} \] Input:

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

Output:

(Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])] 
*(4*(A - B)*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (1 
 + Sin[e + f*x])^(-1)]*(-Sin[e + f*x])^n*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin 
[e + f*x])] - (A + B)*(1 + 2*n)*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x]) 
/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 - (1 + Sin[e + f*x])^(-1 
))^n))/(4*a*f*(1 + 2*n)*(-1 + Sin[e + f*x])*(-Sin[e + f*x]^2)^n*(1 - (1 + 
Sin[e + f*x])^(-1))^n)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {3042, 3466, 3042, 3255, 77, 75, 3266, 3042, 3265, 3042, 3264, 148, 333}

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.

\(\displaystyle \int \frac {(A+B \sin (e+f x)) (d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(A+B \sin (e+f x)) (d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}}dx\)

\(\Big \downarrow \) 3466

\(\displaystyle (A-B) \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x) a+a}}dx+\frac {B \int (d \sin (e+f x))^n \sqrt {\sin (e+f x) a+a}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle (A-B) \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x) a+a}}dx+\frac {B \int (d \sin (e+f x))^n \sqrt {\sin (e+f x) a+a}dx}{a}\)

\(\Big \downarrow \) 3255

\(\displaystyle (A-B) \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x) a+a}}dx+\frac {a B \cos (e+f x) \int \frac {(d \sin (e+f x))^n}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 77

\(\displaystyle (A-B) \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x) a+a}}dx+\frac {a B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin ^n(e+f x)}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\)

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 3266

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3265

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3264

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

\(\Big \downarrow \) 148

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

\(\Big \downarrow \) 333

\(\displaystyle -\frac {(A-B) \cos (e+f x) \sin ^{-n}(e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{f \sqrt {a \sin (e+f x)+a}}-\frac {2 B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}}\)

Input:

Int[((d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]],x]
 

Output:

-(((A - B)*AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/ 
2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f 
*x]])) - (2*B*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*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 148

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), 
x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1) - 1)*(c 
 + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, 
 d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[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]
 

rule 3264

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-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 3265

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[(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] &&  !I 
ntegerQ[m] && GtQ[a, 0] &&  !GtQ[d/b, 0]
 

rule 3266

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Sin[e + f*x])^FracPart[m 
]/(1 + (b/a)*Sin[e + f*x])^FracPart[m])   Int[(1 + (b/a)*Sin[e + f*x])^m*(d 
*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]
 

rule 3466

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**n*sqrt(a)*(int((sin(e + f*x)**n*sqrt(sin(e + f*x) + 1)*sin(e + f*x))/( 
sin(e + f*x) + 1),x)*b + int((sin(e + f*x)**n*sqrt(sin(e + f*x) + 1))/(sin 
(e + f*x) + 1),x)*a))/a