3.1.22 \(\int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x)+C \sin ^2(e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 48, antiderivative size = 216 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B+2 A m+2 B m+C (9+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt {c-c \sin (e+f x)}}+\frac {(A (1-2 m)-B (3+2 m)-C (7+2 m)) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt {c-c \sin (e+f x)}} \]

output

1/4*(A+B+C)*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)/a/f/(c-c*sin(f*x+e))^(3/2)+1 
/4*(A+B+2*A*m+2*B*m+C*(9+2*m))*cos(f*x+e)*(a+a*sin(f*x+e))^m/c/f/(1+2*m)/( 
c-c*sin(f*x+e))^(1/2)+1/4*(A*(1-2*m)-B*(3+2*m)-C*(7+2*m))*cos(f*x+e)*hyper 
geom([1, 1/2+m],[3/2+m],1/2+1/2*sin(f*x+e))*(a+a*sin(f*x+e))^m/c/f/(1+2*m) 
/(c-c*sin(f*x+e))^(1/2)
 

3.1.22.2 Mathematica [A] (verified)

Time = 34.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.75 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\cos (e+f x) (a (1+\sin (e+f x)))^m \left ((8 C+B (3+2 m)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (-1+\sin (e+f x))+2 \left (B+4 C+2 B m-(A+C) \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (-1+\sin (e+f x))-4 C \sin (e+f x)\right )\right )}{4 c f (1+2 m) (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \]

input

Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)) 
/(c - c*Sin[e + f*x])^(3/2),x]
 

output

-1/4*(Cos[e + f*x]*(a*(1 + Sin[e + f*x]))^m*((8*C + B*(3 + 2*m))*Hypergeom 
etric2F1[1, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(-1 + Sin[e + f*x]) + 
2*(B + 4*C + 2*B*m - (A + C)*Hypergeometric2F1[2, 1/2 + m, 3/2 + m, (1 + S 
in[e + f*x])/2]*(-1 + Sin[e + f*x]) - 4*C*Sin[e + f*x])))/(c*f*(1 + 2*m)*( 
-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])
 

3.1.22.3 Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 3514, 27, 3042, 3452, 3042, 3224, 3042, 3146, 78}

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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin (e+f x)^2\right )}{(c-c \sin (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 3514

\(\displaystyle \frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\int -\frac {(\sin (e+f x) a+a)^m \left (a^2 (A (3-2 m)-(B+C) (2 m+5))-a^2 (2 m A+A+B+2 B m+C (2 m+9)) \sin (e+f x)\right )}{2 \sqrt {c-c \sin (e+f x)}}dx}{4 a^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left (a^2 (A (3-2 m)-(B+C) (2 m+5))-a^2 (2 m A+A+B+2 B m+C (2 m+9)) \sin (e+f x)\right )}{\sqrt {c-c \sin (e+f x)}}dx}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left (a^2 (A (3-2 m)-(B+C) (2 m+5))-a^2 (2 m A+A+B+2 B m+C (2 m+9)) \sin (e+f x)\right )}{\sqrt {c-c \sin (e+f x)}}dx}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3452

\(\displaystyle \frac {2 a^2 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \int \frac {(\sin (e+f x) a+a)^m}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \int \frac {(\sin (e+f x) a+a)^m}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3224

\(\displaystyle \frac {\frac {2 a^2 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) \int \sec (e+f x) (\sin (e+f x) a+a)^{m+\frac {1}{2}}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 a^2 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3146

\(\displaystyle \frac {\frac {2 a^3 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{a-a \sin (e+f x)}d(a \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\frac {2 a^2 (A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (1,m+\frac {1}{2},m+\frac {3}{2},\frac {\sin (e+f x) a+a}{2 a}\right )}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}}}{8 a^2 c}+\frac {(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}\)

input

Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c - 
c*Sin[e + f*x])^(3/2),x]
 

output

((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(c - c*Sin[ 
e + f*x])^(3/2)) + ((2*a^2*(A + B + 2*A*m + 2*B*m + C*(9 + 2*m))*Cos[e + f 
*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*a^ 
2*(A*(1 - 2*m) - B*(3 + 2*m) - C*(7 + 2*m))*Cos[e + f*x]*Hypergeometric2F1 
[1, 1/2 + m, 3/2 + m, (a + a*Sin[e + f*x])/(2*a)]*(a + a*Sin[e + f*x])^m)/ 
(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]))/(8*a^2*c)
 

3.1.22.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 78

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

rule 3042

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

rule 3146

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x 
)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I 
ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/ 
2])
 

rule 3224

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPart[m]*c^IntPart[m]*(a + b*Sin[e 
+ f*x])^FracPart[m]*((c + d*Sin[e + f*x])^FracPart[m]/Cos[e + f*x]^(2*FracP 
art[m]))   Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; F 
reeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] 
 && (FractionQ[m] ||  !FractionQ[n])
 

rule 3452

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

rule 3514

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*A - b*B + a*C)*Cos[e + f*x]*(a + b* 
Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(2*b*c*f*(2*m + 1))), x] - Si 
mp[1/(2*b*c*d*(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + 
f*x])^n*Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - B*c*d*(m - n - 1) - C*(c 
^2*m - d^2*(n + 1)) + d*((A*c + B*d)*(m + n + 2) - c*C*(3*m - n))*Sin[e + f 
*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + 
 a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -2^(-1)] || (EqQ[m + n + 2, 0] && 
NeQ[2*m + 1, 0]))
 

3.1.22.4 Maple [F]

\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

input

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(3 
/2),x)
 

output

int((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(3 
/2),x)
 

3.1.22.5 Fricas [F]

\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c-c*sin(f*x+ 
e))^(3/2),x, algorithm="fricas")
 

output

integral((C*cos(f*x + e)^2 - B*sin(f*x + e) - A - C)*sqrt(-c*sin(f*x + e) 
+ c)*(a*sin(f*x + e) + a)^m/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c 
^2), x)
 

3.1.22.6 Sympy [F]

\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )} + C \sin ^{2}{\left (e + f x \right )}\right )}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]

input

integrate((a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)+C*sin(f*x+e)**2)/(c-c*sin(f* 
x+e))**(3/2),x)
 

output

Integral((a*(sin(e + f*x) + 1))**m*(A + B*sin(e + f*x) + C*sin(e + f*x)**2 
)/(-c*(sin(e + f*x) - 1))**(3/2), x)
 

3.1.22.7 Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c-c*sin(f*x+ 
e))^(3/2),x, algorithm="maxima")
 

output

integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m/( 
-c*sin(f*x + e) + c)^(3/2), x)
 

3.1.22.8 Giac [F(-2)]

Exception generated. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c-c*sin(f*x+ 
e))^(3/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,1,1,0,0,0,0,0,0]%%%}+%%%{1,[0,0,1,1,1,0,0,0,0,0]%%% 
}+%%%{1,[
 

3.1.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (C\,{\sin \left (e+f\,x\right )}^2+B\,\sin \left (e+f\,x\right )+A\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

input

int(((a + a*sin(e + f*x))^m*(A + B*sin(e + f*x) + C*sin(e + f*x)^2))/(c - 
c*sin(e + f*x))^(3/2),x)
 

output

int(((a + a*sin(e + f*x))^m*(A + B*sin(e + f*x) + C*sin(e + f*x)^2))/(c - 
c*sin(e + f*x))^(3/2), x)