3.1.6 \(\int \frac {(a+a \sin (e+f x))^m (A+C \sin ^2(e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 40, antiderivative size = 207 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {(A+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{8 a f (c-c \sin (e+f x))^{5/2}}+\frac {(A (5-2 m)-C (11+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{16 c f (c-c \sin (e+f x))^{3/2}}+\frac {\left (A \left (3-8 m+4 m^2\right )+C \left (19+24 m+4 m^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{32 c^2 f (1+2 m) \sqrt {c-c \sin (e+f x)}} \]

output

1/8*(A+C)*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)/a/f/(c-c*sin(f*x+e))^(5/2)+1/1 
6*(A*(5-2*m)-C*(11+2*m))*cos(f*x+e)*(a+a*sin(f*x+e))^m/c/f/(c-c*sin(f*x+e) 
)^(3/2)+1/32*(A*(4*m^2-8*m+3)+C*(4*m^2+24*m+19))*cos(f*x+e)*hypergeom([1, 
1/2+m],[3/2+m],1/2+1/2*sin(f*x+e))*(a+a*sin(f*x+e))^m/c^2/f/(1+2*m)/(c-c*s 
in(f*x+e))^(1/2)
 

3.1.6.2 Mathematica [A] (verified)

Time = 36.83 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.63 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\cos (e+f x) \left (4 C \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right )-4 C \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right )+(A+C) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right )\right ) (a (1+\sin (e+f x)))^m}{4 c^2 (f+2 f m) \sqrt {c-c \sin (e+f x)}} \]

input

Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f 
*x])^(5/2),x]
 

output

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

3.1.6.3 Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3515, 27, 3042, 3451, 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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+C \sin (e+f x)^2\right )}{(c-c \sin (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 3515

\(\displaystyle \frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}-\frac {\int -\frac {(\sin (e+f x) a+a)^m \left ((A (9-2 m)-C (2 m+7)) a^2+(A (1-2 m)-C (2 m+15)) \sin (e+f x) a^2\right )}{2 (c-c \sin (e+f x))^{3/2}}dx}{8 a^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left ((A (9-2 m)-C (2 m+7)) a^2+(A (1-2 m)-C (2 m+15)) \sin (e+f x) a^2\right )}{(c-c \sin (e+f x))^{3/2}}dx}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left ((A (9-2 m)-C (2 m+7)) a^2+(A (1-2 m)-C (2 m+15)) \sin (e+f x) a^2\right )}{(c-c \sin (e+f x))^{3/2}}dx}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3451

\(\displaystyle \frac {\frac {a^2 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \int \frac {(\sin (e+f x) a+a)^m}{\sqrt {c-c \sin (e+f x)}}dx}{2 c}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \int \frac {(\sin (e+f x) a+a)^m}{\sqrt {c-c \sin (e+f x)}}dx}{2 c}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3224

\(\displaystyle \frac {\frac {a^2 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \cos (e+f x) \int \sec (e+f x) (\sin (e+f x) a+a)^{m+\frac {1}{2}}dx}{2 c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\cos (e+f x)}dx}{2 c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3146

\(\displaystyle \frac {\frac {a^3 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{a-a \sin (e+f x)}d(a \sin (e+f x))}{2 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\frac {a^2 \left (A \left (4 m^2-8 m+3\right )+C \left (4 m^2+24 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (1,m+\frac {1}{2},m+\frac {3}{2},\frac {\sin (e+f x) a+a}{2 a}\right )}{2 c f (2 m+1) \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (c-c \sin (e+f x))^{3/2}}}{16 a^2 c}+\frac {(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}\)

input

Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^( 
5/2),x]
 

output

((A + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(8*a*f*(c - c*Sin[e + 
f*x])^(5/2)) + ((a^2*(A*(5 - 2*m) - C*(11 + 2*m))*Cos[e + f*x]*(a + a*Sin[ 
e + f*x])^m)/(f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*(A*(3 - 8*m + 4*m^2) + 
C*(19 + 24*m + 4*m^2))*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)/(2*c*f*(1 + 2*m)*Sqrt 
[c - c*Sin[e + f*x]]))/(16*a^2*c)
 

3.1.6.3.1 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 3451

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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( 
a*f*(2*m + 1))), x] + Simp[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)) 
  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] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 
] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] &&  !SumSimplerQ[n, 1])) && NeQ[2* 
m + 1, 0]
 

rule 3515

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(a*A + 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] - Simp[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)) - C*(c^2*m - d^2*(n + 1)) + d*(A*c*(m + n + 2) - c*C*(3*m - n))* 
Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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]))
 

3.1.6.4 Maple [F]

\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]

input

int((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(5/2),x)
 

output

int((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(5/2),x)
 

3.1.6.5 Fricas [F]

\[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(5/2),x, 
algorithm="fricas")
 

output

integral((C*cos(f*x + e)^2 - A - C)*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + 
 e) + a)^m/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*si 
n(f*x + e)), x)
 

3.1.6.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

input

integrate((a+a*sin(f*x+e))**m*(A+C*sin(f*x+e)**2)/(c-c*sin(f*x+e))**(5/2), 
x)
 

output

Timed out
 

3.1.6.7 Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(5/2),x, 
algorithm="maxima")
 

output

integrate((C*sin(f*x + e)^2 + A)*(a*sin(f*x + e) + a)^m/(-c*sin(f*x + e) + 
 c)^(5/2), x)
 

3.1.6.8 Giac [F(-2)]

Exception generated. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2)/(c-c*sin(f*x+e))^(5/2),x, 
algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error index.cc index_gcd Error: Bad 
 Argument Value
 

3.1.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (C\,{\sin \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

input

int(((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^( 
5/2),x)
 

output

int(((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^( 
5/2), x)