\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx\) [107]

Optimal result
Mathematica [C] (warning: unable to verify)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [F(-2)]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 38, antiderivative size = 266 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a^3 (3 A-17 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \] Output:

-1/1024*a^3*(3*A-17*B)*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x 
+e))^(1/2))*2^(1/2)/c^(11/2)/f+1/10*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin( 
f*x+e))^(17/2)+1/80*a^3*(3*A-17*B)*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(13/2 
)-1/96*a^3*(3*A-17*B)*cos(f*x+e)^3/c/f/(c-c*sin(f*x+e))^(9/2)+1/128*a^3*(3 
*A-17*B)*cos(f*x+e)/c^3/f/(c-c*sin(f*x+e))^(5/2)-1/512*a^3*(3*A-17*B)*cos( 
f*x+e)/c^4/f/(c-c*sin(f*x+e))^(3/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 17.25 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.82 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {\left (\frac {1}{512}+\frac {i}{512}\right ) \sqrt [4]{-1} (3 A-17 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \sec \left (\frac {1}{4} (e+f x)\right ) \left (\cos \left (\frac {1}{4} (e+f x)\right )+\sin \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a+a \sin (e+f x))^3}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a+a \sin (e+f x))^3 \left (56370 A \cos \left (\frac {1}{2} (e+f x)\right )+38970 B \cos \left (\frac {1}{2} (e+f x)\right )-31140 A \cos \left (\frac {3}{2} (e+f x)\right )-38580 B \cos \left (\frac {3}{2} (e+f x)\right )-10404 A \cos \left (\frac {5}{2} (e+f x)\right )-12724 B \cos \left (\frac {5}{2} (e+f x)\right )+435 A \cos \left (\frac {7}{2} (e+f x)\right )+7775 B \cos \left (\frac {7}{2} (e+f x)\right )-45 A \cos \left (\frac {9}{2} (e+f x)\right )+255 B \cos \left (\frac {9}{2} (e+f x)\right )+56370 A \sin \left (\frac {1}{2} (e+f x)\right )+38970 B \sin \left (\frac {1}{2} (e+f x)\right )+31140 A \sin \left (\frac {3}{2} (e+f x)\right )+38580 B \sin \left (\frac {3}{2} (e+f x)\right )-10404 A \sin \left (\frac {5}{2} (e+f x)\right )-12724 B \sin \left (\frac {5}{2} (e+f x)\right )-435 A \sin \left (\frac {7}{2} (e+f x)\right )-7775 B \sin \left (\frac {7}{2} (e+f x)\right )-45 A \sin \left (\frac {9}{2} (e+f x)\right )+255 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{122880 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}} \] Input:

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x 
])^(11/2),x]
 

Output:

((1/512 + I/512)*(-1)^(1/4)*(3*A - 17*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*Sec 
[(e + f*x)/4]*(Cos[(e + f*x)/4] + Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] - S 
in[(e + f*x)/2])^11*(a + a*Sin[e + f*x])^3)/(f*(Cos[(e + f*x)/2] + Sin[(e 
+ f*x)/2])^6*(c - c*Sin[e + f*x])^(11/2)) + ((Cos[(e + f*x)/2] - Sin[(e + 
f*x)/2])*(a + a*Sin[e + f*x])^3*(56370*A*Cos[(e + f*x)/2] + 38970*B*Cos[(e 
 + f*x)/2] - 31140*A*Cos[(3*(e + f*x))/2] - 38580*B*Cos[(3*(e + f*x))/2] - 
 10404*A*Cos[(5*(e + f*x))/2] - 12724*B*Cos[(5*(e + f*x))/2] + 435*A*Cos[( 
7*(e + f*x))/2] + 7775*B*Cos[(7*(e + f*x))/2] - 45*A*Cos[(9*(e + f*x))/2] 
+ 255*B*Cos[(9*(e + f*x))/2] + 56370*A*Sin[(e + f*x)/2] + 38970*B*Sin[(e + 
 f*x)/2] + 31140*A*Sin[(3*(e + f*x))/2] + 38580*B*Sin[(3*(e + f*x))/2] - 1 
0404*A*Sin[(5*(e + f*x))/2] - 12724*B*Sin[(5*(e + f*x))/2] - 435*A*Sin[(7* 
(e + f*x))/2] - 7775*B*Sin[(7*(e + f*x))/2] - 45*A*Sin[(9*(e + f*x))/2] + 
255*B*Sin[(9*(e + f*x))/2]))/(122880*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 
])^6*(c - c*Sin[e + f*x])^(11/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 3446, 3042, 3338, 3042, 3159, 3042, 3159, 3042, 3159, 3042, 3129, 3042, 3128, 219}

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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}}dx\)

\(\Big \downarrow \) 3446

\(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}}dx\)

\(\Big \downarrow \) 3338

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{15/2}}dx}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{15/2}}dx}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3159

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}}dx}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{11/2}}dx}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3159

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}}dx}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{7/2}}dx}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3159

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3129

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 3128

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{2 c f}}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\)

Input:

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11 
/2),x]
 

Output:

a^3*c^3*(((A + B)*Cos[e + f*x]^7)/(10*f*(c - c*Sin[e + f*x])^(17/2)) + ((3 
*A - 17*B)*(Cos[e + f*x]^5/(4*c*f*(c - c*Sin[e + f*x])^(13/2)) - (5*(Cos[e 
 + f*x]^3/(3*c*f*(c - c*Sin[e + f*x])^(9/2)) - (Cos[e + f*x]/(2*c*f*(c - c 
*Sin[e + f*x])^(5/2)) - (ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - 
c*Sin[e + f*x]])]/(2*Sqrt[2]*c^(3/2)*f) + Cos[e + f*x]/(2*f*(c - c*Sin[e + 
 f*x])^(3/2)))/(4*c^2))/(2*c^2)))/(8*c^2)))/(20*c))
 

Defintions of rubi rules used

rule 219
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3128
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d 
Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], 
 x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
 

rule 3129
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c 
+ d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n 
+ 1))   Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] 
&& EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
 

rule 3159
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f 
*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 
)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; 
FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & 
& NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
 

rule 3338
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - 
 a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) 
)), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1))   Int[(g*Cos[e 
+ f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 
]) && NeQ[2*m + p + 1, 0]
 

rule 3446
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^m*c^m   Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin 
[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* 
d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] 
&& GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(235)=470\).

Time = 1.89 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.98

method result size
default \(\frac {a^{3} \left (15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )-75 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \cos \left (f x +e \right )^{4}-180 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+300 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \cos \left (f x +e \right )^{2}+240 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6} \left (3 A -17 B \right ) \sin \left (f x +e \right )-90 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+840 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}+3072 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}-3360 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}+1440 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}+510 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+5480 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}-17408 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}+19040 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}-8160 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}-720 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}+4080 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{15360 c^{\frac {23}{2}} \left (\sin \left (f x +e \right )-1\right )^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) \(526\)
parts \(\text {Expression too large to display}\) \(1807\)

Input:

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x,method=_ 
RETURNVERBOSE)
 

Output:

1/15360/c^(23/2)*a^3*(15*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2 
)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^4*sin(f*x+e)-75*2^(1/2)*arctanh(1/2*( 
c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^4-180*2^( 
1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*co 
s(f*x+e)^2*sin(f*x+e)+300*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/ 
2)/c^(1/2))*c^6*(3*A-17*B)*cos(f*x+e)^2+240*2^(1/2)*arctanh(1/2*(c+c*sin(f 
*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6*(3*A-17*B)*sin(f*x+e)-90*A*(c+c*sin(f*x+ 
e))^(9/2)*c^(3/2)+840*A*(c+c*sin(f*x+e))^(7/2)*c^(5/2)+3072*A*(c+c*sin(f*x 
+e))^(5/2)*c^(7/2)-3360*A*(c+c*sin(f*x+e))^(3/2)*c^(9/2)+1440*A*(c+c*sin(f 
*x+e))^(1/2)*c^(11/2)+510*B*(c+c*sin(f*x+e))^(9/2)*c^(3/2)+5480*B*(c+c*sin 
(f*x+e))^(7/2)*c^(5/2)-17408*B*(c+c*sin(f*x+e))^(5/2)*c^(7/2)+19040*B*(c+c 
*sin(f*x+e))^(3/2)*c^(9/2)-8160*B*(c+c*sin(f*x+e))^(1/2)*c^(11/2)-720*A*2^ 
(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6+4080*B*2^(1/ 
2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6)*(c*(1+sin(f*x+ 
e)))^(1/2)/(sin(f*x+e)-1)^4/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (235) = 470\).

Time = 0.15 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.86 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, a 
lgorithm="fricas")
 

Output:

-1/30720*(15*sqrt(2)*((3*A - 17*B)*a^3*cos(f*x + e)^6 - 5*(3*A - 17*B)*a^3 
*cos(f*x + e)^5 - 18*(3*A - 17*B)*a^3*cos(f*x + e)^4 + 20*(3*A - 17*B)*a^3 
*cos(f*x + e)^3 + 48*(3*A - 17*B)*a^3*cos(f*x + e)^2 - 16*(3*A - 17*B)*a^3 
*cos(f*x + e) - 32*(3*A - 17*B)*a^3 + ((3*A - 17*B)*a^3*cos(f*x + e)^5 + 6 
*(3*A - 17*B)*a^3*cos(f*x + e)^4 - 12*(3*A - 17*B)*a^3*cos(f*x + e)^3 - 32 
*(3*A - 17*B)*a^3*cos(f*x + e)^2 + 16*(3*A - 17*B)*a^3*cos(f*x + e) + 32*( 
3*A - 17*B)*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)* 
sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c* 
cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 
+ (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(15*(3*A - 17*B 
)*a^3*cos(f*x + e)^5 - 5*(39*A + 803*B)*a^3*cos(f*x + e)^4 + 4*(609*A + 38 
9*B)*a^3*cos(f*x + e)^3 + 12*(449*A + 869*B)*a^3*cos(f*x + e)^2 - 24*(143* 
A + 43*B)*a^3*cos(f*x + e) - 6144*(A + B)*a^3 + (15*(3*A - 17*B)*a^3*cos(f 
*x + e)^4 + 80*(3*A + 47*B)*a^3*cos(f*x + e)^3 + 12*(223*A + 443*B)*a^3*co 
s(f*x + e)^2 - 24*(113*A + 213*B)*a^3*cos(f*x + e) - 6144*(A + B)*a^3)*sin 
(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos( 
f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f* 
cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 
+ 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e) 
^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, a 
lgorithm="maxima")
 

Output:

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3/(-c*sin(f*x + e) + c 
)^(11/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \] Input:

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(11 
/2),x)
                                                                                    
                                                                                    
 

Output:

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(11 
/2), x)
 

Reduce [F]

\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx =\text {Too large to display} \] Input:

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x)
 

Output:

(sqrt(c)*a**3*(int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x)**6 - 6*sin(e + 
f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6 
*sin(e + f*x) + 1),x)*a + int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/ 
(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x 
)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x)*b + int((sqrt( - sin(e 
+ f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin 
(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 
1),x)*a + 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)* 
*6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin( 
e + f*x)**2 - 6*sin(e + f*x) + 1),x)*b + 3*int((sqrt( - sin(e + f*x) + 1)* 
sin(e + f*x)**2)/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 
 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x)*a + 3* 
int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**6 - 6*sin(e 
 + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 
- 6*sin(e + f*x) + 1),x)*b + 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x) 
)/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f 
*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x)*a + int((sqrt( - sin( 
e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin( 
e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1 
),x)*b))/c**6