Integrand size = 38, antiderivative size = 266 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a^3 (3 A-17 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \]
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)-1/1024*a^3*(3*A-17*B)*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin (f*x+e))^(1/2))/c^(11/2)/f*2^(1/2)
Result contains complex when optimal does not.
Time = 15.20 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.54 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (56370 A \cos \left (\frac {1}{2} (e+f x)\right )+38970 B \cos \left (\frac {1}{2} (e+f x)\right )-31140 A \cos \left (\frac {3}{2} (e+f x)\right )-38580 B \cos \left (\frac {3}{2} (e+f x)\right )-10404 A \cos \left (\frac {5}{2} (e+f x)\right )-12724 B \cos \left (\frac {5}{2} (e+f x)\right )+435 A \cos \left (\frac {7}{2} (e+f x)\right )+7775 B \cos \left (\frac {7}{2} (e+f x)\right )-45 A \cos \left (\frac {9}{2} (e+f x)\right )+255 B \cos \left (\frac {9}{2} (e+f x)\right )+(240+240 i) \sqrt [4]{-1} (3 A-17 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}+56370 A \sin \left (\frac {1}{2} (e+f x)\right )+38970 B \sin \left (\frac {1}{2} (e+f x)\right )+31140 A \sin \left (\frac {3}{2} (e+f x)\right )+38580 B \sin \left (\frac {3}{2} (e+f x)\right )-10404 A \sin \left (\frac {5}{2} (e+f x)\right )-12724 B \sin \left (\frac {5}{2} (e+f x)\right )-435 A \sin \left (\frac {7}{2} (e+f x)\right )-7775 B \sin \left (\frac {7}{2} (e+f x)\right )-45 A \sin \left (\frac {9}{2} (e+f x)\right )+255 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{122880 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}} \]
(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(56370*A*C os[(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*Co s[(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] + (240 + 240* I)*(-1)^(1/4)*(3*A - 17*B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x )/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10 + 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] - 10404*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))
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 |
| \(\displaystyle a^3 c^3 \left (\frac {(3 A-17 B) \left (\frac {\cos ^5(e+f x)}{4 c f (c-c \sin (e+f x))^{13/2}}-\frac {5 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )}{8 c^2}\right )}{20 c}+\frac {(A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}\right )\) |
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))
3.2.7.3.1 Defintions of rubi rules used
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])
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]
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]
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]
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]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(235)=470\).
Time = 5.05 (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 ) \left (\cos ^{4}\left (f x +e \right )\right ) \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 ) \left (\cos ^{4}\left (f x +e \right )\right )-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 ) \left (\cos ^{2}\left (f x +e \right )\right ) \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 ) \left (\cos ^{2}\left (f x +e \right )\right )+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 | \(\text {Expression too large to display}\) | \(1807\) |
1/15360*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)*arct anh(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*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)*arc tanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^6+4080*B*2^(1/2)*arctan h(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 )/c^(23/2)/(sin(f*x+e)-1)^4/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (235) = 470\).
Time = 0.29 (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=-\frac {15 \, \sqrt {2} {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 18 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 20 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 48 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} + {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 6 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 12 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) + 32 \, {\left (3 \, A - 17 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (39 \, A + 803 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 4 \, {\left (609 \, A + 389 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (449 \, A + 869 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (143 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3} + {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 80 \, {\left (3 \, A + 47 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (223 \, A + 443 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (113 \, A + 213 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{30720 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
-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))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (235) = 470\).
Time = 0.56 (sec) , antiderivative size = 990, normalized size of antiderivative = 3.72 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Too large to display} \]
-1/245760*(120*sqrt(2)*(3*A*a^3*sqrt(c) - 17*B*a^3*sqrt(c))*log(-(cos(-1/4 *pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))/(c^6*sgn (sin(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*(6*A*a^3*sqrt(c) + 6*B*a^3*sqr t(c) + 15*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 75*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 30*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 290*B* a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 120*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3 /(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 360*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 60*A*a^3 *sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1 /2*e) + 1)^4 - 900*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(c os(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4 + 822*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/ 2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5 - 4658*B*a^3* sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5/(cos(-1/4*pi + 1/2*f*x + 1/ 2*e) + 1)^5)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^5/(c^6*(cos(-1/4*pi + 1/ 2*f*x + 1/2*e) - 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + sqrt(2)*(60*A *a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 900*B*a^3*c^(49/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)...
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]