Integrand size = 40, antiderivative size = 247 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {a B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {a^4 B \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
1/8*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(9/2)-1/3*a *B*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c/f/(c-c*sin(f*x+e))^(7/2)+1/2*a^2*B* cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^2/f/(c-c*sin(f*x+e))^(5/2)-a^3*B*cos(f *x+e)*(a+a*sin(f*x+e))^(1/2)/c^3/f/(c-c*sin(f*x+e))^(3/2)-a^4*B*cos(f*x+e) *ln(1-sin(f*x+e))/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 13.58 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.96 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {\left (6 (A+B)-4 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+9 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-3 (A+7 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6-6 B \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{9/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]
Output:
((6*(A + B) - 4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + 9*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 - 3*(A + 7*B)*(Cos[(e + f* x)/2] - Sin[(e + f*x)/2])^6 - 6*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] *(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8)*(Cos[(e + f*x)/2] - Sin[(e + f*x )/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x) /2])^7*(c - c*Sin[e + f*x])^(9/2))
Time = 1.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {3042, 3451, 3042, 3218, 3042, 3218, 3042, 3218, 3042, 3216, 3042, 3146, 16}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}}dx}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}}dx}{c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}\right )}{c}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}-\frac {B \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}\right )}{c}\right )}{c}\right )}{c}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(9/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*f*(c - c*Sin[e + f*x] )^(9/2)) - (B*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*(c - c*Sin [e + f*x])^(7/2)) - (a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*( c - c*Sin[e + f*x])^(5/2)) - (a*((a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]) /(f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x] ])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])))/c))/c))/c
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 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] :> 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]
Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(221)=442\).
Time = 7.90 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(-\frac {A \sqrt {4}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3} a^{3} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right ) \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}}{16 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{4}}+\frac {2 B \left (\left (\left (-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-72 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3\right ) \ln \left (\frac {2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\left (\left (96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+72 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+3\right ) \ln \left (\frac {2}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (-64 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+64 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-44 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, a^{3}}{3 f \left (16 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+36 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+6 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sqrt {-\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}\, c^{4}}\) | \(659\) |
| parts | \(-\frac {A \sqrt {4}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3} a^{3} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right ) \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}}{16 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{4}}+\frac {2 B \left (\left (\left (-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-72 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3\right ) \ln \left (\frac {2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\left (\left (96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+72 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+3\right ) \ln \left (\frac {2}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (-64 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+64 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-44 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right ) \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, a^{3}}{3 f \left (16 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+36 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+6 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sqrt {-\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}\, c^{4}}\) | \(659\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x,metho d=_RETURNVERBOSE)
Output:
-1/16*A/f*4^(1/2)*(cos(1/4*Pi+1/2*f*x+1/2*e)-1)^3*a^3*(a*sin(1/4*Pi+1/2*f* x+1/2*e)^2)^(1/2)*(cos(1/4*Pi+1/2*f*x+1/2*e)+1)^3/(c*cos(1/4*Pi+1/2*f*x+1/ 2*e)^2)^(1/2)/c^4*tan(1/4*Pi+1/2*f*x+1/2*e)*sec(1/4*Pi+1/2*f*x+1/2*e)^6+2/ 3*B/f*(((-96*cos(1/2*f*x+1/2*e)^5+96*cos(1/2*f*x+1/2*e)^3+24*cos(1/2*f*x+1 /2*e))*sin(1/2*f*x+1/2*e)-48*cos(1/2*f*x+1/2*e)^8+96*cos(1/2*f*x+1/2*e)^6+ 24*cos(1/2*f*x+1/2*e)^4-72*cos(1/2*f*x+1/2*e)^2-3)*ln(2*(sin(1/2*f*x+1/2*e )-cos(1/2*f*x+1/2*e))/(cos(1/2*f*x+1/2*e)+1))+((96*cos(1/2*f*x+1/2*e)^5-96 *cos(1/2*f*x+1/2*e)^3-24*cos(1/2*f*x+1/2*e))*sin(1/2*f*x+1/2*e)+48*cos(1/2 *f*x+1/2*e)^8-96*cos(1/2*f*x+1/2*e)^6-24*cos(1/2*f*x+1/2*e)^4+72*cos(1/2*f *x+1/2*e)^2+3)*ln(2/(cos(1/2*f*x+1/2*e)+1))+cos(1/2*f*x+1/2*e)*((-64*cos(1 /2*f*x+1/2*e)^5+64*cos(1/2*f*x+1/2*e)^3-44*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+ 1/2*e)^2+24*cos(1/2*f*x+1/2*e))*sin(1/2*f*x+1/2*e)^2-3*sin(1/2*f*x+1/2*e)) )*((2*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)+1)*a)^(1/2)*a^3/(16*sin(1/2*f* x+1/2*e)*cos(1/2*f*x+1/2*e)^7-24*cos(1/2*f*x+1/2*e)^6-24*cos(1/2*f*x+1/2*e )^5*sin(1/2*f*x+1/2*e)+36*cos(1/2*f*x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^3*sin( 1/2*f*x+1/2*e)-10*cos(1/2*f*x+1/2*e)^2+6*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/ 2*e)-1)/(-(2*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)-1)*c)^(1/2)/c^4
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x , algorithm="fricas")
Output:
integral((B*a^3*cos(f*x + e)^4 - (3*A + 5*B)*a^3*cos(f*x + e)^2 + 4*(A + B )*a^3 - ((A + 3*B)*a^3*cos(f*x + e)^2 - 4*(A + B)*a^3)*sin(f*x + e))*sqrt( a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^5*cos(f*x + e)^4 - 20*c ^5*cos(f*x + e)^2 + 16*c^5 - (c^5*cos(f*x + e)^4 - 12*c^5*cos(f*x + e)^2 + 16*c^5)*sin(f*x + e)), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(9/2) ,x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x , algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(9/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x , algorithm="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
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(9/2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(9/2), x)
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x)
Output:
(sqrt(c)*sqrt(a)*a**3*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f *x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)*b - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**5 - 5 *sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x ) - 1),x)*a - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin( e + f*x)**3)/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 1 0*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)*b - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x )*a - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x) **2)/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)*b - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)*a - int(( sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f* x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*si n(e + f*x) - 1),x)*b - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)*a))/c**5