Integrand size = 40, antiderivative size = 239 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {8 a^4 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \] Output:
-8*a^4*(A+B)*cos(f*x+e)*ln(1-sin(f*x+e))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin (f*x+e))^(1/2)-4*a^3*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f/(c-c*sin(f* x+e))^(1/2)-a^2*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f/(c-c*sin(f*x+e)) ^(1/2)-1/3*a*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c*sin(f*x+e))^(1 /2)-1/4*B*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(1/2)
Time = 13.90 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.77 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \left (-12 (8 A+15 B) \cos (2 (e+f x))+3 B \cos (4 (e+f x))+1536 (A+B) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+24 (29 A+36 B) \sin (e+f x)-8 (A+4 B) \sin (3 (e+f x))\right )}{96 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin [e + f*x]],x]
Output:
-1/96*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt [a*(1 + Sin[e + f*x])]*(-12*(8*A + 15*B)*Cos[2*(e + f*x)] + 3*B*Cos[4*(e + f*x)] + 1536*(A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 24*(29*A + 36*B)*Sin[e + f*x] - 8*(A + 4*B)*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])
Time = 1.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {3042, 3452, 3042, 3219, 3042, 3219, 3042, 3219, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle (A+B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A+B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle (A+B) \left (2 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A+B) \left (2 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (-\frac {2 a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle (A+B) \left (2 a \left (2 a \left (-\frac {2 a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\right )-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f *x]],x]
Output:
-1/4*(B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(f*Sqrt[c - c*Sin[e + f*x ]]) + (A + B)*(-1/3*(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(f*Sqrt[c - c*Sin[e + f*x]]) + 2*a*(-1/2*(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)) /(f*Sqrt[c - c*Sin[e + f*x]]) + 2*a*((-2*a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]))))
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[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( 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[(-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]
Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(215)=430\).
Time = 7.50 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.21
| method | result | size |
| default | \(\frac {2 A \sqrt {4}\, \left (2 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}-9 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+18 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+12 \ln \left (\frac {2}{\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1}\right )-12 \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )-1\right )-12 \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1\right )-11\right ) \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, a^{3} \cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )}{3 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}}-\frac {2 B \,a^{3} \left (\left (-24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \ln \left (-\frac {2 \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+\left (24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\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 (-6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+22 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-24\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \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}}{3 f \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\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 ) c}}\) | \(528\) |
| parts | \(-\frac {2 A \sqrt {4}\, \left (-2 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+9 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}-18 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+12 \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )-1\right )-12 \ln \left (\frac {2}{\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1}\right )+12 \ln \left (-\cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+\csc \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )+1\right )+11\right ) \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, a^{3} \cot \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )}{3 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}}-\frac {2 B \,a^{3} \left (\left (-24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\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 (24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\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 (-6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+22 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-24\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+24 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \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}}{3 f \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\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 ) c}}\) | \(528\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x,metho d=_RETURNVERBOSE)
Output:
2/3*A/f*4^(1/2)*(2*cos(1/4*Pi+1/2*f*x+1/2*e)^6-9*cos(1/4*Pi+1/2*f*x+1/2*e) ^4+18*cos(1/4*Pi+1/2*f*x+1/2*e)^2+12*ln(2/(cos(1/4*Pi+1/2*f*x+1/2*e)+1))-1 2*ln(-cot(1/4*Pi+1/2*f*x+1/2*e)+csc(1/4*Pi+1/2*f*x+1/2*e)-1)-12*ln(-cot(1/ 4*Pi+1/2*f*x+1/2*e)+csc(1/4*Pi+1/2*f*x+1/2*e)+1)-11)*(a*sin(1/4*Pi+1/2*f*x +1/2*e)^2)^(1/2)*a^3/(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)*cot(1/4*Pi+1/2* f*x+1/2*e)-2/3*B*a^3/f*((-24*sin(1/2*f*x+1/2*e)+24*cos(1/2*f*x+1/2*e))*ln( -2*(cos(1/2*f*x+1/2*e)-sin(1/2*f*x+1/2*e))/(cos(1/2*f*x+1/2*e)+1))+(24*sin (1/2*f*x+1/2*e)-24*cos(1/2*f*x+1/2*e))*ln(2/(cos(1/2*f*x+1/2*e)+1))+cos(1/ 2*f*x+1/2*e)*((-6*cos(1/2*f*x+1/2*e)^6+6*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+ 1/2*e)+22*cos(1/2*f*x+1/2*e)^4+10*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)+ 5*cos(1/2*f*x+1/2*e)^2-21*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)-24)*sin(1/ 2*f*x+1/2*e)^2+24*sin(1/2*f*x+1/2*e)*cos(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)/(cos(1/2*f*x+1/2*e)+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)*c)^(1/2)
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),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)/(c*sin(f*x + e) - c), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(1/2) ,x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x , algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/sqrt(-c*sin(f*x + e) + c), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),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. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(1/2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(1/2), x)
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{3} \left (-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}}{\sin \left (f x +e \right )-1}d x \right ) b -\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )-1}d x \right ) a -3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )-1}d x \right ) b -3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right ) a -3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right ) b -3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right ) a -\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right ) b -\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )-1}d x \right ) a \right )}{c} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),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) - 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) - 1),x)*a - 3* int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(si n(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) - 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) - 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) - 1),x)*a - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1) *sin(e + f*x))/(sin(e + f*x) - 1),x)*b - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(sin(e + f*x) - 1),x)*a))/c