Integrand size = 30, antiderivative size = 140 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{60 c^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}} \] Output:
1/6*a*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f/(c-c*sin(f*x+e))^(13/2)-1/15*a^2 *cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c/f/(c-c*sin(f*x+e))^(11/2)+1/60*a^3*co s(f*x+e)/c^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2)
Time = 10.61 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (29-15 \cos (2 (e+f x))+36 \sin (e+f x))}{120 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^6 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(13/2),x]
Output:
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(29 - 15*Cos[2*(e + f*x)] + 36*Sin[e + f*x]))/(120*c^6*f*(Cos[(e + f*x)/2] + S in[(e + f*x)/2])*(-1 + Sin[e + f*x])^6*Sqrt[c - c*Sin[e + f*x]])
Time = 0.66 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3218, 3042, 3218, 3042, 3217}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c-c \sin (e+f x))^{13/2}}dx\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{3 c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}\right )}{3 c}\) |
Input:
Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(13/2),x]
Output:
(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(13/ 2)) - (a*((a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*f*(c - c*Sin[e + f* x])^(11/2)) - (a^2*Cos[e + f*x])/(20*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*S in[e + f*x])^(9/2))))/(3*c)
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
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])
Time = 1.91 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{5} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+3 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+6 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+10\right ) a^{2} \sqrt {4}}{1920 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{6}}\) | \(127\) |
Input:
int((a+sin(f*x+e)*a)^(5/2)/(c-c*sin(f*x+e))^(13/2),x,method=_RETURNVERBOSE )
Output:
1/1920/f*tan(1/4*Pi+1/2*f*x+1/2*e)^5*sec(1/4*Pi+1/2*f*x+1/2*e)^6*(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)^6+3*cos(1/4*Pi+1/2 *f*x+1/2*e)^4+6*cos(1/4*Pi+1/2*f*x+1/2*e)^2+10)*a^2/(c*cos(1/4*Pi+1/2*f*x+ 1/2*e)^2)^(1/2)/c^6*4^(1/2)
Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {{\left (15 \, a^{2} \cos \left (f x + e\right )^{2} - 18 \, a^{2} \sin \left (f x + e\right ) - 22 \, a^{2}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="fri cas")
Output:
1/60*(15*a^2*cos(f*x + e)^2 - 18*a^2*sin(f*x + e) - 22*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos( f*x + e)^5 + 48*c^7*f*cos(f*x + e)^3 - 32*c^7*f*cos(f*x + e) + 2*(3*c^7*f* cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^7*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(13/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="max ima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2)/(-c*sin(f*x + e) + c)^(13/2), x)
Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {{\left (15 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 24 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 10 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{960 \, c^{\frac {13}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="gia c")
Output:
-1/960*(15*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 - 24*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f *x + 1/2*e)^2 + 10*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^(13 /2)*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^1 2)
Time = 21.83 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.05 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {464\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,c^7\,f}+\frac {192\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}-\frac {16\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \] Input:
int((a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(13/2),x)
Output:
-((c - c*sin(e + f*x))^(1/2)*((464*a^2*exp(e*7i + f*x*7i)*(a + a*sin(e + f *x))^(1/2))/(15*c^7*f) + (192*a^2*exp(e*7i + f*x*7i)*sin(e + f*x)*(a + a*s in(e + f*x))^(1/2))/(5*c^7*f) - (16*a^2*exp(e*7i + f*x*7i)*cos(2*e + 2*f*x )*(a + a*sin(e + f*x))^(1/2))/(c^7*f)))/(858*exp(e*7i + f*x*7i)*cos(3*e + 3*f*x) - 858*cos(e + f*x)*exp(e*7i + f*x*7i) - 130*exp(e*7i + f*x*7i)*cos( 5*e + 5*f*x) + 2*exp(e*7i + f*x*7i)*cos(7*e + 7*f*x) + 1144*exp(e*7i + f*x *7i)*sin(2*e + 2*f*x) - 416*exp(e*7i + f*x*7i)*sin(4*e + 4*f*x) + 24*exp(e *7i + f*x*7i)*sin(6*e + 6*f*x))
\[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{2} \left (-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{7}-7 \sin \left (f x +e \right )^{6}+21 \sin \left (f x +e \right )^{5}-35 \sin \left (f x +e \right )^{4}+35 \sin \left (f x +e \right )^{3}-21 \sin \left (f x +e \right )^{2}+7 \sin \left (f x +e \right )-1}d x \right )-2 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{7}-7 \sin \left (f x +e \right )^{6}+21 \sin \left (f x +e \right )^{5}-35 \sin \left (f x +e \right )^{4}+35 \sin \left (f x +e \right )^{3}-21 \sin \left (f x +e \right )^{2}+7 \sin \left (f x +e \right )-1}d x \right )-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{7}-7 \sin \left (f x +e \right )^{6}+21 \sin \left (f x +e \right )^{5}-35 \sin \left (f x +e \right )^{4}+35 \sin \left (f x +e \right )^{3}-21 \sin \left (f x +e \right )^{2}+7 \sin \left (f x +e \right )-1}d x \right )\right )}{c^{7}} \] Input:
int((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x)
Output:
(sqrt(c)*sqrt(a)*a**2*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f *x)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7* sin(e + f*x) - 1),x) - 2*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x)* *5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*sin( e + f*x) - 1),x) - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/ (sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x)**5 - 35*sin(e + f*x )**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*sin(e + f*x) - 1),x)))/ c**7