Integrand size = 22, antiderivative size = 233 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=-\frac {199 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{101250 e}-\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{3375 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}}+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{3375 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{101250 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \]
1/75*(-5*cos(e*x+d)+3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2)+8/ 3375*(5*cos(e*x+d)-3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2)-199 /101250*(5*cos(e*x+d)-3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2)- 8/3375*(cos(1/2*d+1/2*e*x-1/2*arctan(5/3))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2* arctan(5/3))*EllipticF(sin(1/2*d+1/2*e*x-1/2*arctan(5/3)),1/15*(510-30*34^ (1/2))^(1/2))/e/(2+34^(1/2))^(1/2)-199/101250*(cos(1/2*d+1/2*e*x-1/2*arcta n(5/3))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(5/3))*EllipticE(sin(1/2*d+1/ 2*e*x-1/2*arctan(5/3)),1/15*(510-30*34^(1/2))^(1/2))*(2+34^(1/2))^(1/2)/e
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 3.00 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\frac {-13532 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {597 (12+43 \cos (d+e x)+15 \sin (d+e x))}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}+\frac {27000 (5+17 \sin (d+e x))}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}}-\frac {300 (305+272 \sin (d+e x))}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {20 (1595+3383 \sin (d+e x))}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-638 \sqrt {30} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}+\frac {2985 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{3037500 e} \]
(-13532*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] + (597*(12 + 43*Cos[d + e*x] + 15*Sin[d + e*x]))/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] + ( 27000*(5 + 17*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2) - (300*(305 + 272*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2 ) + (20*(1595 + 3383*Sin[d + e*x]))/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e* x]] - 638*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5 ]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcT an[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] + (2985*Sqrt[30]*Ap pellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/( -17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34] )]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]])/(3037500*e)
Time = 1.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3608, 27, 3042, 3635, 25, 3042, 3635, 27, 3042, 3628, 3042, 3597, 3042, 3132, 3605, 3042, 3140}
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.
| \(\displaystyle \int \frac {1}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{7/2}}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {1}{75} \int -\frac {-9 \cos (d+e x)-15 \sin (d+e x)+10}{2 (3 \cos (d+e x)+5 \sin (d+e x)+2)^{5/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{150} \int \frac {-9 \cos (d+e x)-15 \sin (d+e x)+10}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{5/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{150} \int \frac {-9 \cos (d+e x)-15 \sin (d+e x)+10}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{5/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {1}{150} \left (\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}-\frac {1}{45} \int -\frac {-24 \cos (d+e x)-40 \sin (d+e x)+183}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \int \frac {-24 \cos (d+e x)-40 \sin (d+e x)+183}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \int \frac {-24 \cos (d+e x)-40 \sin (d+e x)+183}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{15} \int -\frac {597 \cos (d+e x)+995 \sin (d+e x)+638}{2 \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (-\frac {1}{30} \int \frac {597 \cos (d+e x)+995 \sin (d+e x)+638}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (-\frac {1}{30} \int \frac {597 \cos (d+e x)+995 \sin (d+e x)+638}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-199 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-199 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3597 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-199 \int \sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}dx-240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-199 \int \sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}dx-240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3605 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-240 \int \frac {1}{\sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}}dx-\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-240 \int \frac {1}{\sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}}dx-\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{150} \left (\frac {1}{45} \left (\frac {1}{30} \left (-\frac {480 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}-\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {199 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )+\frac {16 (5 \cos (d+e x)-3 \sin (d+e x))}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}\) |
-1/75*(5*Cos[d + e*x] - 3*Sin[d + e*x])/(e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2)) + ((16*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(45*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2)) + (((-398*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/e - (480*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e))/30 - (199*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(15*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]]))/45)/150
3.5.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Int[Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]] , x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[ b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c *(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x ] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Result contains complex when optimal does not.
Time = 1.70 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.79
17/4*(-(-34^(1/2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2 )/(17*sin(e*x+d+arctan(3/5))+34^(1/2))^3*(51*34^(1/2)*sin(e*x+d+arctan(3/5 ))^2+289*sin(e*x+d+arctan(3/5))^3+2*34^(1/2)+102*sin(e*x+d+arctan(3/5)))*( -2/1275*(-(-34^(1/2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^( 1/2)/(sin(e*x+d+arctan(3/5))+1/17*34^(1/2))^3+16/57375*34^(1/2)*(-(-34^(1/ 2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2)/(sin(e*x+d+ar ctan(3/5))+1/17*34^(1/2))^2-6766/50625*34^(1/2)*cos(e*x+d+arctan(3/5))^2/( -cos(e*x+d+arctan(3/5))^2*34^(1/2)*(-289*sin(e*x+d+arctan(3/5))-17*34^(1/2 )))^(1/2)-1276/50625*(-1+1/17*34^(1/2))*(1/(-34^(1/2)+17)*(-17*sin(e*x+d+a rctan(3/5))-34^(1/2)))^(1/2)*((-17*sin(e*x+d+arctan(3/5))+17)/(34^(1/2)+17 ))^(1/2)*(1/(-34^(1/2)+17)*(17*sin(e*x+d+arctan(3/5))+17))^(1/2)/(-(-34^(1 /2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2)*EllipticF((1 /(-34^(1/2)+17)*(-17*sin(e*x+d+arctan(3/5))-34^(1/2)))^(1/2),I*((-34^(1/2) +17)/(34^(1/2)+17))^(1/2))-398/50625*34^(1/2)*(-1+1/17*34^(1/2))*(1/(-34^( 1/2)+17)*(-17*sin(e*x+d+arctan(3/5))-34^(1/2)))^(1/2)*((-17*sin(e*x+d+arct an(3/5))+17)/(34^(1/2)+17))^(1/2)*(1/(-34^(1/2)+17)*(17*sin(e*x+d+arctan(3 /5))+17))^(1/2)/(-(-34^(1/2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/ 5))^2)^(1/2)*((-1/17*34^(1/2)-1)*EllipticE((1/(-34^(1/2)+17)*(-17*sin(e*x+ d+arctan(3/5))-34^(1/2)))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))+El lipticF((1/(-34^(1/2)+17)*(-17*sin(e*x+d+arctan(3/5))-34^(1/2)))^(1/2),...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.48 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\text {Too large to display} \]
1/10327500*(559*sqrt(5*I + 3)*(-(594*I + 990)*sqrt(2)*cos(e*x + d)^3 - (28 8*I + 480)*sqrt(2)*cos(e*x + d)^2 + 5*((6*I + 10)*sqrt(2)*cos(e*x + d)^2 + (108*I + 180)*sqrt(2)*cos(e*x + d) + (111*I + 185)*sqrt(2))*sin(e*x + d) + (783*I + 1305)*sqrt(2)*cos(e*x + d) + (474*I + 790)*sqrt(2))*weierstrass PInverse(860/289*I + 1376/867, -5480/132651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17) + 559*sqrt(-5*I + 3)*((594*I - 990)*sqr t(2)*cos(e*x + d)^3 + (288*I - 480)*sqrt(2)*cos(e*x + d)^2 + 5*(-(6*I - 10 )*sqrt(2)*cos(e*x + d)^2 - (108*I - 180)*sqrt(2)*cos(e*x + d) - (111*I - 1 85)*sqrt(2))*sin(e*x + d) - (783*I - 1305)*sqrt(2)*cos(e*x + d) - (474*I - 790)*sqrt(2))*weierstrassPInverse(-860/289*I + 1376/867, 5480/132651*I - 12056/14739, cos(e*x + d) + I*sin(e*x + d) + 10/51*I + 2/17) + 10149*sqrt( 5*I + 3)*(198*I*sqrt(2)*cos(e*x + d)^3 + 96*I*sqrt(2)*cos(e*x + d)^2 + 5*( -2*I*sqrt(2)*cos(e*x + d)^2 - 36*I*sqrt(2)*cos(e*x + d) - 37*I*sqrt(2))*si n(e*x + d) - 261*I*sqrt(2)*cos(e*x + d) - 158*I*sqrt(2))*weierstrassZeta(8 60/289*I + 1376/867, -5480/132651*I - 12056/14739, weierstrassPInverse(860 /289*I + 1376/867, -5480/132651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17)) + 10149*sqrt(-5*I + 3)*(-198*I*sqrt(2)*cos(e*x + d )^3 - 96*I*sqrt(2)*cos(e*x + d)^2 + 5*(2*I*sqrt(2)*cos(e*x + d)^2 + 36*I*s qrt(2)*cos(e*x + d) + 37*I*sqrt(2))*sin(e*x + d) + 261*I*sqrt(2)*cos(e*x + d) + 158*I*sqrt(2))*weierstrassZeta(-860/289*I + 1376/867, 5480/132651...
Timed out. \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx=\int \frac {1}{{\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{7/2}} \,d x \]