Integrand size = 22, antiderivative size = 45 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {2 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e} \]
2*(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*arcta n(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 = 1.94 (sec) , antiderivative size = 326, normalized size of antiderivative = 7.24 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {-15 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+\left (-75 \cos (d+e x)+45 \sin (d+e x)+2 \sqrt {30} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{15 e \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}} \]
(-15*Sqrt[30]*AppellF1[-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])]*Sin[d + e*x - ArcTan[5/3]] + (-75*Cos[d + e*x] + 45*Sin [d + e*x] + 2*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]*Sqrt[2 + Sqrt[ 34]*Cos[d + e*x - ArcTan[5/3]]]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[3 4]*Sin[d + e*x + ArcTan[3/5]]])*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/(15*e* Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sqrt[Sin[d + e*x - ArcTan[5/ 3]]^2])
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3597, 3042, 3132}
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.
| \(\displaystyle \int \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}dx\) |
| \(\Big \downarrow \) 3597 |
| \(\displaystyle \int \sqrt {\sqrt {34} \cos \left (-\arctan \left (\frac {5}{3}\right )+d+e x\right )+2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sqrt {34} \sin \left (-\arctan \left (\frac {5}{3}\right )+d+e x+\frac {\pi }{2}\right )+2}dx\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\) |
3.5.5.3.1 Defintions of rubi rules used
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[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]
Result contains complex when optimal does not.
Time = 2.76 (sec) , antiderivative size = 461, normalized size of antiderivative = 10.24
| method | result | size |
| default | \(\frac {2 \sqrt {17}\, \sqrt {\frac {\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+1}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \left (2 \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right ) \sqrt {34}+15 \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right ) \sqrt {34}-17 \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right ) \sqrt {34}+34 \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right )+34 \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right )\right )}{17 \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right ) \sqrt {\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2}\, e}\) | \(461\) |
| risch | \(\text {Expression too large to display}\) | \(1003\) |
2/17*17^(1/2)*((sin(e*x+d+arctan(3/5))+1)/(-34^(1/2)+17))^(1/2)*(-17*(sin( e*x+d+arctan(3/5))-1)/(34^(1/2)+17))^(1/2)*(2*((17*sin(e*x+d+arctan(3/5))+ 34^(1/2))/(34^(1/2)+17))^(1/2)*EllipticF(((17*sin(e*x+d+arctan(3/5))+34^(1 /2))/(34^(1/2)+17))^(1/2),I*(1/(-34^(1/2)+17)*(34^(1/2)+17))^(1/2))*34^(1/ 2)+15*(-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2)*Ellipti cE((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1 /2)+17)/(34^(1/2)+17))^(1/2))*34^(1/2)-17*(-(17*sin(e*x+d+arctan(3/5))+34^ (1/2))/(-34^(1/2)+17))^(1/2)*EllipticF((-(17*sin(e*x+d+arctan(3/5))+34^(1/ 2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))*34^(1/2) +34*((17*sin(e*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+17))^(1/2)*EllipticF(( (17*sin(e*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+17))^(1/2),I*(1/(-34^(1/2)+ 17)*(34^(1/2)+17))^(1/2))+34*(-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^( 1/2)+17))^(1/2)*EllipticF((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2 )+17))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2)))/cos(e*x+d+arctan(3/5 ))/(34^(1/2)*sin(e*x+d+arctan(3/5))+2)^(1/2)/e
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.67 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {\left (3 i + 5\right ) \, \sqrt {5 i + 3} \sqrt {2} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) - \left (3 i - 5\right ) \, \sqrt {-5 i + 3} \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) - 51 i \, \sqrt {5 i + 3} \sqrt {2} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) + 51 i \, \sqrt {-5 i + 3} \sqrt {2} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right )}{51 \, e} \]
1/51*((3*I + 5)*sqrt(5*I + 3)*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) - (3*I - 5)*sqrt(-5*I + 3)*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) - 51*I*sqrt(5*I + 3)*sqrt(2)*weierstrassZeta(860/289*I + 1376/867, -5480/132651*I - 12056/14739, weierstrassPInverse(860/289*I + 13 76/867, -5480/132651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/5 1*I + 2/17)) + 51*I*sqrt(-5*I + 3)*sqrt(2)*weierstrassZeta(-860/289*I + 13 76/867, 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)))/e
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int \sqrt {5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2}\, dx \]
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int { \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2} \,d x } \]
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int { \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2} \,d x } \]
Timed out. \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int \sqrt {3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2} \,d x \]