Integrand size = 22, antiderivative size = 45 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {2 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e} \] Output:
2*(2+34^(1/2))^(1/2)*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(5/3)),1/15*(51 0-30*34^(1/2))^(1/2))/e
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 1.90 (sec) , antiderivative size = 326, normalized size of antiderivative = 7.24 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {-15 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+\left (-75 \cos (d+e x)+45 \sin (d+e x)+2 \sqrt {30} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{15 e \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}} \] Input:
Integrate[Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]
Output:
(-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 = 0.25 (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 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\) |
Input:
Int[Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]
Output:
(2*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[3 4]))/15])/e
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 = 1.16 (sec) , antiderivative size = 461, normalized size of antiderivative = 10.24
| method | result | size |
| default | \(\frac {2 \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \sqrt {17}\, \sqrt {\frac {\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+1}{-\sqrt {34}+17}}\, \left (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}+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}+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 | \(\text {Expression too large to display}\) | \(1003\) |
Input:
int((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/17*(-17*(sin(e*x+d+arctan(3/5))-1)/(34^(1/2)+17))^(1/2)*17^(1/2)*((sin(e *x+d+arctan(3/5))+1)/(-34^(1/2)+17))^(1/2)*(15*(-(17*sin(e*x+d+arctan(3/5) )+34^(1/2))/(-34^(1/2)+17))^(1/2)*EllipticE((-(17*sin(e*x+d+arctan(3/5))+3 4^(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)*Elli pticF((-(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)+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) +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))+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)))/cos(e*x+d+arctan(3/5 ))/(34^(1/2)*sin(e*x+d+arctan(3/5))+2)^(1/2)/e
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.40 \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=-\frac {2 \, {\left (-\left (3 i + 5\right ) \, \sqrt {\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) + \left (3 i - 5\right ) \, \sqrt {-\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) + 51 i \, \sqrt {\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) - 51 i \, \sqrt {-\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right )\right )}}{51 \, e} \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="fricas")
Output:
-2/51*(-(3*I + 5)*sqrt(5/2*I + 3/2)*weierstrassPInverse(860/289*I + 1376/8 67, -5480/132651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17) + (3*I - 5)*sqrt(-5/2*I + 3/2)*weierstrassPInverse(-860/289*I + 13 76/867, 5480/132651*I - 12056/14739, cos(e*x + d) + I*sin(e*x + d) + 10/51 *I + 2/17) + 51*I*sqrt(5/2*I + 3/2)*weierstrassZeta(860/289*I + 1376/867, -5480/132651*I - 12056/14739, weierstrassPInverse(860/289*I + 1376/867, -5 480/132651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17 )) - 51*I*sqrt(-5/2*I + 3/2)*weierstrassZeta(-860/289*I + 1376/867, 5480/1 32651*I - 12056/14739, weierstrassPInverse(-860/289*I + 1376/867, 5480/132 651*I - 12056/14739, cos(e*x + d) + I*sin(e*x + d) + 10/51*I + 2/17)))/e
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int \sqrt {5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2}\, dx \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))**(1/2),x)
Output:
Integral(sqrt(5*sin(d + e*x) + 3*cos(d + e*x) + 2), x)
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int { \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2} \,d x } \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2), x)
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int { \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2} \,d x } \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2), x)
Timed out. \[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\int \sqrt {3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2} \,d x \] Input:
int((3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(1/2),x)
Output:
int((3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(1/2), x)
\[ \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx=\frac {-\frac {10 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}}{3}+2 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e +\frac {34 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \cos \left (e x +d \right )}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e}{3}}{e} \] Input:
int((2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2),x)
Output:
(2*( - 5*sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2) + 3*int(sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)/(3*cos(d + e*x) + 5*sin(d + e*x) + 2),x)*e + 17 *int((sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*cos(d + e*x))/(3*cos(d + e *x) + 5*sin(d + e*x) + 2),x)*e))/(3*e)