\(\int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx\) [407]

Optimal result

Integrand size = 22, antiderivative size = 94 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=-\frac {\sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \]

[Out]

1/15*(-5*cos(e*x+d)+3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2)-1/15*(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))*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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3207, 3197, 2732} \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=-\frac {\sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}} \]

[In]

Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-3/2),x]

[Out]

-1/15*(Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/e - (5*Cos[d + e*x] -
3*Sin[d + e*x])/(15*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])

Rule 2732

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]

Rule 3197

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]

Rule 3207

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-3/2), x_Symbol] :> Simp[2*((c*Cos
[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Dist[1/(a^2
- b^2 - c^2), Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 -
b^2 - c^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-\frac {1}{30} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx \\ & = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-\frac {1}{30} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \, dx \\ & = -\frac {\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 )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 4.79 (sec) , antiderivative size = 390, normalized size of antiderivative = 4.15 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\frac {18 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}-68 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {20 (5+17 \sin (d+e x))}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-2 \sqrt {30} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}-\frac {15 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}+\frac {15 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{450 e} \]

[In]

Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-3/2),x]

[Out]

(18*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] - 68*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] + (20*(5 + 17
*Sin[d + e*x]))/Sqrt[2 + 3*Cos[d + e*x] + 5*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]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] -
(15*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - ArcTan[5/3]]/Sqrt[34]] + (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 - A
rcTan[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]]])/(450*e)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.35 (sec) , antiderivative size = 425, normalized size of antiderivative = 4.52

method	result	size
default	\(\frac {\sqrt {34}\, \left (255 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+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}}\, \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 )-255 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+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}}\, \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 )+289 \sqrt {\left (\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}-289 \sqrt {\left (\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\right ) \sqrt {17}}{4335 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \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}\)	\(425\)

[In]

int(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4335*34^(1/2)*(255*((17*sin(e*x+d+arctan(3/5))+34^(1/2))*34^(1/2)*cos(e*x+d+arctan(3/5))^2)^(1/2)*((17*sin(e
*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+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)*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))-255*((17*sin(e*x+d+arctan(3/5))+34^(1/2))*34^(1/2)*cos(e*x+d+arctan
(3/5))^2)^(1/2)*((17*sin(e*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+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)*EllipticE(((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))+289*((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))^2-289*((34^(1/2)*sin(e*x+d+arctan(3/5))+2)*cos(e*x+d+ar
ctan(3/5))^2)^(1/2))*17^(1/2)/((17*sin(e*x+d+arctan(3/5))+34^(1/2))*34^(1/2)*cos(e*x+d+arctan(3/5))^2)^(1/2)/c
os(e*x+d+arctan(3/5))/(34^(1/2)*sin(e*x+d+arctan(3/5))+2)^(1/2)/e

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\frac {\sqrt {5 i + 3} {\left (-\left (9 i + 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (15 i + 25\right ) \, \sqrt {2} \sin \left (e x + d\right ) - \left (6 i + 10\right ) \, \sqrt {2}\right )} {\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 ) + \sqrt {-5 i + 3} {\left (\left (9 i - 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (15 i - 25\right ) \, \sqrt {2} \sin \left (e x + d\right ) + \left (6 i - 10\right ) \, \sqrt {2}\right )} {\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 \, \sqrt {5 i + 3} {\left (-3 i \, \sqrt {2} \cos \left (e x + d\right ) - 5 i \, \sqrt {2} \sin \left (e x + d\right ) - 2 i \, \sqrt {2}\right )} {\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 \, \sqrt {-5 i + 3} {\left (3 i \, \sqrt {2} \cos \left (e x + d\right ) + 5 i \, \sqrt {2} \sin \left (e x + d\right ) + 2 i \, \sqrt {2}\right )} {\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 ) - 102 \, {\left (5 \, \cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right )\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{1530 \, {\left (3 \, e \cos \left (e x + d\right ) + 5 \, e \sin \left (e x + d\right ) + 2 \, e\right )}} \]

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2),x, algorithm="fricas")

[Out]

1/1530*(sqrt(5*I + 3)*(-(9*I + 15)*sqrt(2)*cos(e*x + d) - (15*I + 25)*sqrt(2)*sin(e*x + d) - (6*I + 10)*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) + sqrt(-5*I + 3)*((9*I - 15)*sqrt(2)*cos(e*x + d) + (15*I - 25)*sqrt(2)*sin(e*x + d) + (6*I - 10)
*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*sqrt(5*I + 3)*(-3*I*sqrt(2)*cos(e*x + d) - 5*I*sqrt(2)*sin(e*x + d) - 2*I*sqrt(2))*we
ierstrassZeta(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*sqrt(-5*I + 3)*(3*I*sqrt(2)*
cos(e*x + d) + 5*I*sqrt(2)*sin(e*x + d) + 2*I*sqrt(2))*weierstrassZeta(-860/289*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)) - 102*(5*cos(e*x + d) - 3*sin(e*x + d))*sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2))/(3*
e*cos(e*x + d) + 5*e*sin(e*x + d) + 2*e)

Sympy [F]

\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{\left (5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))**(3/2),x)

[Out]

Integral((5*sin(d + e*x) + 3*cos(d + e*x) + 2)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2),x, algorithm="maxima")

[Out]

integrate((3*cos(e*x + d) + 5*sin(e*x + d) + 2)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2),x, algorithm="giac")

[Out]

integrate((3*cos(e*x + d) + 5*sin(e*x + d) + 2)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{{\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{3/2}} \,d x \]

[In]

int(1/(3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(3/2),x)

[Out]

int(1/(3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(3/2), x)