\(\int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx\) [408]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \]

[Out]

1/45*(-5*cos(e*x+d)+3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2)+4/675*(5*cos(e*x+d)-3*sin(e*x+d))/e/(2
+3*cos(e*x+d)+5*sin(e*x+d))^(1/2)+1/45*(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*arct
an(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)+4/675
*(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 = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3208, 3235, 3228, 3197, 2732, 3205, 2740} \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}+\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}} \]

[In]

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

[Out]

(4*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(675*e) + EllipticF[(d + e
*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15]/(45*Sqrt[2 + Sqrt[34]]*e) - (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)) + (4*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(675*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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 3205

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]

Rule 3208

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] +
Dist[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*C
os[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]

Rule 3228

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] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(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]

Rule 3235

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] + Dist[
1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {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}}+\frac {1}{45} \int \frac {-3+\frac {3}{2} \cos (d+e x)+\frac {5}{2} \sin (d+e x)}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx \\ & = -\frac {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}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {1}{675} \int \frac {\frac {23}{2}+6 \cos (d+e x)+10 \sin (d+e x)}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx \\ & = -\frac {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}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx \\ & = -\frac {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}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}} \, dx \\ & = \frac {4 \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 )}{675 e}+\frac {\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 )}{45 \sqrt {2+\sqrt {34}} e}-\frac {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}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 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 = 2.47 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {-24 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}+\frac {272}{3} \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {100 (5+17 \sin (d+e x))}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}-\frac {10 (115+136 \sin (d+e x))}{3 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+23 \sqrt {\frac {10}{3}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}+\frac {20 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}-\frac {20 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{6750 e} \]

[In]

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

[Out]

(-24*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] + (272*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/3 + (100*
(5 + 17*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2) - (10*(115 + 136*Sin[d + e*x]))/(3*Sqrt[2 +
3*Cos[d + e*x] + 5*Sin[d + e*x]]) + 23*Sqrt[10/3]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + A
rcTan[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]]] + (20*Sin[d + e*x - A
rcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - ArcTan[5/3]]/Sqrt[34]] - (20*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 + S
qrt[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 - ArcTa
n[5/3]]])/(6750*e)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.62 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.13

method	result	size
default	\(\frac {17 \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}}\, \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}+2 \sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \left (-\frac {\sqrt {34}\, \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}}}{765 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\frac {\sqrt {34}}{17}\right )^{2}}+\frac {136 \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}{675 \sqrt {-\cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2} \sqrt {34}\, \left (-289 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-17 \sqrt {34}\right )}}+\frac {46 \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\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 )}{675 \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}}}+\frac {8 \sqrt {34}\, \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\sqrt {34}+17}}\, \left (\left (-\frac {\sqrt {34}}{17}-1\right ) \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 )+\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 )}{675 \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 )}{2 \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\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}\)	\(586\)

[In]

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

[Out]

17/2*(-(-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
))^2*(17*sin(e*x+d+arctan(3/5))^2+2*34^(1/2)*sin(e*x+d+arctan(3/5))+2)*(-1/765*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+arctan(3/5))+1/17*34^(1/2))^2+136/675*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)+46/
675*(-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+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+a
rctan(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))+8/675*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+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)*((
-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))+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))))/cos(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 = 431, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {53 \, \sqrt {5 i + 3} {\left (\left (48 i + 80\right ) \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (-\left (9 i + 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (6 i + 10\right ) \, \sqrt {2}\right )} \sin \left (e x + d\right ) - \left (36 i + 60\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (87 i + 145\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 ) + 53 \, \sqrt {-5 i + 3} {\left (-\left (48 i - 80\right ) \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (\left (9 i - 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (6 i - 10\right ) \, \sqrt {2}\right )} \sin \left (e x + d\right ) + \left (36 i - 60\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (87 i - 145\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 ) + 408 \, \sqrt {5 i + 3} {\left (-16 i \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (3 i \, \sqrt {2} \cos \left (e x + d\right ) + 2 i \, \sqrt {2}\right )} \sin \left (e x + d\right ) + 12 i \, \sqrt {2} \cos \left (e x + d\right ) + 29 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 ) + 408 \, \sqrt {-5 i + 3} {\left (16 i \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (-3 i \, \sqrt {2} \cos \left (e x + d\right ) - 2 i \, \sqrt {2}\right )} \sin \left (e x + d\right ) - 12 i \, \sqrt {2} \cos \left (e x + d\right ) - 29 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 ) - 204 \, {\left (120 \, \cos \left (e x + d\right )^{2} + {\left (64 \, \cos \left (e x + d\right ) + 21\right )} \sin \left (e x + d\right ) - 35 \, \cos \left (e x + d\right ) - 60\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{137700 \, {\left (16 \, e \cos \left (e x + d\right )^{2} - 12 \, e \cos \left (e x + d\right ) - 10 \, {\left (3 \, e \cos \left (e x + d\right ) + 2 \, e\right )} \sin \left (e x + d\right ) - 29 \, e\right )}} \]

[In]

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

[Out]

1/137700*(53*sqrt(5*I + 3)*((48*I + 80)*sqrt(2)*cos(e*x + d)^2 + 10*(-(9*I + 15)*sqrt(2)*cos(e*x + d) - (6*I +
10)*sqrt(2))*sin(e*x + d) - (36*I + 60)*sqrt(2)*cos(e*x + d) - (87*I + 145)*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) + 53*sqrt(-5*I
+ 3)*(-(48*I - 80)*sqrt(2)*cos(e*x + d)^2 + 10*((9*I - 15)*sqrt(2)*cos(e*x + d) + (6*I - 10)*sqrt(2))*sin(e*x
+ d) + (36*I - 60)*sqrt(2)*cos(e*x + d) + (87*I - 145)*sqrt(2))*weierstrassPInverse(-860/289*I + 1376/867, 54
80/132651*I - 12056/14739, cos(e*x + d) + I*sin(e*x + d) + 10/51*I + 2/17) + 408*sqrt(5*I + 3)*(-16*I*sqrt(2)*
cos(e*x + d)^2 + 10*(3*I*sqrt(2)*cos(e*x + d) + 2*I*sqrt(2))*sin(e*x + d) + 12*I*sqrt(2)*cos(e*x + d) + 29*I*s
qrt(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) - 10/51*I + 2/17)) + 408*sqrt(-5*I + 3)*(1
6*I*sqrt(2)*cos(e*x + d)^2 + 10*(-3*I*sqrt(2)*cos(e*x + d) - 2*I*sqrt(2))*sin(e*x + d) - 12*I*sqrt(2)*cos(e*x
+ d) - 29*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)) - 204*(120
*cos(e*x + d)^2 + (64*cos(e*x + d) + 21)*sin(e*x + d) - 35*cos(e*x + d) - 60)*sqrt(3*cos(e*x + d) + 5*sin(e*x
+ d) + 2))/(16*e*cos(e*x + d)^2 - 12*e*cos(e*x + d) - 10*(3*e*cos(e*x + d) + 2*e)*sin(e*x + d) - 29*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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