\(\int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx\) [415]

Optimal result

Integrand size = 22, antiderivative size = 382 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {8 a E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

2/3*(c*cos(e*x+d)-b*sin(e*x+d))/(a^2-b^2-c^2)/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^(3/2)+8/3*(a*c*cos(e*x+d)-a*b*si
n(e*x+d))/(a^2-b^2-c^2)^2/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)+8/3*a*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2)^(
1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(b,c))*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/
(a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)/(a^2-b^2-c^2)^2/e/((a+b*cos(e*x+d)+c*sin(e*x+d
))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(b,
c))*EllipticF(sin(1/2*d+1/2*e*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*co
s(e*x+d)+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a^2-b^2-c^2)/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 382, 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, 3198, 2732, 3206, 2740} \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=-\frac {2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{3 e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{3 e \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \]

[In]

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

[Out]

(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (8
*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) +
(8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x]
+ c*Sin[d + e*x]])/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])
- (2*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x
] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*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 3198

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

Rule 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && 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 {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}-\frac {2 \int \frac {-\frac {3 a}{2}+\frac {1}{2} b \cos (d+e x)+\frac {1}{2} c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx}{3 \left (a^2-b^2-c^2\right )} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2+b^2+c^2\right )+a b \cos (d+e x)+a c \sin (d+e x)}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{3 \left (a^2-b^2-c^2\right )^2} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {(4 a) \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{3 \left (a^2-b^2-c^2\right )^2}-\frac {\int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \, dx}{3 \left (a^2-b^2-c^2\right )} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {\left (4 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}\right ) \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}} \, dx}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {8 a E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 \left (a^2-b^2-c^2\right )^2 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 6.69 (sec) , antiderivative size = 2408, normalized size of antiderivative = 6.30 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((8*a*(b^2 + c^2))/(3*b*c*(a^2 - b^2 - c^2)^2) + (2*(a*c + b^2*Sin[
d + e*x] + c^2*Sin[d + e*x]))/(3*b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(3*a^2*c +
b^2*c + c^3 + 4*a*b^2*Sin[d + e*x] + 4*a*c^2*Sin[d + e*x]))/(3*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c
*Sin[d + e*x]))))/e + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]
])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]]
)/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c
^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2
+ c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x +
ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^2*e) + (2*b^2*AppellF1[
1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 +
b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 +
b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d +
e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]
]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)
/c^2])])/(3*Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^2*e) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^
2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2
/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTa
n[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 +
c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sq
rt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*(-a^2 +
b^2 + c^2)^2*e) + (4*a*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcT
an[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTa
n[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^
2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)
/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b
^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*
Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sq
rt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*c*(-a^2 + b^2 + c^2)^2*e) + (4*a*c*(-((c*AppellF1[-1/2, -1/2,
-1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2
/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2
/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)
/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x
- ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sq
rt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d +
e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*(-a^
2 + b^2 + c^2)^2*e)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3641\) vs. \(2(428)=856\).

Time = 9.28 (sec) , antiderivative size = 3642, normalized size of antiderivative = 9.53

method	result	size
default	\(\text {Expression too large to display}\)	\(3642\)

[In]

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

[Out]

-(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos(e*x+d-arctan(-b,c))^2/(b^2
+c^2)^(1/2))^(1/2)*(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))+a*(b^2+c^2)^(1/2))*(cos(e*x+d-arct
an(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*(b^4*sin(e*x+d-arctan(-b,c))^4+2*b^2*
c^2*sin(e*x+d-arctan(-b,c))^4+c^4*sin(e*x+d-arctan(-b,c))^4-2*a^2*b^2*sin(e*x+d-arctan(-b,c))^2-2*a^2*c^2*sin(
e*x+d-arctan(-b,c))^2+a^4)/(b^4*sin(e*x+d-arctan(-b,c))^3+2*b^2*c^2*sin(e*x+d-arctan(-b,c))^3+c^4*sin(e*x+d-ar
ctan(-b,c))^3+3*(b^2+c^2)^(1/2)*a*b^2*sin(e*x+d-arctan(-b,c))^2+3*(b^2+c^2)^(1/2)*a*c^2*sin(e*x+d-arctan(-b,c)
)^2+3*a^2*b^2*sin(e*x+d-arctan(-b,c))+3*a^2*c^2*sin(e*x+d-arctan(-b,c))+(b^2+c^2)^(1/2)*a^3)/(2*(((b^2+c^2)^(1
/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)*sin(e*x+d-arctan(-b,c))*a*b^2+2*(((b^2+c^2)^(1
/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)*sin(e*x+d-arctan(-b,c))*a*c^2-(cos(e*x+d-arcta
n(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*sin(e*x+d-arctan(-b,c))^2*b^2-(cos(e*x
+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*sin(e*x+d-arctan(-b,c))^2*c^2-
(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*a^2)*(-1/4/a/(a^2-b^2-
c^2)*(b^2+c^2)^(1/2)*(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)/(
b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1/2))+1/3/(a^2-b^2-c^2)/(b^2+c^2)*(cos(e*
x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b
^2+c^2)^(1/2)*a)^2-4/3*(-b^2-c^2)*cos(e*x+d-arctan(-b,c))^2/(a^2-b^2-c^2)^2*a/(cos(e*x+d-arctan(-b,c))^2*((b^2
+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)+2*(-1/24*(b^2+c^2)^(1/2)/(a^2-b^2-c^2)+2/3*a^2*(b^2+c^
2)^(1/2)/(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1
/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c)
)+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-
b,c))+a)*(b^2+c^2))^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((
-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*(13*a^2*b^2+13*a^2*c^2+3*b^4+6*b^2*c^2+3*c^4)/(24*a^5-48*a^
3*b^2-48*a^3*c^2+24*a*b^4+48*a*b^2*c^2+24*a*c^4)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b
,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((
-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(
1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(e*x
+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF(((
(b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2
)))^(1/2)))-1/8*(5*a^2*b^2+5*a^2*c^2-b^4-2*b^2*c^2-c^4)/a^2/(a^2-b^2-c^2)/(b^2+c^2)^(1/2)*(1/(b^2+c^2)^(1/2)*a
+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+
c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2
)/(cos(e*x+d-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*EllipticPi((((b^2+c^
2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a-1)*(b^2+c^2)^(1/2)/a
,((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+1/4*(b^2+c^2)/a/(a^2-b^2-c^2)*(((b^2+c^2)^(1/2)*sin(e*x+d-
arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)/(b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))-a*(
b^2+c^2)^(1/2))+1/3/(a^2-b^2-c^2)/(b^2+c^2)^(1/2)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arcta
n(-b,c))^2)^(1/2)/(sin(e*x+d-arctan(-b,c))+1/(b^2+c^2)^(1/2)*a)^2+4/3*(b^2+c^2)^(1/2)*cos(e*x+d-arctan(-b,c))^
2/(a^2-b^2-c^2)^2*a/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)+2*(-7/24/(a^
2-b^2-c^2)+2/3*a^2/(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b
^2+c^2)^(1/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-ar
ctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x
+d-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((
-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*(1/8/a/(a^2-b^2-c^2)*(b^2+c^2)^(1/2)+2/3*a*(b^2+c^2)^(1/2)/
(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/
2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2
+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)
^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(
1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)
/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2)))+1/8*(5*a^2-b^2-c^2)/a^2/(a^2-b
^2-c^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(
e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/
2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)*El
lipticPi((((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a-1)
*(b^2+c^2)^(1/2)/a,((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2)))/cos(e*x+d-arctan(-b,c))/((b^2*sin(e*x+d
-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))+a*(b^2+c^2)^(1/2))/(b^2+c^2)^(1/2))^(1/2)/e

Fricas [C] (verification not implemented)

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

Time = 0.21 (sec) , antiderivative size = 2805, normalized size of antiderivative = 7.34 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/9*((sqrt(2)*(I*a^2*b^3 + 3*I*b^5 - I*a^2*b*c^2 - a^2*c^3 - 3*I*b*c^4 - 3*c^5 + (a^2*b^2 + 3*b^4)*c)*cos(e*x
+ d)^2 - 2*sqrt(2)*(-I*a^3*b^2 - 3*I*a*b^4 - 3*I*a*b^2*c^2 - 3*a*b*c^3 - (a^3*b + 3*a*b^3)*c)*cos(e*x + d) - 2
*(sqrt(2)*(-3*I*b^2*c^3 - 3*b*c^4 - (a^2*b + 3*b^3)*c^2 - I*(a^2*b^2 + 3*b^4)*c)*cos(e*x + d) + sqrt(2)*(-3*I*
a*b*c^3 - 3*a*c^4 - (a^3 + 3*a*b^2)*c^2 - I*(a^3*b + 3*a*b^3)*c))*sin(e*x + d) + sqrt(2)*(I*a^4*b + 3*I*a^2*b^
3 + 3*I*b*c^4 + 3*c^5 + (4*a^2 + 3*b^2)*c^3 + I*(4*a^2*b + 3*b^3)*c^2 + (a^4 + 3*a^2*b^2)*c))*sqrt(b + I*c)*we
ierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b
^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b -
3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2
+ c^2)*cos(e*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2)) + (sqrt(2)*(-I*a^2*b^3 - 3*I*b^5 + I*a^2*b
*c^2 - a^2*c^3 + 3*I*b*c^4 - 3*c^5 + (a^2*b^2 + 3*b^4)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(I*a^3*b^2 + 3*I*a*b^4 +
3*I*a*b^2*c^2 - 3*a*b*c^3 - (a^3*b + 3*a*b^3)*c)*cos(e*x + d) - 2*(sqrt(2)*(3*I*b^2*c^3 - 3*b*c^4 - (a^2*b + 3
*b^3)*c^2 + I*(a^2*b^2 + 3*b^4)*c)*cos(e*x + d) + sqrt(2)*(3*I*a*b*c^3 - 3*a*c^4 - (a^3 + 3*a*b^2)*c^2 + I*(a^
3*b + 3*a*b^3)*c))*sin(e*x + d) + sqrt(2)*(-I*a^4*b - 3*I*a^2*b^3 - 3*I*b*c^4 + 3*c^5 + (4*a^2 + 3*b^2)*c^3 -
I*(4*a^2*b + 3*b^3)*c^2 + (a^4 + 3*a^2*b^2)*c))*sqrt(b - I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a
^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27
*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b
^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(-I*b^2 - I*c^2)*sin(
e*x + d))/(b^2 + c^2)) - 12*(sqrt(2)*(I*a*b^4 - I*a*c^4)*cos(e*x + d)^2 + 2*sqrt(2)*(I*a^2*b^3 + I*a^2*b*c^2)*
cos(e*x + d) + 2*(sqrt(2)*(I*a*b^3*c + I*a*b*c^3)*cos(e*x + d) + sqrt(2)*(I*a^2*b^2*c + I*a^2*c^3))*sin(e*x +
d) + sqrt(2)*(I*a^3*b^2 + I*a*c^4 + I*(a^3 + a*b^2)*c^2))*sqrt(b + I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4
- 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^
5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)
*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 +
3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5
+ 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^
2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2))
) - 12*(sqrt(2)*(-I*a*b^4 + I*a*c^4)*cos(e*x + d)^2 + 2*sqrt(2)*(-I*a^2*b^3 - I*a^2*b*c^2)*cos(e*x + d) + 2*(s
qrt(2)*(-I*a*b^3*c - I*a*b*c^3)*cos(e*x + d) + sqrt(2)*(-I*a^2*b^2*c - I*a^2*c^3))*sin(e*x + d) + sqrt(2)*(-I*
a^3*b^2 - I*a*c^4 - I*(a^3 + a*b^2)*c^2))*sqrt(b - I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6
*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 +
9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4
*c^2 + 3*b^2*c^4 + c^6), weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a
^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9
*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/
3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(-I*b^2 - I*c^2)*sin(e*x + d))/(b^2 + c^2))) - 6*(4*a*b^3*
c + 4*a*b*c^3 - 8*(a*b^3*c + a*b*c^3)*cos(e*x + d)^2 + (c^5 - (5*a^2 - 2*b^2)*c^3 - (5*a^2*b^2 - b^4)*c)*cos(e
*x + d) + (5*a^2*b^3 - b^5 - b*c^4 + (5*a^2*b - 2*b^3)*c^2 + 4*(a*b^4 - a*c^4)*cos(e*x + d))*sin(e*x + d))*sqr
t(b*cos(e*x + d) + c*sin(e*x + d) + a))/((a^4*b^4 - 2*a^2*b^6 + b^8 - c^8 + 2*(a^2 - b^2)*c^6 - (a^4 - 2*a^2*b
^2)*c^4 - 2*(a^2*b^4 - b^6)*c^2)*e*cos(e*x + d)^2 + 2*(a^5*b^3 - 2*a^3*b^5 + a*b^7 + a*b*c^6 - (2*a^3*b - 3*a*
b^3)*c^4 + (a^5*b - 4*a^3*b^3 + 3*a*b^5)*c^2)*e*cos(e*x + d) + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + c^8 - (a^2 - 3
*b^2)*c^6 - (a^4 + a^2*b^2 - 3*b^4)*c^4 + (a^6 - 3*a^4*b^2 + a^2*b^4 + b^6)*c^2)*e + 2*((b*c^7 - (2*a^2*b - 3*
b^3)*c^5 + (a^4*b - 4*a^2*b^3 + 3*b^5)*c^3 + (a^4*b^3 - 2*a^2*b^5 + b^7)*c)*e*cos(e*x + d) + (a*c^7 - (2*a^3 -
3*a*b^2)*c^5 + (a^5 - 4*a^3*b^2 + 3*a*b^4)*c^3 + (a^5*b^2 - 2*a^3*b^4 + a*b^6)*c)*e)*sin(e*x + d))

Sympy [F]

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

[In]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**(5/2),x)

[Out]

Integral((a + b*cos(d + e*x) + c*sin(d + e*x))**(-5/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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