\(\int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx\) [414]

Optimal result

Integrand size = 22, antiderivative size = 186 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx=\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{\left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{\left (a^2-b^2-c^2\right ) e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \]

[Out]

2*(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))^(1/2)+2*(cos(1/2*d+1/2*e*x-1/2*arc
tan(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)/e/((a+b*cos(e*x+d)
+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 186, 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, 3198, 2732} \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx=\frac {2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]

[In]

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

[Out]

(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/((a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*Ell
ipticE[(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^2 - b^2 - c^2)*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])

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 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 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 {2 (c \cos (d+e x)-b \sin (d+e x))}{\left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {\int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{a^2-b^2-c^2} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{\left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {\sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \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}{\left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \\ & = \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{\left (a^2-b^2-c^2\right ) e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {2 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)}}{\left (a^2-b^2-c^2\right ) e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 6.44 (sec) , antiderivative size = 1540, normalized size of antiderivative = 8.28 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx=\frac {\sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \left (-\frac {2 \left (b^2+c^2\right )}{b c \left (-a^2+b^2+c^2\right )}+\frac {2 \left (a c+b^2 \sin (d+e x)+c^2 \sin (d+e x)\right )}{b \left (-a^2+b^2+c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}\right )}{e}-\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {a+\sqrt {1+\frac {b^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{\sqrt {1+\frac {b^2}{c^2}} \left (1-\frac {a}{\sqrt {1+\frac {b^2}{c^2}} c}\right ) c},-\frac {a+\sqrt {1+\frac {b^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{\sqrt {1+\frac {b^2}{c^2}} \left (-1-\frac {a}{\sqrt {1+\frac {b^2}{c^2}} c}\right ) c}\right ) \sec \left (d+e x+\arctan \left (\frac {b}{c}\right )\right ) \sqrt {\frac {c \sqrt {\frac {b^2+c^2}{c^2}}-c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{a+c \sqrt {\frac {b^2+c^2}{c^2}}}} \sqrt {a+c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {b^2+c^2}{c^2}}+c \sqrt {\frac {b^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {b}{c}\right )\right )}{-a+c \sqrt {\frac {b^2+c^2}{c^2}}}}}{\sqrt {1+\frac {b^2}{c^2}} c \left (-a^2+b^2+c^2\right ) e}-\frac {b^2 \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (-1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}-b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2+c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2+c^2}{b^2}}}}}-\frac {\frac {2 b \left (a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}}\right )}{c \left (-a^2+b^2+c^2\right ) e}-\frac {c \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )},-\frac {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \left (-1-\frac {a}{b \sqrt {1+\frac {c^2}{b^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}-b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2+c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2+c^2}{b^2}}+b \sqrt {\frac {b^2+c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2+c^2}{b^2}}}}}-\frac {\frac {2 b \left (a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )\right )}{b^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}{b \sqrt {1+\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1+\frac {c^2}{b^2}} \cos \left (d+e x-\arctan \left (\frac {c}{b}\right )\right )}}\right )}{\left (-a^2+b^2+c^2\right ) e} \]

[In]

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

[Out]

(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((-2*(b^2 + c^2))/(b*c*(-a^2 + b^2 + c^2)) + (2*(a*c + b^2*Sin[d +
e*x] + c^2*Sin[d + e*x]))/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e - (2*a*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]]]*S
qrt[(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)*e) - (b^2*(-((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 - Arc
Tan[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*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)*e) - (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 - Arc
Tan[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*Sqrt[1 + c^2/b^2]*Cos[d + e*x - Ar
cTan[c/b]]]))/((-a^2 + b^2 + c^2)*e)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2644\) vs. \(2(213)=426\).

Time = 2.44 (sec) , antiderivative size = 2645, normalized size of antiderivative = 14.22

method	result	size
default	\(\text {Expression too large to display}\)	\(2645\)

[In]

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(3/2),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+c^2)^(3/2)*(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-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*b^2+sin(e*x+d-arctan(-b,c))^2*c^2-a^2)/(sin(e*x+d-arctan(-b,c))^2*b^2+sin(e*x+d-arctan(-b,c))^2*c^2+2*si
n(e*x+d-arctan(-b,c))*a*(b^2+c^2)^(1/2)+a^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)-sin(e*x+d-arctan(-b,c))*(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+c^2)^(1/2)*(-b^2-c^2)*cos(e*x+d-arctan(-b,c))^2/(a^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)+a*(b^2+c^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)/(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*(-(b^2+c^2)^(3/2)+2*(b^2+c^2)^(1/2)*b^2+2*(b^2+c^2)^(1/2)*c^2)/(2*a^2-2*b^2-2*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
)/(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)))+1/2*(b^2+c^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*s
in(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)/a*EllipticPi((((b^2+c^2)^(1/2)*sin(e*x+d-ar
ctan(-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))+(b^2+c^2)*cos(e*x+d-arctan(-b,c))^2/(a^2-b^2-c^2)/(((b^2+c^2)^(1/2)*sin(e*x+d-a
rctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)+1/(a^2-b^2-c^2)*(b^2+c^2)^(1/2)*a*(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)*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/(a^2-b^2-c^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
)*((-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)))-(1/2*b^2+1/2*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-arc
tan(-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)/a*EllipticP
i((((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 = 0.15 (sec) , antiderivative size = 1737, normalized size of antiderivative = 9.34 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/3*((sqrt(2)*(-I*a*b^2 - a*b*c)*cos(e*x + d) + sqrt(2)*(-I*a*b*c - a*c^2)*sin(e*x + d) + sqrt(2)*(-I*a^2*b -
a^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)) + (sqrt(2)*(I*a*b
^2 - a*b*c)*cos(e*x + d) + sqrt(2)*(I*a*b*c - a*c^2)*sin(e*x + d) + sqrt(2)*(I*a^2*b - a^2*c))*sqrt(b - I*c)*w
eierstrassPInverse(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)) - 3*(sqrt(2)*(-I*b^3 - I*b*c^2)*cos(e*x
+ d) + sqrt(2)*(-I*b^2*c - I*c^3)*sin(e*x + d) + sqrt(2)*(-I*a*b^2 - I*a*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), -8/2
7*(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))) - 3*(sqrt(2)*(I*b^3 + I*b*c^2)*cos(e*x + d) + sqrt(2)*(I*b^2*c + I*c^3)*sin(e*x + d) +
sqrt(2)*(I*a*b^2 + I*a*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*((b^2*c + c^3)*cos(e*x
+ d) - (b^3 + b*c^2)*sin(e*x + d))*sqrt(b*cos(e*x + d) + c*sin(e*x + d) + a))/((a^2*b^3 - b^5 - b*c^4 + (a^2*
b - 2*b^3)*c^2)*e*cos(e*x + d) - (c^5 - (a^2 - 2*b^2)*c^3 - (a^2*b^2 - b^4)*c)*e*sin(e*x + d) + (a^3*b^2 - a*b
^4 - a*c^4 + (a^3 - 2*a*b^2)*c^2)*e)

Sympy [F]

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

[In]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**(3/2),x)

[Out]

Integral((a + b*cos(d + e*x) + c*sin(d + e*x))**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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