3.401 \(\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^3} \, dx\)

Optimal. Leaf size=197 \[ \frac{\left (2 a^2+b^2+c^2\right ) \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{5/2}}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2} \]

[Out]

((2*a^2 + b^2 + c^2)*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(5/2)*e)
+ (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c
*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))

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Rubi [A]  time = 0.197937, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3129, 3153, 3124, 618, 204} \[ \frac{\left (2 a^2+b^2+c^2\right ) \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{5/2}}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^2 + b^2 + c^2)*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(5/2)*e)
+ (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c
*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))

Rule 3129

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 3153

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

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^3} \, dx &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^2}-\frac{\int \frac{-2 a+b \cos (d+e x)+c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^2} \, dx}{2 \left (a^2-b^2-c^2\right )}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{\left (2 a^2+b^2+c^2\right ) \int \frac{1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{2 \left (a^2-b^2-c^2\right )^2}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{\left (2 a^2+b^2+c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (a^2-b^2-c^2\right )^2 e}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{\left (2 \left (2 a^2+b^2+c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (a^2-b^2-c^2\right )^2 e}\\ &=\frac{\left (2 a^2+b^2+c^2\right ) \tan ^{-1}\left (\frac{c+(a-b) \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{5/2} e}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))}\\ \end{align*}

Mathematica [A]  time = 0.908594, size = 200, normalized size = 1.02 \[ \frac{\frac{a c+\left (b^2+c^2\right ) \sin (d+e x)}{b \left (-a^2+b^2+c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}-\frac{c \left (2 a^2+b^2+c^2\right )+3 a \left (b^2+c^2\right ) \sin (d+e x)}{b \left (-a^2+b^2+c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{2 \left (2 a^2+b^2+c^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{-a^2+b^2+c^2}}\right )}{\left (-a^2+b^2+c^2\right )^{5/2}}}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(2*a^2 + b^2 + c^2)*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5
/2) + (a*c + (b^2 + c^2)*Sin[d + e*x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(2*
a^2 + b^2 + c^2) + 3*a*(b^2 + c^2)*Sin[d + e*x])/(b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x])
))/(2*e)

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Maple [B]  time = 0.191, size = 3933, normalized size = 20. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^
4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*a^4-1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)
+a+b)^2*c^3/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*a^2-1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(
1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*b
^4-1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c^3/(a^4-2*a^2*b^2-2*a^2*c
^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*b^2-1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2
*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*b^4-2/e/(a*tan(1/2*d+1/2*e
*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)*t
an(1/2*d+1/2*e*x)^3*c^4-2/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c^5/(
a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2+1/e/(a*tan(1/2*d+1/2*e*x)^2-b*
tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)
*tan(1/2*d+1/2*e*x)*b^5+1/e/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b
)*tan(1/2*d+1/2*e*x)+2*c)/(a^2-b^2-c^2)^(1/2))*b^2+1/e/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-b^2-c^
2)^(1/2)*arctan(1/2*(2*(a-b)*tan(1/2*d+1/2*e*x)+2*c)/(a^2-b^2-c^2)^(1/2))*c^2+13/e/(a*tan(1/2*d+1/2*e*x)^2-b*t
an(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2
)*tan(1/2*d+1/2*e*x)^2*a^2*b^2-2/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^
2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*a*b*c^2-6/e/(a*tan(1/2*d+1/2*e*x)^2-b
*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b
^2)*tan(1/2*d+1/2*e*x)^2*a*b^3-12/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)
^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2*a^3*b-6/e/(a*tan(1/2*d+1
/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c^3/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)
/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2*a*b-3/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2
*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*a^2*b*c^2-7/e/(a*t
an(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^
2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*a*b^2*c^2+1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan
(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2*b^4-
1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c^3/(a^4-2*a^2*b^2-2*a^2*c^2+
b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2*b^2-4/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2
+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*
a^4*b+5/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^
2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*a^3*b^2+11/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e
*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*
e*x)*a^3*c^2+3/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2
*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*a^2*b^3-5/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d
+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*
d+1/2*e*x)*a*b^4-2/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b
^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*a*c^4-1/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2
*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/
2*d+1/2*e*x)*b^3*c^2-2/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a^4-2*a
^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)*b*c^4-3/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan
(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*
a^2*b^2-4/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-
2*a^2*c^2+b^4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*a^3*b+7/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c
*tan(1/2*d+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*a^2*b^2+5/e/
(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^
4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*a^2*c^2-2/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d
+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)*tan(1/2*d+1/2*e*x)^3*a*b^3-3/e/(a*tan(1/2*d
+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2/(a-b)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+
c^4)*tan(1/2*d+1/2*e*x)^3*b^2*c^2+2/e/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-b^2-c^2)^(1/2)*arctan(1
/2*(2*(a-b)*tan(1/2*d+1/2*e*x)+2*c)/(a^2-b^2-c^2)^(1/2))*a^2+4/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^
2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*
x)^2*a^4+7/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+2*c*tan(1/2*d+1/2*e*x)+a+b)^2*c^3/(a^4-2*a^2*b^2-2
*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)^2*a^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.45739, size = 4077, normalized size = 20.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(6*a*b*c^3 - 12*(a*b*c^3 - (a^3*b - a*b^3)*c)*cos(e*x + d)^2 - (2*a^4 + a^2*b^2 + c^4 + (3*a^2 + b^2)*c^2
+ (2*a^2*b^2 + b^4 - 2*a^2*c^2 - c^4)*cos(e*x + d)^2 + 2*(2*a^3*b + a*b^3 + a*b*c^2)*cos(e*x + d) + 2*(a*c^3
+ (2*a^3 + a*b^2)*c + (b*c^3 + (2*a^2*b + b^3)*c)*cos(e*x + d))*sin(e*x + d))*sqrt(-a^2 + b^2 + c^2)*log(-(a^2
*b^2 - 2*b^4 - c^4 - (a^2 + 3*b^2)*c^2 - (2*a^2*b^2 - b^4 - 2*a^2*c^2 + c^4)*cos(e*x + d)^2 - 2*(a*b^3 + a*b*c
^2)*cos(e*x + d) - 2*(a*b^2*c + a*c^3 - (b*c^3 - (2*a^2*b - b^3)*c)*cos(e*x + d))*sin(e*x + d) + 2*(2*a*b*c*co
s(e*x + d)^2 - a*b*c + (b^2*c + c^3)*cos(e*x + d) - (b^3 + b*c^2 + (a*b^2 - a*c^2)*cos(e*x + d))*sin(e*x + d))
*sqrt(-a^2 + b^2 + c^2))/(2*a*b*cos(e*x + d) + (b^2 - c^2)*cos(e*x + d)^2 + a^2 + c^2 + 2*(b*c*cos(e*x + d) +
a*c)*sin(e*x + d))) - 6*(a^3*b - a*b^3)*c + 2*(c^5 - (5*a^2 - 2*b^2)*c^3 + (4*a^4 - 5*a^2*b^2 + b^4)*c)*cos(e*
x + d) - 2*(4*a^4*b - 5*a^2*b^3 + b^5 + b*c^4 - (5*a^2*b - 2*b^3)*c^2 + 3*(a^3*b^2 - a*b^4 - a^3*c^2 + a*c^4)*
cos(e*x + d))*sin(e*x + d))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + c^8 - (3*a^2 - 2*b^2)*c^6 + 3*(a^4 - a^2
*b^2)*c^4 - (a^6 - 3*a^2*b^4 + 2*b^6)*c^2)*e*cos(e*x + d)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7 - a*b*c
^6 + 3*(a^3*b - a*b^3)*c^4 - 3*(a^5*b - 2*a^3*b^3 + a*b^5)*c^2)*e*cos(e*x + d) + (a^8 - 3*a^6*b^2 + 3*a^4*b^4
- a^2*b^6 - c^8 + (2*a^2 - 3*b^2)*c^6 + 3*(a^2*b^2 - b^4)*c^4 - (2*a^6 - 3*a^4*b^2 + b^6)*c^2)*e - 2*((b*c^7 -
3*(a^2*b - b^3)*c^5 + 3*(a^4*b - 2*a^2*b^3 + b^5)*c^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c)*e*cos(e*x +
d) + (a*c^7 - 3*(a^3 - a*b^2)*c^5 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^3 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c)
*e)*sin(e*x + d)), 1/2*(3*a*b*c^3 - 6*(a*b*c^3 - (a^3*b - a*b^3)*c)*cos(e*x + d)^2 + (2*a^4 + a^2*b^2 + c^4 +
(3*a^2 + b^2)*c^2 + (2*a^2*b^2 + b^4 - 2*a^2*c^2 - c^4)*cos(e*x + d)^2 + 2*(2*a^3*b + a*b^3 + a*b*c^2)*cos(e*x
+ d) + 2*(a*c^3 + (2*a^3 + a*b^2)*c + (b*c^3 + (2*a^2*b + b^3)*c)*cos(e*x + d))*sin(e*x + d))*sqrt(a^2 - b^2
- c^2)*arctan(-(a*b*cos(e*x + d) + a*c*sin(e*x + d) + b^2 + c^2)*sqrt(a^2 - b^2 - c^2)/((c^3 - (a^2 - b^2)*c)*
cos(e*x + d) + (a^2*b - b^3 - b*c^2)*sin(e*x + d))) - 3*(a^3*b - a*b^3)*c + (c^5 - (5*a^2 - 2*b^2)*c^3 + (4*a^
4 - 5*a^2*b^2 + b^4)*c)*cos(e*x + d) - (4*a^4*b - 5*a^2*b^3 + b^5 + b*c^4 - (5*a^2*b - 2*b^3)*c^2 + 3*(a^3*b^2
- a*b^4 - a^3*c^2 + a*c^4)*cos(e*x + d))*sin(e*x + d))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + c^8 - (3*a^2
- 2*b^2)*c^6 + 3*(a^4 - a^2*b^2)*c^4 - (a^6 - 3*a^2*b^4 + 2*b^6)*c^2)*e*cos(e*x + d)^2 + 2*(a^7*b - 3*a^5*b^3
+ 3*a^3*b^5 - a*b^7 - a*b*c^6 + 3*(a^3*b - a*b^3)*c^4 - 3*(a^5*b - 2*a^3*b^3 + a*b^5)*c^2)*e*cos(e*x + d) + (
a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - c^8 + (2*a^2 - 3*b^2)*c^6 + 3*(a^2*b^2 - b^4)*c^4 - (2*a^6 - 3*a^4*b^2
+ b^6)*c^2)*e - 2*((b*c^7 - 3*(a^2*b - b^3)*c^5 + 3*(a^4*b - 2*a^2*b^3 + b^5)*c^3 - (a^6*b - 3*a^4*b^3 + 3*a^
2*b^5 - b^7)*c)*e*cos(e*x + d) + (a*c^7 - 3*(a^3 - a*b^2)*c^5 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^3 - (a^7 - 3*a^5
*b^2 + 3*a^3*b^4 - a*b^6)*c)*e)*sin(e*x + d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.17864, size = 1204, normalized size = 6.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((pi*floor(1/2*(x*e + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*x*e + 1/2*d) - b*tan(1/2*x*e + 1/2*d)
+ c)/sqrt(a^2 - b^2 - c^2)))*(2*a^2 + b^2 + c^2)/((a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*sqrt(
a^2 - b^2 - c^2)) + (4*a^4*b*tan(1/2*x*e + 1/2*d)^3 - 11*a^3*b^2*tan(1/2*x*e + 1/2*d)^3 + 9*a^2*b^3*tan(1/2*x*
e + 1/2*d)^3 - a*b^4*tan(1/2*x*e + 1/2*d)^3 - b^5*tan(1/2*x*e + 1/2*d)^3 - 5*a^3*c^2*tan(1/2*x*e + 1/2*d)^3 +
7*a^2*b*c^2*tan(1/2*x*e + 1/2*d)^3 + a*b^2*c^2*tan(1/2*x*e + 1/2*d)^3 - 3*b^3*c^2*tan(1/2*x*e + 1/2*d)^3 + 2*a
*c^4*tan(1/2*x*e + 1/2*d)^3 - 2*b*c^4*tan(1/2*x*e + 1/2*d)^3 - 4*a^4*c*tan(1/2*x*e + 1/2*d)^2 + 12*a^3*b*c*tan
(1/2*x*e + 1/2*d)^2 - 13*a^2*b^2*c*tan(1/2*x*e + 1/2*d)^2 + 6*a*b^3*c*tan(1/2*x*e + 1/2*d)^2 - b^4*c*tan(1/2*x
*e + 1/2*d)^2 - 7*a^2*c^3*tan(1/2*x*e + 1/2*d)^2 + 6*a*b*c^3*tan(1/2*x*e + 1/2*d)^2 + b^2*c^3*tan(1/2*x*e + 1/
2*d)^2 + 2*c^5*tan(1/2*x*e + 1/2*d)^2 + 4*a^4*b*tan(1/2*x*e + 1/2*d) - 5*a^3*b^2*tan(1/2*x*e + 1/2*d) - 3*a^2*
b^3*tan(1/2*x*e + 1/2*d) + 5*a*b^4*tan(1/2*x*e + 1/2*d) - b^5*tan(1/2*x*e + 1/2*d) - 11*a^3*c^2*tan(1/2*x*e +
1/2*d) + 3*a^2*b*c^2*tan(1/2*x*e + 1/2*d) + 7*a*b^2*c^2*tan(1/2*x*e + 1/2*d) + b^3*c^2*tan(1/2*x*e + 1/2*d) +
2*a*c^4*tan(1/2*x*e + 1/2*d) + 2*b*c^4*tan(1/2*x*e + 1/2*d) - 4*a^4*c + 3*a^2*b^2*c + b^4*c + a^2*c^3 + b^2*c^
3)/((a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^4*c^2 + 4*a^3*b*c^2 - 4*a*b^3*c^2 + 2
*b^4*c^2 + a^2*c^4 - 2*a*b*c^4 + b^2*c^4)*(a*tan(1/2*x*e + 1/2*d)^2 - b*tan(1/2*x*e + 1/2*d)^2 + 2*c*tan(1/2*x
*e + 1/2*d) + a + b)^2))*e^(-1)