3.5.0 ∫ 1 ___________________ (a+b cos(d+ex)+c sin(d+ex))4 dx [402]

Integrand size = 20, antiderivative size = 292 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\frac {a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \arctan \left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{7/2} e}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))} \]

a*(2*a^2+3*b^2+3*c^2)*arctan((c+(a-b)*tan(1/2*e*x+1/2*d))/(a^2-b^2-c^2)^(1/2))/(a^2-b^2-c^2)^(7/2)/e+1/3*(c*co

s(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+5/6*(a*c*cos(e*x+d)-a*b*sin(e*x+d))/(a^

2-b^2-c^2)^2/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^2+1/6*(c*(11*a^2+4*b^2+4*c^2)*cos(e*x+d)-b*(11*a^2+4*b^2+4*c^2)*s

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

Rubi [A] (veriﬁed)

Time = 0.43 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3208, 3235, 3232, 3203, 632, 210} \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\frac {a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} (d+e x)\right )+c}{\sqrt {a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{7/2}}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 e \left (a^2-b^2-c^2\right )^3 (a+b \cos (d+e x)+c \sin (d+e x))}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 e \left (a^2-b^2-c^2\right )^2 (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3} \]

Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

(a*(2*a^2 + 3*(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)^(7

/2)*e) + (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) + (

5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(6*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (

c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x] - b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*(a^2 - b^2 - c^2)^3*e*(a

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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]

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]

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

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]

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 {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}-\frac {\int \frac {-3 a+2 b \cos (d+e x)+2 c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^3} \, dx}{3 \left (a^2-b^2-c^2\right )} \\ & = \frac {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}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {2 \left (3 a^2+2 \left (b^2+c^2\right )\right )-5 a b \cos (d+e x)-5 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^2} \, dx}{6 \left (a^2-b^2-c^2\right )^2} \\ & = \frac {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}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac {\left (a \left (2 a^2+3 \left (b^2+c^2\right )\right )\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 )^3} \\ & = \frac {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}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac {\left (a \left (2 a^2+3 \left (b^2+c^2\right )\right )\right ) \text {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 )^3 e} \\ & = \frac {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}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))}-\frac {\left (2 a \left (2 a^2+3 \left (b^2+c^2\right )\right )\right ) \text {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 )^3 e} \\ & = \frac {a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \arctan \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 )^{7/2} e}+\frac {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}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(606\) vs. \(2(292)=584\).

Time = 1.71 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\frac {\frac {24 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \text {arctanh}\left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2+c^2}}\right )}{\left (-a^2+b^2+c^2\right )^{7/2}}+\frac {44 a^5 c+82 a^3 b^2 c+24 a b^4 c+82 a^3 c^3+48 a b^2 c^3+24 a c^5+30 a^2 b c \left (2 a^2+3 \left (b^2+c^2\right )\right ) \cos (d+e x)-6 a c \left (-2 b^4+2 b^2 c^2+4 c^4+a^2 \left (7 b^2+11 c^2\right )\right ) \cos (2 (d+e x))-22 a^2 b^3 c \cos (3 (d+e x))-8 b^5 c \cos (3 (d+e x))-22 a^2 b c^3 \cos (3 (d+e x))-16 b^3 c^3 \cos (3 (d+e x))-8 b c^5 \cos (3 (d+e x))+72 a^4 b^2 \sin (d+e x)-9 a^2 b^4 \sin (d+e x)+12 b^6 \sin (d+e x)+132 a^4 c^2 \sin (d+e x)+72 a^2 b^2 c^2 \sin (d+e x)+36 b^4 c^2 \sin (d+e x)+81 a^2 c^4 \sin (d+e x)+36 b^2 c^4 \sin (d+e x)+12 c^6 \sin (d+e x)+54 a^3 b^3 \sin (2 (d+e x))+6 a b^5 \sin (2 (d+e x))+78 a^3 b c^2 \sin (2 (d+e x))+48 a b^3 c^2 \sin (2 (d+e x))+42 a b c^4 \sin (2 (d+e x))+11 a^2 b^4 \sin (3 (d+e x))+4 b^6 \sin (3 (d+e x))+4 b^4 c^2 \sin (3 (d+e x))-11 a^2 c^4 \sin (3 (d+e x))-4 b^2 c^4 \sin (3 (d+e x))-4 c^6 \sin (3 (d+e x))}{b \left (-a^2+b^2+c^2\right )^3 (a+b \cos (d+e x)+c \sin (d+e x))^3}}{24 e} \]

Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]

((24*a*(2*a^2 + 3*(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)^(7/2) + (44*a^5*c + 82*a^3*b^2*c + 24*a*b^4*c + 82*a^3*c^3 + 48*a*b^2*c^3 + 24*a*c^5 + 30*a^2*b*c*(2*a^2 +

3*(b^2 + c^2))*Cos[d + e*x] - 6*a*c*(-2*b^4 + 2*b^2*c^2 + 4*c^4 + a^2*(7*b^2 + 11*c^2))*Cos[2*(d + e*x)] - 22

*a^2*b^3*c*Cos[3*(d + e*x)] - 8*b^5*c*Cos[3*(d + e*x)] - 22*a^2*b*c^3*Cos[3*(d + e*x)] - 16*b^3*c^3*Cos[3*(d +

e*x)] - 8*b*c^5*Cos[3*(d + e*x)] + 72*a^4*b^2*Sin[d + e*x] - 9*a^2*b^4*Sin[d + e*x] + 12*b^6*Sin[d + e*x] + 1

32*a^4*c^2*Sin[d + e*x] + 72*a^2*b^2*c^2*Sin[d + e*x] + 36*b^4*c^2*Sin[d + e*x] + 81*a^2*c^4*Sin[d + e*x] + 36

*b^2*c^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 54*a^3*b^3*Sin[2*(d + e*x)] + 6*a*b^5*Sin[2*(d + e*x)] + 78*a^3*

b*c^2*Sin[2*(d + e*x)] + 48*a*b^3*c^2*Sin[2*(d + e*x)] + 42*a*b*c^4*Sin[2*(d + e*x)] + 11*a^2*b^4*Sin[3*(d + e

*x)] + 4*b^6*Sin[3*(d + e*x)] + 4*b^4*c^2*Sin[3*(d + e*x)] - 11*a^2*c^4*Sin[3*(d + e*x)] - 4*b^2*c^4*Sin[3*(d

+ e*x)] - 4*c^6*Sin[3*(d + e*x)])/(b*(-a^2 + b^2 + c^2)^3*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3))/(24*e)

Maple [C] (veriﬁed)

Time = 4.41 (sec) , antiderivative size = 1655, normalized size of antiderivative = 5.67


method	result	size

risch	\(\text {Expression too large to display}\)	\(1655\)
derivativedivides	\(\text {Expression too large to display}\)	\(1656\)
default	\(\text {Expression too large to display}\)	\(1656\)

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x,method=_RETURNVERBOSE)

1/3*I*(-12*I*b*c^4*exp(2*I*(e*x+d))-82*I*a^3*b^2*exp(3*I*(e*x+d))-24*I*a*b^4*exp(3*I*(e*x+d))+120*a^3*b*c*exp(

I*(e*x+d))+8*I*b^3*c^2+12*I*b*c^4+30*a*b^3*c*exp(I*(e*x+d))-24*I*b^3*c^2*exp(2*I*(e*x+d))+33*a^2*b^2*c+60*I*a^

3*c^2*exp(I*(e*x+d))+15*I*a*c^4*exp(I*(e*x+d))-4*c^5-30*a^4*c*exp(4*I*(e*x+d))+102*a^4*c*exp(2*I*(e*x+d))-45*a

^2*c^3*exp(4*I*(e*x+d))+36*a^2*c^3*exp(2*I*(e*x+d))-6*I*a^3*b^2*exp(5*I*(e*x+d))-30*I*a^4*b*exp(4*I*(e*x+d))-2

4*I*a*c^4*exp(3*I*(e*x+d))-44*I*exp(3*I*(e*x+d))*a^5-12*I*b^5*exp(2*I*(e*x+d))-48*I*a*b^2*c^2*exp(3*I*(e*x+d))

-11*a^2*c^3+9*I*a*c^4*exp(5*I*(e*x+d))-60*I*a^3*b^2*exp(I*(e*x+d))-15*I*a*b^4*exp(I*(e*x+d))+24*b^2*c^3*exp(2*

I*(e*x+d))+12*b^4*c*exp(2*I*(e*x+d))-11*I*a^2*b^3-4*I*b^5+12*c^5*exp(2*I*(e*x+d))+12*b^4*c+8*b^2*c^3+6*I*a^3*c

^2*exp(5*I*(e*x+d))+33*I*a^2*b*c^2-82*I*a^3*c^2*exp(3*I*(e*x+d))-9*I*a*b^4*exp(5*I*(e*x+d))-36*I*a^2*b^3*exp(2

*I*(e*x+d))-12*a^3*b*c*exp(5*I*(e*x+d))-36*I*a^2*b*c^2*exp(2*I*(e*x+d))+36*a^2*b^2*c*exp(2*I*(e*x+d))-102*I*a^

4*b*exp(2*I*(e*x+d))-45*I*a^2*b*c^2*exp(4*I*(e*x+d))-45*a^2*b^2*c*exp(4*I*(e*x+d))+30*a*b*c^3*exp(I*(e*x+d))-4

5*I*a^2*b^3*exp(4*I*(e*x+d))-18*a*b*c^3*exp(5*I*(e*x+d))-18*a*b^3*c*exp(5*I*(e*x+d)))/(c*exp(2*I*(e*x+d))+I*b*

exp(2*I*(e*x+d))-c+2*I*a*exp(I*(e*x+d))+I*b)^3/(-a^2+b^2+c^2)^3/e-1/(-a^2+b^2+c^2)^(1/2)*a^3/(a^2-b^2-c^2)^3/e

*ln(exp(I*(e*x+d))+(I*a*c*(-a^2+b^2+c^2)^(1/2)+I*a^2*b-I*b^3-I*b*c^2+a*b*(-a^2+b^2+c^2)^(1/2)-a^2*c+b^2*c+c^3)

/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))-3/2/(-a^2+b^2+c^2)^(1/2)*a/(a^2-b^2-c^2)^3/e*ln(exp(I*(e*x+d))+(I*a*c*(-a^2+b

^2+c^2)^(1/2)+I*a^2*b-I*b^3-I*b*c^2+a*b*(-a^2+b^2+c^2)^(1/2)-a^2*c+b^2*c+c^3)/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))*

b^2-3/2/(-a^2+b^2+c^2)^(1/2)*a/(a^2-b^2-c^2)^3/e*ln(exp(I*(e*x+d))+(I*a*c*(-a^2+b^2+c^2)^(1/2)+I*a^2*b-I*b^3-I

*b*c^2+a*b*(-a^2+b^2+c^2)^(1/2)-a^2*c+b^2*c+c^3)/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))*c^2+1/(-a^2+b^2+c^2)^(1/2)*a^

3/(a^2-b^2-c^2)^3/e*ln(exp(I*(e*x+d))+(I*a*c*(-a^2+b^2+c^2)^(1/2)-I*a^2*b+I*b^3+I*b*c^2+a*b*(-a^2+b^2+c^2)^(1/

2)+a^2*c-b^2*c-c^3)/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))+3/2/(-a^2+b^2+c^2)^(1/2)*a/(a^2-b^2-c^2)^3/e*ln(exp(I*(e*x

+d))+(I*a*c*(-a^2+b^2+c^2)^(1/2)-I*a^2*b+I*b^3+I*b*c^2+a*b*(-a^2+b^2+c^2)^(1/2)+a^2*c-b^2*c-c^3)/(b^2+c^2)/(-a

^2+b^2+c^2)^(1/2))*b^2+3/2/(-a^2+b^2+c^2)^(1/2)*a/(a^2-b^2-c^2)^3/e*ln(exp(I*(e*x+d))+(I*a*c*(-a^2+b^2+c^2)^(1

/2)-I*a^2*b+I*b^3+I*b*c^2+a*b*(-a^2+b^2+c^2)^(1/2)+a^2*c-b^2*c-c^3)/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))*c^2

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1940 vs. \(2 (284) = 568\).

Time = 0.47 (sec) , antiderivative size = 4069, normalized size of antiderivative = 13.93 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="fricas")

[1/12*(6*a*b*c^5 + 12*(4*a^3*b + a*b^3)*c^3 + 2*(4*c^7 + (7*a^2 - 4*b^2)*c^5 - (11*a^4 + 14*a^2*b^2 + 20*b^4)*

c^3 + 3*(11*a^4*b^2 - 7*a^2*b^4 - 4*b^6)*c)*cos(e*x + d)^3 - 12*(a*b*c^5 + 2*(4*a^3*b + a*b^3)*c^3 - (9*a^5*b

- 8*a^3*b^3 - a*b^5)*c)*cos(e*x + d)^2 + 3*(2*a^6 + 3*a^4*b^2 + 9*a^2*c^4 + (2*a^3*b^3 + 3*a*b^5 - 9*a*b*c^4 -

6*(a^3*b + a*b^3)*c^2)*cos(e*x + d)^3 + 9*(a^4 + a^2*b^2)*c^2 + 3*(2*a^4*b^2 + 3*a^2*b^4 - 2*a^4*c^2 - 3*a^2*

c^4)*cos(e*x + d)^2 + 3*(2*a^5*b + 3*a^3*b^3 + 3*a*b*c^4 + (5*a^3*b + 3*a*b^3)*c^2)*cos(e*x + d) + (3*a*c^5 +

(11*a^3 + 3*a*b^2)*c^3 - (3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 - 3*(2*a^3*b^2 + 3*a*b^4)*c)*cos(e*x + d)^2 + 3*(2*a

^5 + 3*a^3*b^2)*c + 6*(3*a^2*b*c^3 + (2*a^4*b + 3*a^2*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*cos(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))*si

n(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*(9*a^5*b - 8*a^3*b^3 - a*b^5)*c - 6*(2*b^2*c^5 + 2*c^7 + (4*a^4 - 7*a^2*b^2

- 2*b^4)*c^3 - (6*a^6 - 15*a^4*b^2 + 7*a^2*b^4 + 2*b^6)*c)*cos(e*x + d) - 2*(18*a^6*b - 23*a^4*b^3 + 7*a^2*b^

5 - 2*b^7 - 14*b^3*c^4 - 6*b*c^6 - (12*a^4*b - 7*a^2*b^3 + 10*b^5)*c^2 + (11*a^4*b^3 - 7*a^2*b^5 - 4*b^7 + 12*

b*c^6 + (21*a^2*b + 20*b^3)*c^4 - (33*a^4*b - 14*a^2*b^3 - 4*b^5)*c^2)*cos(e*x + d)^2 + 3*(9*a^5*b^2 - 8*a^3*b

^4 - a*b^6 + a*c^6 + (8*a^3 + a*b^2)*c^4 - (9*a^5 + a*b^4)*c^2)*cos(e*x + d))*sin(e*x + d))/((a^8*b^3 - 4*a^6*

b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11 - 3*b*c^10 + (12*a^2*b - 11*b^3)*c^8 - 2*(9*a^4*b - 16*a^2*b^3 + 7*b^5)*c^6

+ 6*(2*a^6*b - 5*a^4*b^3 + 4*a^2*b^5 - b^7)*c^4 - (3*a^8*b - 8*a^6*b^3 + 6*a^4*b^5 - b^9)*c^2)*e*cos(e*x + d)

^3 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10 - a*c^10 + (4*a^3 - 3*a*b^2)*c^8 - 2*(3*a^5 - 4*a

^3*b^2 + a*b^4)*c^6 + 2*(2*a^7 - 3*a^5*b^2 + a*b^6)*c^4 - (a^9 - 6*a^5*b^4 + 8*a^3*b^6 - 3*a*b^8)*c^2)*e*cos(e

*x + d)^2 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9 + b*c^10 - (3*a^2*b - 4*b^3)*c^8 + 2*(a^4*

b - 4*a^2*b^3 + 3*b^5)*c^6 + 2*(a^6*b - 3*a^2*b^5 + 2*b^7)*c^4 - (3*a^8*b - 8*a^6*b^3 + 6*a^4*b^5 - b^9)*c^2)*

e*cos(e*x + d) + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8 + 3*a*c^10 - (11*a^3 - 12*a*b^2)*c^8 + 2*

(7*a^5 - 16*a^3*b^2 + 9*a*b^4)*c^6 - 6*(a^7 - 4*a^5*b^2 + 5*a^3*b^4 - 2*a*b^6)*c^4 - (a^9 - 6*a^5*b^4 + 8*a^3*

b^6 - 3*a*b^8)*c^2)*e - ((c^11 - (4*a^2 - b^2)*c^9 + 6*(a^4 - b^4)*c^7 - 2*(2*a^6 + 3*a^4*b^2 - 12*a^2*b^4 + 7

*b^6)*c^5 + (a^8 + 8*a^6*b^2 - 30*a^4*b^4 + 32*a^2*b^6 - 11*b^8)*c^3 - 3*(a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*

a^2*b^8 + b^10)*c)*e*cos(e*x + d)^2 - 6*(a*b*c^9 - 4*(a^3*b - a*b^3)*c^7 + 6*(a^5*b - 2*a^3*b^3 + a*b^5)*c^5 -

4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*c^3 + (a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*c)*e*cos(

e*x + d) - (c^11 - (a^2 - 4*b^2)*c^9 - 6*(a^4 - b^4)*c^7 + 2*(7*a^6 - 12*a^4*b^2 + 3*a^2*b^4 + 2*b^6)*c^5 - (1

1*a^8 - 32*a^6*b^2 + 30*a^4*b^4 - 8*a^2*b^6 - b^8)*c^3 + 3*(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8

)*c)*e)*sin(e*x + d)), 1/6*(3*a*b*c^5 + 6*(4*a^3*b + a*b^3)*c^3 + (4*c^7 + (7*a^2 - 4*b^2)*c^5 - (11*a^4 + 14*

a^2*b^2 + 20*b^4)*c^3 + 3*(11*a^4*b^2 - 7*a^2*b^4 - 4*b^6)*c)*cos(e*x + d)^3 - 6*(a*b*c^5 + 2*(4*a^3*b + a*b^3

)*c^3 - (9*a^5*b - 8*a^3*b^3 - a*b^5)*c)*cos(e*x + d)^2 + 3*(2*a^6 + 3*a^4*b^2 + 9*a^2*c^4 + (2*a^3*b^3 + 3*a*

b^5 - 9*a*b*c^4 - 6*(a^3*b + a*b^3)*c^2)*cos(e*x + d)^3 + 9*(a^4 + a^2*b^2)*c^2 + 3*(2*a^4*b^2 + 3*a^2*b^4 - 2

*a^4*c^2 - 3*a^2*c^4)*cos(e*x + d)^2 + 3*(2*a^5*b + 3*a^3*b^3 + 3*a*b*c^4 + (5*a^3*b + 3*a*b^3)*c^2)*cos(e*x +

d) + (3*a*c^5 + (11*a^3 + 3*a*b^2)*c^3 - (3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 - 3*(2*a^3*b^2 + 3*a*b^4)*c)*cos(e*

x + d)^2 + 3*(2*a^5 + 3*a^3*b^2)*c + 6*(3*a^2*b*c^3 + (2*a^4*b + 3*a^2*b^3)*c)*cos(e*x + d))*sin(e*x + d))*sqr

t(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*(9*a^5*b - 8*a^3*b^3 - a*b^5)*c - 3*(2*b^2

*c^5 + 2*c^7 + (4*a^4 - 7*a^2*b^2 - 2*b^4)*c^3 - (6*a^6 - 15*a^4*b^2 + 7*a^2*b^4 + 2*b^6)*c)*cos(e*x + d) - (1

8*a^6*b - 23*a^4*b^3 + 7*a^2*b^5 - 2*b^7 - 14*b^3*c^4 - 6*b*c^6 - (12*a^4*b - 7*a^2*b^3 + 10*b^5)*c^2 + (11*a^

4*b^3 - 7*a^2*b^5 - 4*b^7 + 12*b*c^6 + (21*a^2*b + 20*b^3)*c^4 - (33*a^4*b - 14*a^2*b^3 - 4*b^5)*c^2)*cos(e*x

+ d)^2 + 3*(9*a^5*b^2 - 8*a^3*b^4 - a*b^6 + a*c^6 + (8*a^3 + a*b^2)*c^4 - (9*a^5 + a*b^4)*c^2)*cos(e*x + d))*s

in(e*x + d))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11 - 3*b*c^10 + (12*a^2*b - 11*b^3)*c^8 - 2*(9*

a^4*b - 16*a^2*b^3 + 7*b^5)*c^6 + 6*(2*a^6*b - 5*a^4*b^3 + 4*a^2*b^5 - b^7)*c^4 - (3*a^8*b - 8*a^6*b^3 + 6*a^4

*b^5 - b^9)*c^2)*e*cos(e*x + d)^3 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10 - a*c^10 + (4*a^3

- 3*a*b^2)*c^8 - 2*(3*a^5 - 4*a^3*b^2 + a*b^4)*c^6 + 2*(2*a^7 - 3*a^5*b^2 + a*b^6)*c^4 - (a^9 - 6*a^5*b^4 + 8*

a^3*b^6 - 3*a*b^8)*c^2)*e*cos(e*x + d)^2 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9 + b*c^10 -

(3*a^2*b - 4*b^3)*c^8 + 2*(a^4*b - 4*a^2*b^3 + 3*b^5)*c^6 + 2*(a^6*b - 3*a^2*b^5 + 2*b^7)*c^4 - (3*a^8*b - 8*a

^6*b^3 + 6*a^4*b^5 - b^9)*c^2)*e*cos(e*x + d) + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8 + 3*a*c^10

- (11*a^3 - 12*a*b^2)*c^8 + 2*(7*a^5 - 16*a^3*b^2 + 9*a*b^4)*c^6 - 6*(a^7 - 4*a^5*b^2 + 5*a^3*b^4 - 2*a*b^6)*

c^4 - (a^9 - 6*a^5*b^4 + 8*a^3*b^6 - 3*a*b^8)*c^2)*e - ((c^11 - (4*a^2 - b^2)*c^9 + 6*(a^4 - b^4)*c^7 - 2*(2*a

^6 + 3*a^4*b^2 - 12*a^2*b^4 + 7*b^6)*c^5 + (a^8 + 8*a^6*b^2 - 30*a^4*b^4 + 32*a^2*b^6 - 11*b^8)*c^3 - 3*(a^8*b

^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*c)*e*cos(e*x + d)^2 - 6*(a*b*c^9 - 4*(a^3*b - a*b^3)*c^7 + 6*(a

^5*b - 2*a^3*b^3 + a*b^5)*c^5 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*c^3 + (a^9*b - 4*a^7*b^3 + 6*a^5*b^5

- 4*a^3*b^7 + a*b^9)*c)*e*cos(e*x + d) - (c^11 - (a^2 - 4*b^2)*c^9 - 6*(a^4 - b^4)*c^7 + 2*(7*a^6 - 12*a^4*b^

2 + 3*a^2*b^4 + 2*b^6)*c^5 - (11*a^8 - 32*a^6*b^2 + 30*a^4*b^4 - 8*a^2*b^6 - b^8)*c^3 + 3*(a^10 - 4*a^8*b^2 +

6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*c)*e)*sin(e*x + d))]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Timed out} \]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**4,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Exception raised: ValueError} \]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="maxima")

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2+b^2-a^2>0)', see `assume?`

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2558 vs. \(2 (284) = 568\).

Time = 0.42 (sec) , antiderivative size = 2558, normalized size of antiderivative = 8.76 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \]

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x, algorithm="giac")

-1/3*(3*(2*a^3 + 3*a*b^2 + 3*a*c^2)*(pi*floor(1/2*(e*x + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*e*x

+ 1/2*d) - b*tan(1/2*e*x + 1/2*d) + c)/sqrt(a^2 - b^2 - c^2)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^

2 + 6*a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 - c^6)*sqrt(a^2 - b^2 - c^2)) + (18*a^7*b*tan(1/2*e*x +

1/2*d)^5 - 81*a^6*b^2*tan(1/2*e*x + 1/2*d)^5 + 141*a^5*b^3*tan(1/2*e*x + 1/2*d)^5 - 120*a^4*b^4*tan(1/2*e*x +

1/2*d)^5 + 60*a^3*b^5*tan(1/2*e*x + 1/2*d)^5 - 33*a^2*b^6*tan(1/2*e*x + 1/2*d)^5 + 21*a*b^7*tan(1/2*e*x + 1/2*

d)^5 - 6*b^8*tan(1/2*e*x + 1/2*d)^5 - 27*a^6*c^2*tan(1/2*e*x + 1/2*d)^5 + 81*a^5*b*c^2*tan(1/2*e*x + 1/2*d)^5

- 72*a^4*b^2*c^2*tan(1/2*e*x + 1/2*d)^5 + 18*a^3*b^3*c^2*tan(1/2*e*x + 1/2*d)^5 - 27*a^2*b^4*c^2*tan(1/2*e*x +

1/2*d)^5 + 45*a*b^5*c^2*tan(1/2*e*x + 1/2*d)^5 - 18*b^6*c^2*tan(1/2*e*x + 1/2*d)^5 + 18*a^4*c^4*tan(1/2*e*x +

1/2*d)^5 - 36*a^3*b*c^4*tan(1/2*e*x + 1/2*d)^5 + 36*a*b^3*c^4*tan(1/2*e*x + 1/2*d)^5 - 18*b^4*c^4*tan(1/2*e*x

+ 1/2*d)^5 - 6*a^2*c^6*tan(1/2*e*x + 1/2*d)^5 + 12*a*b*c^6*tan(1/2*e*x + 1/2*d)^5 - 6*b^2*c^6*tan(1/2*e*x + 1

/2*d)^5 - 18*a^7*c*tan(1/2*e*x + 1/2*d)^4 + 108*a^6*b*c*tan(1/2*e*x + 1/2*d)^4 - 261*a^5*b^2*c*tan(1/2*e*x + 1

/2*d)^4 + 336*a^4*b^3*c*tan(1/2*e*x + 1/2*d)^4 - 264*a^3*b^4*c*tan(1/2*e*x + 1/2*d)^4 + 144*a^2*b^5*c*tan(1/2*

e*x + 1/2*d)^4 - 57*a*b^6*c*tan(1/2*e*x + 1/2*d)^4 + 12*b^7*c*tan(1/2*e*x + 1/2*d)^4 - 81*a^5*c^3*tan(1/2*e*x

+ 1/2*d)^4 + 216*a^4*b*c^3*tan(1/2*e*x + 1/2*d)^4 - 198*a^3*b^2*c^3*tan(1/2*e*x + 1/2*d)^4 + 108*a^2*b^3*c^3*t

an(1/2*e*x + 1/2*d)^4 - 81*a*b^4*c^3*tan(1/2*e*x + 1/2*d)^4 + 36*b^5*c^3*tan(1/2*e*x + 1/2*d)^4 + 36*a^3*c^5*t

an(1/2*e*x + 1/2*d)^4 - 36*a^2*b*c^5*tan(1/2*e*x + 1/2*d)^4 - 36*a*b^2*c^5*tan(1/2*e*x + 1/2*d)^4 + 36*b^3*c^5

*tan(1/2*e*x + 1/2*d)^4 - 12*a*c^7*tan(1/2*e*x + 1/2*d)^4 + 12*b*c^7*tan(1/2*e*x + 1/2*d)^4 + 36*a^7*b*tan(1/2

*e*x + 1/2*d)^3 - 108*a^6*b^2*tan(1/2*e*x + 1/2*d)^3 + 76*a^5*b^3*tan(1/2*e*x + 1/2*d)^3 + 60*a^4*b^4*tan(1/2*

e*x + 1/2*d)^3 - 100*a^3*b^5*tan(1/2*e*x + 1/2*d)^3 + 44*a^2*b^6*tan(1/2*e*x + 1/2*d)^3 - 12*a*b^7*tan(1/2*e*x

+ 1/2*d)^3 + 4*b^8*tan(1/2*e*x + 1/2*d)^3 - 108*a^6*c^2*tan(1/2*e*x + 1/2*d)^3 + 240*a^5*b*c^2*tan(1/2*e*x +

1/2*d)^3 - 162*a^4*b^2*c^2*tan(1/2*e*x + 1/2*d)^3 + 122*a^3*b^3*c^2*tan(1/2*e*x + 1/2*d)^3 - 174*a^2*b^4*c^2*t

an(1/2*e*x + 1/2*d)^3 + 78*a*b^5*c^2*tan(1/2*e*x + 1/2*d)^3 + 4*b^6*c^2*tan(1/2*e*x + 1/2*d)^3 - 42*a^4*c^4*ta

n(1/2*e*x + 1/2*d)^3 + 162*a^3*b*c^4*tan(1/2*e*x + 1/2*d)^3 - 210*a^2*b^2*c^4*tan(1/2*e*x + 1/2*d)^3 + 102*a*b

^3*c^4*tan(1/2*e*x + 1/2*d)^3 - 12*b^4*c^4*tan(1/2*e*x + 1/2*d)^3 + 8*a^2*c^6*tan(1/2*e*x + 1/2*d)^3 + 12*a*b*

c^6*tan(1/2*e*x + 1/2*d)^3 - 20*b^2*c^6*tan(1/2*e*x + 1/2*d)^3 - 8*c^8*tan(1/2*e*x + 1/2*d)^3 - 36*a^7*c*tan(1

/2*e*x + 1/2*d)^2 + 108*a^6*b*c*tan(1/2*e*x + 1/2*d)^2 - 108*a^5*b^2*c*tan(1/2*e*x + 1/2*d)^2 + 12*a^4*b^3*c*t

an(1/2*e*x + 1/2*d)^2 + 84*a^3*b^4*c*tan(1/2*e*x + 1/2*d)^2 - 108*a^2*b^5*c*tan(1/2*e*x + 1/2*d)^2 + 60*a*b^6*

c*tan(1/2*e*x + 1/2*d)^2 - 12*b^7*c*tan(1/2*e*x + 1/2*d)^2 - 120*a^5*c^3*tan(1/2*e*x + 1/2*d)^2 + 132*a^4*b*c^

3*tan(1/2*e*x + 1/2*d)^2 + 42*a^3*b^2*c^3*tan(1/2*e*x + 1/2*d)^2 - 36*a^2*b^3*c^3*tan(1/2*e*x + 1/2*d)^2 + 18*

a*b^4*c^3*tan(1/2*e*x + 1/2*d)^2 - 36*b^5*c^3*tan(1/2*e*x + 1/2*d)^2 + 18*a^3*c^5*tan(1/2*e*x + 1/2*d)^2 + 72*

a^2*b*c^5*tan(1/2*e*x + 1/2*d)^2 - 54*a*b^2*c^5*tan(1/2*e*x + 1/2*d)^2 - 36*b^3*c^5*tan(1/2*e*x + 1/2*d)^2 - 1

2*a*c^7*tan(1/2*e*x + 1/2*d)^2 - 12*b*c^7*tan(1/2*e*x + 1/2*d)^2 + 18*a^7*b*tan(1/2*e*x + 1/2*d) - 27*a^6*b^2*

tan(1/2*e*x + 1/2*d) - 21*a^5*b^3*tan(1/2*e*x + 1/2*d) + 48*a^4*b^4*tan(1/2*e*x + 1/2*d) - 12*a^3*b^5*tan(1/2*

e*x + 1/2*d) - 15*a^2*b^6*tan(1/2*e*x + 1/2*d) + 15*a*b^7*tan(1/2*e*x + 1/2*d) - 6*b^8*tan(1/2*e*x + 1/2*d) -

81*a^6*c^2*tan(1/2*e*x + 1/2*d) + 27*a^5*b*c^2*tan(1/2*e*x + 1/2*d) + 90*a^4*b^2*c^2*tan(1/2*e*x + 1/2*d) + 9*

a^2*b^4*c^2*tan(1/2*e*x + 1/2*d) - 27*a*b^5*c^2*tan(1/2*e*x + 1/2*d) - 18*b^6*c^2*tan(1/2*e*x + 1/2*d) + 12*a^

4*c^4*tan(1/2*e*x + 1/2*d) + 42*a^3*b*c^4*tan(1/2*e*x + 1/2*d) + 18*a^2*b^2*c^4*tan(1/2*e*x + 1/2*d) - 54*a*b^

3*c^4*tan(1/2*e*x + 1/2*d) - 18*b^4*c^4*tan(1/2*e*x + 1/2*d) - 6*a^2*c^6*tan(1/2*e*x + 1/2*d) - 12*a*b*c^6*tan

(1/2*e*x + 1/2*d) - 6*b^2*c^6*tan(1/2*e*x + 1/2*d) - 18*a^7*c + 21*a^5*b^2*c + 12*a^3*b^4*c - 15*a*b^6*c + 5*a

^5*c^3 + 16*a^3*b^2*c^3 - 21*a*b^4*c^3 - 2*a^3*c^5 - 6*a*b^2*c^5)/((a^9 - 3*a^8*b + 8*a^6*b^3 - 6*a^5*b^4 - 6*

a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 + b^9 - 3*a^7*c^2 + 9*a^6*b*c^2 - 3*a^5*b^2*c^2 - 15*a^4*b^3*c^2 + 15*a^3*b^4*c^

2 + 3*a^2*b^5*c^2 - 9*a*b^6*c^2 + 3*b^7*c^2 + 3*a^5*c^4 - 9*a^4*b*c^4 + 6*a^3*b^2*c^4 + 6*a^2*b^3*c^4 - 9*a*b^

4*c^4 + 3*b^5*c^4 - a^3*c^6 + 3*a^2*b*c^6 - 3*a*b^2*c^6 + b^3*c^6)*(a*tan(1/2*e*x + 1/2*d)^2 - b*tan(1/2*e*x +

1/2*d)^2 + 2*c*tan(1/2*e*x + 1/2*d) + a + b)^3))/e

Mupad [B] (veriﬁcation not implemented)

Time = 29.88 (sec) , antiderivative size = 1946, normalized size of antiderivative = 6.66 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \]

int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^4,x)

(a*atanh((a*(2*a^2 + 3*b^2 + 3*c^2)*(2*b^6*c - 2*a^6*c + 2*c^7 - 6*a^2*c^5 + 6*a^4*c^3 + 6*b^2*c^5 + 6*b^4*c^3

- 6*a^2*b^4*c + 6*a^4*b^2*c - 12*a^2*b^2*c^3))/(2*(b^2 - a^2 + c^2)^(7/2)*(3*a*b^2 + 3*a*c^2 + 2*a^3)) + (a*t

an(d/2 + (e*x)/2)*(2*a - 2*b)*(2*a^2 + 3*b^2 + 3*c^2)*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3

*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2))/(2*(b^2 - a^2 + c^2)^(7/2)*(3*a*b^2 + 3*a*c^2 + 2*a^3)))*(2

*a^2 + 3*b^2 + 3*c^2))/(e*(b^2 - a^2 + c^2)^(7/2)) - ((18*a^7*c + 2*a^3*c^5 - 5*a^5*c^3 + 6*a*b^2*c^5 + 21*a*b

^4*c^3 - 12*a^3*b^4*c - 21*a^5*b^2*c - 16*a^3*b^2*c^3 + 15*a*b^6*c)/(3*(a - b)^3*(b^6 - a^6 + c^6 - 3*a^2*b^4

+ 3*a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) + (tan(d/2 + (e*x)/2)*(2*b^8 - 6

*a^7*b - 5*a*b^7 + 5*a^2*b^6 + 4*a^3*b^5 - 16*a^4*b^4 + 7*a^5*b^3 + 9*a^6*b^2 + 2*a^2*c^6 - 4*a^4*c^4 + 27*a^6

*c^2 + 2*b^2*c^6 + 6*b^4*c^4 + 6*b^6*c^2 + 18*a*b^3*c^4 + 9*a*b^5*c^2 - 14*a^3*b*c^4 - 9*a^5*b*c^2 - 6*a^2*b^2

*c^4 - 3*a^2*b^4*c^2 - 30*a^4*b^2*c^2 + 4*a*b*c^6))/((a - b)^3*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^

2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) + (tan(d/2 + (e*x)/2)^4*(6*a^6*c + 4*b^6*c + 4*c^7

- 12*a^2*c^5 + 27*a^4*c^3 + 12*b^2*c^5 + 12*b^4*c^3 - 15*a*b^3*c^3 + 33*a^2*b^4*c - 45*a^3*b*c^3 - 55*a^3*b^3

*c + 57*a^4*b^2*c + 21*a^2*b^2*c^3 - 15*a*b^5*c - 30*a^5*b*c))/((a - b)^2*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^4

*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) - (2*tan(d/2 + (e*x)/2)^3*(18*a^7*b - 6

*a*b^7 + 2*b^8 - 4*c^8 + 22*a^2*b^6 - 50*a^3*b^5 + 30*a^4*b^4 + 38*a^5*b^3 - 54*a^6*b^2 + 4*a^2*c^6 - 21*a^4*c

^4 - 54*a^6*c^2 - 10*b^2*c^6 - 6*b^4*c^4 + 2*b^6*c^2 + 51*a*b^3*c^4 + 39*a*b^5*c^2 + 81*a^3*b*c^4 + 120*a^5*b*

c^2 - 105*a^2*b^2*c^4 - 87*a^2*b^4*c^2 + 61*a^3*b^3*c^2 - 81*a^4*b^2*c^2 + 6*a*b*c^6))/(3*(a - b)^3*(b^6 - a^6

+ c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) - (tan(d/2 +

(e*x)/2)^5*(3*a*b^5 + 6*a^5*b - 2*b^6 - 2*c^6 - 3*a^2*b^4 + 11*a^3*b^3 - 15*a^4*b^2 + 6*a^2*c^4 - 9*a^4*c^2 -

6*b^2*c^4 - 6*b^4*c^2 + 3*a*b^3*c^2 + 9*a^3*b*c^2 + 3*a^2*b^2*c^2))/((a - b)*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*

a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) + (2*tan(d/2 + (e*x)/2)^2*(2*a*c^7 +

6*a^7*c + 2*b*c^7 + 2*b^7*c - 3*a^3*c^5 + 20*a^5*c^3 + 6*b^3*c^5 + 6*b^5*c^3 + 9*a*b^2*c^5 - 3*a*b^4*c^3 - 12

*a^2*b*c^5 + 18*a^2*b^5*c - 14*a^3*b^4*c - 22*a^4*b*c^3 - 2*a^4*b^3*c + 18*a^5*b^2*c + 6*a^2*b^3*c^3 - 7*a^3*b

^2*c^3 - 10*a*b^6*c - 18*a^6*b*c))/((a - b)^3*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2

+ 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)))/(e*(tan(d/2 + (e*x)/2)^3*(12*a^2*c - 12*b^2*c + 8*c^3) + tan(d/2 +

(e*x)/2)*(6*a^2*c + 6*b^2*c + 12*a*b*c) + tan(d/2 + (e*x)/2)^2*(3*a^2*b - 3*a*b^2 + 12*a*c^2 + 12*b*c^2 + 3*a

^3 - 3*b^3) - tan(d/2 + (e*x)/2)^4*(3*a*b^2 + 3*a^2*b - 12*a*c^2 + 12*b*c^2 - 3*a^3 - 3*b^3) + tan(d/2 + (e*x)

/2)^5*(6*a^2*c + 6*b^2*c - 12*a*b*c) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 + tan(d/2 + (e*x)/2)^6*(3*a*b^2 - 3*a^2*b

\(\int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx\) [402]

Optimal result