3.369 \(\int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx\)

Optimal. Leaf size=207 \[ \frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]

[Out]

-1/32*a*(5*a^2+3*c^2)*ln(a+c*tan(1/2*e*x+1/2*d))/c^7/e+1/48*(-c*cos(e*x+d)+a*sin(e*x+d))/c^2/e/(a+a*cos(e*x+d)
+c*sin(e*x+d))^3+5/96*(a*c*cos(e*x+d)-a^2*sin(e*x+d))/c^4/e/(a+a*cos(e*x+d)+c*sin(e*x+d))^2+1/96*(-c*(15*a^2+4
*c^2)*cos(e*x+d)+a*(15*a^2+4*c^2)*sin(e*x+d))/c^6/e/(a+a*cos(e*x+d)+c*sin(e*x+d))

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3129, 3156, 3153, 3124, 31} \[ -\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]

[Out]

-(a*(5*a^2 + 3*c^2)*Log[a + c*Tan[(d + e*x)/2]])/(32*c^7*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(48*c^2*e*(a +
a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(96*c^4*e*(a + a*Cos[d + e*x]
+ c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] - a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a + a*C
os[d + e*x] + c*Sin[d + e*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 3156

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*} \int \frac {1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx &=-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {\int \frac {-6 a+4 a \cos (d+e x)+4 c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx}{12 c^2}\\ &=-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {8 \left (5 a^2+2 c^2\right )-20 a^2 \cos (d+e x)-20 a c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{96 c^4}\\ &=-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \int \frac {1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{16 c^6}\\ &=-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a+4 c x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{8 c^6 e}\\ &=-\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 1.73, size = 492, normalized size = 2.38 \[ \frac {\cos \left (\frac {1}{2} (d+e x)\right ) \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac {1}{2} (d+e x)\right ) \log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right ) \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )^3-192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac {1}{2} (d+e x)\right ) \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )^3 \log \left (a \cos \left (\frac {1}{2} (d+e x)\right )+c \sin \left (\frac {1}{2} (d+e x)\right )\right )+\frac {c \left (150 a^6 \sin (d+e x)+120 a^6 \sin (2 (d+e x))+30 a^6 \sin (3 (d+e x))-75 a^5 c \cos (3 (d+e x))+150 a^5 c+255 a^4 c^2 \sin (d+e x)+72 a^4 c^2 \sin (2 (d+e x))-37 a^4 c^2 \sin (3 (d+e x))-35 a^3 c^3 \cos (3 (d+e x))+130 a^3 c^3+129 a^2 c^4 \sin (d+e x)+36 a^2 c^4 \sin (2 (d+e x))-27 a^2 c^4 \sin (3 (d+e x))-6 \left (25 a^5 c+15 a^3 c^3+4 a c^5\right ) \cos (2 (d+e x))+3 a c \left (25 a^4+25 a^2 c^2-4 c^4\right ) \cos (d+e x)-4 a c^5 \cos (3 (d+e x))+24 a c^5+12 c^6 \sin (d+e x)-4 c^6 \sin (3 (d+e x))\right )}{a}\right )}{384 c^7 e (a \cos (d+e x)+a+c \sin (d+e x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]

[Out]

(Cos[(d + e*x)/2]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])*(192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[Cos[
(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 - 192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[a*Cos
[(d + e*x)/2] + c*Sin[(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 + (c*(150*a^5*c + 130*a^3*c^3
+ 24*a*c^5 + 3*a*c*(25*a^4 + 25*a^2*c^2 - 4*c^4)*Cos[d + e*x] - 6*(25*a^5*c + 15*a^3*c^3 + 4*a*c^5)*Cos[2*(d +
e*x)] - 75*a^5*c*Cos[3*(d + e*x)] - 35*a^3*c^3*Cos[3*(d + e*x)] - 4*a*c^5*Cos[3*(d + e*x)] + 150*a^6*Sin[d +
e*x] + 255*a^4*c^2*Sin[d + e*x] + 129*a^2*c^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 120*a^6*Sin[2*(d + e*x)] +
72*a^4*c^2*Sin[2*(d + e*x)] + 36*a^2*c^4*Sin[2*(d + e*x)] + 30*a^6*Sin[3*(d + e*x)] - 37*a^4*c^2*Sin[3*(d + e*
x)] - 27*a^2*c^4*Sin[3*(d + e*x)] - 4*c^6*Sin[3*(d + e*x)]))/a))/(384*c^7*e*(a + a*Cos[d + e*x] + c*Sin[d + e*
x])^4)

________________________________________________________________________________________

fricas [B]  time = 1.05, size = 791, normalized size = 3.82 \[ \frac {60 \, a^{4} c^{2} + 6 \, a^{2} c^{4} - 2 \, {\left (45 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - 4 \, c^{6}\right )} \cos \left (e x + d\right )^{3} - 12 \, {\left (10 \, a^{4} c^{2} + a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{4} c^{2} - 2 \, a^{2} c^{4} - 2 \, c^{6}\right )} \cos \left (e x + d\right ) - 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} + {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} + \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) + 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} + {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) + 2 \, {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 6 \, a c^{5} + {\left (15 \, a^{5} c - 41 \, a^{3} c^{3} - 12 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} + 3 \, {\left (10 \, a^{5} c - 9 \, a^{3} c^{3} - a c^{5}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{192 \, {\left ({\left (a^{3} c^{7} - 3 \, a c^{9}\right )} e \cos \left (e x + d\right )^{3} + 3 \, {\left (a^{3} c^{7} - a c^{9}\right )} e \cos \left (e x + d\right )^{2} + 3 \, {\left (a^{3} c^{7} + a c^{9}\right )} e \cos \left (e x + d\right ) + {\left (a^{3} c^{7} + 3 \, a c^{9}\right )} e + {\left (6 \, a^{2} c^{8} e \cos \left (e x + d\right ) + {\left (3 \, a^{2} c^{8} - c^{10}\right )} e \cos \left (e x + d\right )^{2} + {\left (3 \, a^{2} c^{8} + c^{10}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="fricas")

[Out]

1/192*(60*a^4*c^2 + 6*a^2*c^4 - 2*(45*a^4*c^2 - 3*a^2*c^4 - 4*c^6)*cos(e*x + d)^3 - 12*(10*a^4*c^2 + a^2*c^4)*
cos(e*x + d)^2 + 6*(5*a^4*c^2 - 2*a^2*c^4 - 2*c^6)*cos(e*x + d) - 3*(5*a^6 + 18*a^4*c^2 + 9*a^2*c^4 + (5*a^6 -
12*a^4*c^2 - 9*a^2*c^4)*cos(e*x + d)^3 + 3*(5*a^6 - 2*a^4*c^2 - 3*a^2*c^4)*cos(e*x + d)^2 + 3*(5*a^6 + 8*a^4*
c^2 + 3*a^2*c^4)*cos(e*x + d) + (15*a^5*c + 14*a^3*c^3 + 3*a*c^5 + (15*a^5*c + 4*a^3*c^3 - 3*a*c^5)*cos(e*x +
d)^2 + 6*(5*a^5*c + 3*a^3*c^3)*cos(e*x + d))*sin(e*x + d))*log(a*c*sin(e*x + d) + 1/2*a^2 + 1/2*c^2 + 1/2*(a^2
- c^2)*cos(e*x + d)) + 3*(5*a^6 + 18*a^4*c^2 + 9*a^2*c^4 + (5*a^6 - 12*a^4*c^2 - 9*a^2*c^4)*cos(e*x + d)^3 +
3*(5*a^6 - 2*a^4*c^2 - 3*a^2*c^4)*cos(e*x + d)^2 + 3*(5*a^6 + 8*a^4*c^2 + 3*a^2*c^4)*cos(e*x + d) + (15*a^5*c
+ 14*a^3*c^3 + 3*a*c^5 + (15*a^5*c + 4*a^3*c^3 - 3*a*c^5)*cos(e*x + d)^2 + 6*(5*a^5*c + 3*a^3*c^3)*cos(e*x + d
))*sin(e*x + d))*log(1/2*cos(e*x + d) + 1/2) + 2*(15*a^5*c + 14*a^3*c^3 + 6*a*c^5 + (15*a^5*c - 41*a^3*c^3 - 1
2*a*c^5)*cos(e*x + d)^2 + 3*(10*a^5*c - 9*a^3*c^3 - a*c^5)*cos(e*x + d))*sin(e*x + d))/((a^3*c^7 - 3*a*c^9)*e*
cos(e*x + d)^3 + 3*(a^3*c^7 - a*c^9)*e*cos(e*x + d)^2 + 3*(a^3*c^7 + a*c^9)*e*cos(e*x + d) + (a^3*c^7 + 3*a*c^
9)*e + (6*a^2*c^8*e*cos(e*x + d) + (3*a^2*c^8 - c^10)*e*cos(e*x + d)^2 + (3*a^2*c^8 + c^10)*e)*sin(e*x + d))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 304, normalized size = 1.47 \[ -\frac {1}{384} \, {\left (\frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a \right |}\right )}{c^{7}} - \frac {110 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 66 \, a c^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 285 \, a^{4} c^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 144 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 9 \, c^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 249 \, a^{5} c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 108 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 9 \, a c^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 73 \, a^{6} + 27 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - c^{6}}{{\left (c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a\right )}^{3} c^{7}} - \frac {c^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 6 \, a c^{7} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 30 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 9 \, c^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{c^{12}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="giac")

[Out]

-1/384*(12*(5*a^3 + 3*a*c^2)*log(abs(c*tan(1/2*x*e + 1/2*d) + a))/c^7 - (110*a^3*c^3*tan(1/2*x*e + 1/2*d)^3 +
66*a*c^5*tan(1/2*x*e + 1/2*d)^3 + 285*a^4*c^2*tan(1/2*x*e + 1/2*d)^2 + 144*a^2*c^4*tan(1/2*x*e + 1/2*d)^2 - 9*
c^6*tan(1/2*x*e + 1/2*d)^2 + 249*a^5*c*tan(1/2*x*e + 1/2*d) + 108*a^3*c^3*tan(1/2*x*e + 1/2*d) - 9*a*c^5*tan(1
/2*x*e + 1/2*d) + 73*a^6 + 27*a^4*c^2 - 3*a^2*c^4 - c^6)/((c*tan(1/2*x*e + 1/2*d) + a)^3*c^7) - (c^8*tan(1/2*x
*e + 1/2*d)^3 - 6*a*c^7*tan(1/2*x*e + 1/2*d)^2 + 30*a^2*c^6*tan(1/2*x*e + 1/2*d) + 9*c^8*tan(1/2*x*e + 1/2*d))
/c^12)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.60, size = 378, normalized size = 1.83 \[ \frac {\tan ^{3}\left (\frac {d}{2}+\frac {e x}{2}\right )}{384 e \,c^{4}}-\frac {\left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right ) a}{64 e \,c^{5}}+\frac {5 a^{2} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{64 e \,c^{6}}+\frac {3 \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{128 e \,c^{4}}+\frac {3 a^{5}}{128 e \,c^{7} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {3 a^{3}}{64 e \,c^{5} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {3 a}{128 e \,c^{3} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}-\frac {a^{6}}{384 e \,c^{7} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {a^{4}}{128 e \,c^{5} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {a^{2}}{128 e \,c^{3} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {1}{384 e c \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {15 a^{4}}{128 e \,c^{7} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {9 a^{2}}{64 e \,c^{5} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {3}{128 e \,c^{3} \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {5 a^{3} \ln \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{7}}-\frac {3 a \ln \left (a +c \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x)

[Out]

1/384/e/c^4*tan(1/2*d+1/2*e*x)^3-1/64/e/c^5*tan(1/2*d+1/2*e*x)^2*a+5/64/e/c^6*a^2*tan(1/2*d+1/2*e*x)+3/128/e/c
^4*tan(1/2*d+1/2*e*x)+3/128/e*a^5/c^7/(a+c*tan(1/2*d+1/2*e*x))^2+3/64/e*a^3/c^5/(a+c*tan(1/2*d+1/2*e*x))^2+3/1
28/e*a/c^3/(a+c*tan(1/2*d+1/2*e*x))^2-1/384/e/c^7/(a+c*tan(1/2*d+1/2*e*x))^3*a^6-1/128/e/c^5/(a+c*tan(1/2*d+1/
2*e*x))^3*a^4-1/128/e/c^3/(a+c*tan(1/2*d+1/2*e*x))^3*a^2-1/384/e/c/(a+c*tan(1/2*d+1/2*e*x))^3-15/128/e/c^7/(a+
c*tan(1/2*d+1/2*e*x))*a^4-9/64/e/c^5/(a+c*tan(1/2*d+1/2*e*x))*a^2-3/128/e/c^3/(a+c*tan(1/2*d+1/2*e*x))-5/32/e*
a^3/c^7*ln(a+c*tan(1/2*d+1/2*e*x))-3/32/e*a/c^5*ln(a+c*tan(1/2*d+1/2*e*x))

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 307, normalized size = 1.48 \[ -\frac {\frac {37 \, a^{6} + 39 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6} + \frac {9 \, {\left (9 \, a^{5} c + 10 \, a^{3} c^{3} + a c^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {9 \, {\left (5 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + c^{6}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{a^{3} c^{7} + \frac {3 \, a^{2} c^{8} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, a c^{9} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {c^{10} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} + \frac {\frac {6 \, a c \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} - \frac {3 \, {\left (10 \, a^{2} + 3 \, c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{c^{6}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (a + \frac {c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="maxima")

[Out]

-1/384*((37*a^6 + 39*a^4*c^2 + 3*a^2*c^4 + c^6 + 9*(9*a^5*c + 10*a^3*c^3 + a*c^5)*sin(e*x + d)/(cos(e*x + d) +
1) + 9*(5*a^4*c^2 + 6*a^2*c^4 + c^6)*sin(e*x + d)^2/(cos(e*x + d) + 1)^2)/(a^3*c^7 + 3*a^2*c^8*sin(e*x + d)/(
cos(e*x + d) + 1) + 3*a*c^9*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 + c^10*sin(e*x + d)^3/(cos(e*x + d) + 1)^3) +
(6*a*c*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 - c^2*sin(e*x + d)^3/(cos(e*x + d) + 1)^3 - 3*(10*a^2 + 3*c^2)*sin(
e*x + d)/(cos(e*x + d) + 1))/c^6 + 12*(5*a^3 + 3*a*c^2)*log(a + c*sin(e*x + d)/(cos(e*x + d) + 1))/c^7)/e

________________________________________________________________________________________

mupad [B]  time = 2.53, size = 260, normalized size = 1.26 \[ \frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3}{384\,c^4\,e}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (\frac {3}{128\,c^4}+\frac {5\,a^2}{64\,c^6}\right )}{e}-\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (27\,a^5+30\,a^3\,c^2+3\,a\,c^4\right )+\frac {37\,a^6+39\,a^4\,c^2+3\,a^2\,c^4+c^6}{3\,c}+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (15\,a^4\,c+18\,a^2\,c^3+3\,c^5\right )}{e\,\left (128\,a^3\,c^6+384\,a^2\,c^7\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+384\,a\,c^8\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+128\,c^9\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\right )}-\frac {a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,c^5\,e}-\frac {\ln \left (a+c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (5\,a^3+3\,a\,c^2\right )}{32\,c^7\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*a + 2*a*cos(d + e*x) + 2*c*sin(d + e*x))^4,x)

[Out]

tan(d/2 + (e*x)/2)^3/(384*c^4*e) + (tan(d/2 + (e*x)/2)*(3/(128*c^4) + (5*a^2)/(64*c^6)))/e - (tan(d/2 + (e*x)/
2)*(3*a*c^4 + 27*a^5 + 30*a^3*c^2) + (37*a^6 + c^6 + 3*a^2*c^4 + 39*a^4*c^2)/(3*c) + tan(d/2 + (e*x)/2)^2*(15*
a^4*c + 3*c^5 + 18*a^2*c^3))/(e*(128*c^9*tan(d/2 + (e*x)/2)^3 + 128*a^3*c^6 + 384*a^2*c^7*tan(d/2 + (e*x)/2) +
384*a*c^8*tan(d/2 + (e*x)/2)^2)) - (a*tan(d/2 + (e*x)/2)^2)/(64*c^5*e) - (log(a + c*tan(d/2 + (e*x)/2))*(3*a*
c^2 + 5*a^3))/(32*c^7*e)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))**4,x)

[Out]

Timed out

________________________________________________________________________________________