3.387 \(\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx\)
Optimal. Leaf size=215 \[ \frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))^2}-\frac{a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a \sin (d+e x)+a+b \cos (d+e x))}+\frac{a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right )\right )}{32 b^7 e}-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3} \]
[Out]
(a*(5*a^2 + 3*b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(48*b^
2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[
d + e*x] + a*Sin[d + e*x])^2) - (a*(15*a^2 + 4*b^2)*Cos[d + e*x] - b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*
(a + b*Cos[d + e*x] + a*Sin[d + e*x]))
________________________________________________________________________________________
Rubi [A] time = 0.244099, antiderivative size = 215, normalized size of antiderivative = 1.,
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, 3123, 31} \[ \frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))^2}-\frac{a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a \sin (d+e x)+a+b \cos (d+e x))}+\frac{a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right )\right )}{32 b^7 e}-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3} \]
Antiderivative was successfully verified.
[In]
Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]
[Out]
(a*(5*a^2 + 3*b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(48*b^
2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[
d + e*x] + a*Sin[d + e*x])^2) - (a*(15*a^2 + 4*b^2)*Cos[d + e*x] - b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*
(a + b*Cos[d + e*x] + a*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 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]
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 3123
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Cot[(d + e*x)/2 + Pi/4], x]}, -Dist[f/e, Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f], x
]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx &=-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac{\int \frac{-6 a+4 b \cos (d+e x)+4 a \sin (d+e x)}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx}{12 b^2}\\ &=-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac{\int \frac{8 \left (5 a^2+2 b^2\right )-20 a b \cos (d+e x)-20 a^2 \sin (d+e x)}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{96 b^4}\\ &=-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}-\frac{a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a+b \cos (d+e x)+a \sin (d+e x))}-\frac{\left (a \left (5 a^2+3 b^2\right )\right ) \int \frac{1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx}{16 b^6}\\ &=-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}-\frac{a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a+b \cos (d+e x)+a \sin (d+e x))}+\frac{\left (a \left (5 a^2+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 a+2 b x} \, dx,x,\cot \left (\frac{\pi }{4}+\frac{1}{2} (d+e x)\right )\right )}{16 b^6 e}\\ &=\frac{a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac{d}{2}+\frac{\pi }{4}+\frac{e x}{2}\right )\right )}{32 b^7 e}-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac{5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}-\frac{a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a+b \cos (d+e x)+a \sin (d+e x))}\\ \end{align*}
Mathematica [B] time = 2.9166, size = 632, normalized size = 2.94 \[ \frac{\frac{b \left (180 a^4 b^2 \sin (d+e x)+54 a^4 b^2 \sin (2 (d+e x))-4 a^4 b^2 \sin (3 (d+e x))+15 a^3 b^3 \sin (d+e x)+102 a^3 b^3 \sin (2 (d+e x))+3 a^3 b^3 \sin (3 (d+e x))+27 a^2 b^4 \sin (d+e x)+6 a^2 b^4 \sin (2 (d+e x))+15 a^2 b^4 \sin (3 (d+e x))-41 a^4 b^2 \cos (3 (d+e x))-38 a^3 b^3 \cos (3 (d+e x))-3 a^2 \left (5 a^2 b^2-50 a^3 b+25 a^4-30 a b^3+4 b^4\right ) \cos (d+e x)-6 a^2 \left (9 a^2 b^2+20 a^3 b+15 a^4+2 a b^3-2 b^4\right ) \cos (2 (d+e x))-12 a^2 b^4 \cos (3 (d+e x))+130 a^4 b^2+24 a^2 b^4+75 a^5 b \sin (d+e x)+120 a^5 b \sin (2 (d+e x))-45 a^5 b \sin (3 (d+e x))-30 a^5 b \cos (3 (d+e x))+225 a^6 \sin (d+e x)-60 a^6 \sin (2 (d+e x))-15 a^6 \sin (3 (d+e x))+15 a^6 \cos (3 (d+e x))+150 a^6+12 a b^5 \sin (d+e x)+6 a b^5 \sin (2 (d+e x))+4 a b^5 \sin (3 (d+e x))-8 a b^5 \cos (3 (d+e x))+12 b^6 \sin (d+e x)+4 b^6 \sin (3 (d+e x))\right )}{(a+b) \left (\sin \left (\frac{1}{2} (d+e x)\right )+\cos \left (\frac{1}{2} (d+e x)\right )\right )^3 \left ((a-b) \sin \left (\frac{1}{2} (d+e x)\right )+(a+b) \cos \left (\frac{1}{2} (d+e x)\right )\right )^3}-12 a \left (5 a^2+3 b^2\right ) \log \left (\sin \left (\frac{1}{2} (d+e x)\right )+\cos \left (\frac{1}{2} (d+e x)\right )\right )+12 a \left (5 a^2+3 b^2\right ) \log \left ((a-b) \sin \left (\frac{1}{2} (d+e x)\right )+(a+b) \cos \left (\frac{1}{2} (d+e x)\right )\right )}{384 b^7 e} \]
Antiderivative was successfully verified.
[In]
Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]
[Out]
(-12*a*(5*a^2 + 3*b^2)*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]] + 12*a*(5*a^2 + 3*b^2)*Log[(a + b)*Cos[(d + e*
x)/2] + (a - b)*Sin[(d + e*x)/2]] + (b*(150*a^6 + 130*a^4*b^2 + 24*a^2*b^4 - 3*a^2*(25*a^4 - 50*a^3*b + 5*a^2*
b^2 - 30*a*b^3 + 4*b^4)*Cos[d + e*x] - 6*a^2*(15*a^4 + 20*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 2*b^4)*Cos[2*(d + e*x)
] + 15*a^6*Cos[3*(d + e*x)] - 30*a^5*b*Cos[3*(d + e*x)] - 41*a^4*b^2*Cos[3*(d + e*x)] - 38*a^3*b^3*Cos[3*(d +
e*x)] - 12*a^2*b^4*Cos[3*(d + e*x)] - 8*a*b^5*Cos[3*(d + e*x)] + 225*a^6*Sin[d + e*x] + 75*a^5*b*Sin[d + e*x]
+ 180*a^4*b^2*Sin[d + e*x] + 15*a^3*b^3*Sin[d + e*x] + 27*a^2*b^4*Sin[d + e*x] + 12*a*b^5*Sin[d + e*x] + 12*b^
6*Sin[d + e*x] - 60*a^6*Sin[2*(d + e*x)] + 120*a^5*b*Sin[2*(d + e*x)] + 54*a^4*b^2*Sin[2*(d + e*x)] + 102*a^3*
b^3*Sin[2*(d + e*x)] + 6*a^2*b^4*Sin[2*(d + e*x)] + 6*a*b^5*Sin[2*(d + e*x)] - 15*a^6*Sin[3*(d + e*x)] - 45*a^
5*b*Sin[3*(d + e*x)] - 4*a^4*b^2*Sin[3*(d + e*x)] + 3*a^3*b^3*Sin[3*(d + e*x)] + 15*a^2*b^4*Sin[3*(d + e*x)] +
4*a*b^5*Sin[3*(d + e*x)] + 4*b^6*Sin[3*(d + e*x)]))/((a + b)*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3*((a + b)
*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2])^3))/(384*b^7*e)
________________________________________________________________________________________
Maple [B] time = 0.223, size = 1069, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x)
[Out]
3/32/e/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)^2*a-1/16/e/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(
1/2*d+1/2*e*x)+a+b)^3*a^2+1/16/e/b^5/(1+tan(1/2*d+1/2*e*x))^2*a-5/32/e/b^6/(1+tan(1/2*d+1/2*e*x))*a^2-1/16/e/b
^5/(1+tan(1/2*d+1/2*e*x))*a-5/32/e*a^3/b^7*ln(1+tan(1/2*d+1/2*e*x))-3/32/e*a/b^5*ln(1+tan(1/2*d+1/2*e*x))-1/48
/e*b^2/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)^3+1/32/e*b/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(
1/2*d+1/2*e*x)+a+b)^2-1/16/e/b^4/(1+tan(1/2*d+1/2*e*x))+3/16/e/b^2/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1
/2*e*x)+a+b)^2*a^3-5/32/e/b^6/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)*a^6+3/8/e/b^5/(a-b)^3/(a
*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)*a^5-3/8/e/b^4/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)
+a+b)*a^4+3/8/e/b^3/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)*a^3-9/32/e/b^2/(a-b)^3/(a*tan(1/2*
d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)*a^2+5/32/e*a^4/b^7/(a-b)*ln(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b
)-5/32/e*a^3/b^6/(a-b)*ln(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)+3/32/e*a^2/b^5/(a-b)*ln(a*tan(1/2*d+1
/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)-3/32/e/b^3/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)^2*a^4-3/3
2/e*a/b^4/(a-b)*ln(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)-1/48/e/b^4/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*t
an(1/2*d+1/2*e*x)+a+b)^3*a^6-1/16/e/b^2/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)^3*a^4-1/16/e/b
^5/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)^2*a^6+3/32/e/b^4/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*ta
n(1/2*d+1/2*e*x)+a+b)^2*a^5-1/16/e/(a-b)^3/(a*tan(1/2*d+1/2*e*x)-b*tan(1/2*d+1/2*e*x)+a+b)-1/48/e/b^4/(1+tan(1
/2*d+1/2*e*x))^3+1/32/e/b^4/(1+tan(1/2*d+1/2*e*x))^2
________________________________________________________________________________________
Maxima [B] time = 1.25344, size = 1300, normalized size = 6.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="maxima")
[Out]
-1/96*(2*(15*a^8 - 31*a^6*b^2 + 9*a^4*b^4 + 15*a^2*b^6 + 3*(25*a^8 - 25*a^7*b - 25*a^6*b^2 + 25*a^5*b^3 - 13*a
^4*b^4 + 13*a^3*b^5 + 11*a^2*b^6 - 5*a*b^7 + 2*b^8)*sin(e*x + d)/(cos(e*x + d) + 1) + 6*(25*a^8 - 50*a^7*b + 2
0*a^6*b^2 + 10*a^5*b^3 - 17*a^4*b^4 + 24*a^3*b^5 - 10*a^2*b^6 + 2*a*b^7)*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 +
2*(75*a^8 - 225*a^7*b + 250*a^6*b^2 - 150*a^5*b^3 + 63*a^4*b^4 + 11*a^3*b^5 - 24*a^2*b^6 + 6*a*b^7 - 2*b^8)*s
in(e*x + d)^3/(cos(e*x + d) + 1)^3 + 3*(25*a^8 - 100*a^7*b + 165*a^6*b^2 - 160*a^5*b^3 + 115*a^4*b^4 - 60*a^3*
b^5 + 19*a^2*b^6 - 4*a*b^7)*sin(e*x + d)^4/(cos(e*x + d) + 1)^4 + 3*(5*a^8 - 25*a^7*b + 53*a^6*b^2 - 65*a^5*b^
3 + 55*a^4*b^4 - 35*a^3*b^5 + 17*a^2*b^6 - 7*a*b^7 + 2*b^8)*sin(e*x + d)^5/(cos(e*x + d) + 1)^5)/(a^6*b^6 - 3*
a^4*b^8 + 3*a^2*b^10 - b^12 + 6*(a^6*b^6 - a^5*b^7 - 2*a^4*b^8 + 2*a^3*b^9 + a^2*b^10 - a*b^11)*sin(e*x + d)/(
cos(e*x + d) + 1) + 3*(5*a^6*b^6 - 10*a^5*b^7 - a^4*b^8 + 12*a^3*b^9 - 5*a^2*b^10 - 2*a*b^11 + b^12)*sin(e*x +
d)^2/(cos(e*x + d) + 1)^2 + 4*(5*a^6*b^6 - 15*a^5*b^7 + 12*a^4*b^8 + 4*a^3*b^9 - 9*a^2*b^10 + 3*a*b^11)*sin(e
*x + d)^3/(cos(e*x + d) + 1)^3 + 3*(5*a^6*b^6 - 20*a^5*b^7 + 29*a^4*b^8 - 16*a^3*b^9 - a^2*b^10 + 4*a*b^11 - b
^12)*sin(e*x + d)^4/(cos(e*x + d) + 1)^4 + 6*(a^6*b^6 - 5*a^5*b^7 + 10*a^4*b^8 - 10*a^3*b^9 + 5*a^2*b^10 - a*b
^11)*sin(e*x + d)^5/(cos(e*x + d) + 1)^5 + (a^6*b^6 - 6*a^5*b^7 + 15*a^4*b^8 - 20*a^3*b^9 + 15*a^2*b^10 - 6*a*
b^11 + b^12)*sin(e*x + d)^6/(cos(e*x + d) + 1)^6) - 3*(5*a^3 + 3*a*b^2)*log(-a - b - (a - b)*sin(e*x + d)/(cos
(e*x + d) + 1))/b^7 + 3*(5*a^3 + 3*a*b^2)*log(sin(e*x + d)/(cos(e*x + d) + 1) + 1)/b^7)/e
________________________________________________________________________________________
Fricas [B] time = 2.90969, size = 1632, normalized size = 7.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="fricas")
[Out]
1/192*(60*a^4*b^2 + 6*a^2*b^4 + 2*(15*a^5*b - 41*a^3*b^3 - 12*a*b^5)*cos(e*x + d)^3 - 12*(10*a^4*b^2 + a^2*b^4
)*cos(e*x + d)^2 - 6*(10*a^5*b - 9*a^3*b^3 - 2*a*b^5)*cos(e*x + d) + 3*(20*a^6 + 12*a^4*b^2 - (15*a^5*b + 4*a^
3*b^3 - 3*a*b^5)*cos(e*x + d)^3 - 3*(5*a^6 - 2*a^4*b^2 - 3*a^2*b^4)*cos(e*x + d)^2 + 6*(5*a^5*b + 3*a^3*b^3)*c
os(e*x + d) + (20*a^6 + 12*a^4*b^2 - (5*a^6 - 12*a^4*b^2 - 9*a^2*b^4)*cos(e*x + d)^2 + 6*(5*a^5*b + 3*a^3*b^3)
*cos(e*x + d))*sin(e*x + d))*log(2*a*b*cos(e*x + d) + a^2 + b^2 + (a^2 - b^2)*sin(e*x + d)) - 3*(20*a^6 + 12*a
^4*b^2 - (15*a^5*b + 4*a^3*b^3 - 3*a*b^5)*cos(e*x + d)^3 - 3*(5*a^6 - 2*a^4*b^2 - 3*a^2*b^4)*cos(e*x + d)^2 +
6*(5*a^5*b + 3*a^3*b^3)*cos(e*x + d) + (20*a^6 + 12*a^4*b^2 - (5*a^6 - 12*a^4*b^2 - 9*a^2*b^4)*cos(e*x + d)^2
+ 6*(5*a^5*b + 3*a^3*b^3)*cos(e*x + d))*sin(e*x + d))*log(sin(e*x + d) + 1) + 2*(30*a^4*b^2 + 3*a^2*b^4 + 2*b^
6 - (45*a^4*b^2 - 3*a^2*b^4 - 4*b^6)*cos(e*x + d)^2 - 3*(10*a^5*b - 9*a^3*b^3 - a*b^5)*cos(e*x + d))*sin(e*x +
d))/(6*a^2*b^8*e*cos(e*x + d) + 4*a^3*b^7*e - (3*a^2*b^8 - b^10)*e*cos(e*x + d)^3 - 3*(a^3*b^7 - a*b^9)*e*cos
(e*x + d)^2 + (6*a^2*b^8*e*cos(e*x + d) + 4*a^3*b^7*e - (a^3*b^7 - 3*a*b^9)*e*cos(e*x + d)^2)*sin(e*x + d))
________________________________________________________________________________________
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/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))**4,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.18962, size = 1368, normalized size = 6.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="giac")
[Out]
1/96*(3*(5*a^4 - 5*a^3*b + 3*a^2*b^2 - 3*a*b^3)*log(abs(a*tan(1/2*x*e + 1/2*d) - b*tan(1/2*x*e + 1/2*d) + a +
b))/(a*b^7 - b^8) - 2*(15*a^8*tan(1/2*x*e + 1/2*d)^5 - 75*a^7*b*tan(1/2*x*e + 1/2*d)^5 + 159*a^6*b^2*tan(1/2*x
*e + 1/2*d)^5 - 195*a^5*b^3*tan(1/2*x*e + 1/2*d)^5 + 165*a^4*b^4*tan(1/2*x*e + 1/2*d)^5 - 105*a^3*b^5*tan(1/2*
x*e + 1/2*d)^5 + 51*a^2*b^6*tan(1/2*x*e + 1/2*d)^5 - 21*a*b^7*tan(1/2*x*e + 1/2*d)^5 + 6*b^8*tan(1/2*x*e + 1/2
*d)^5 + 75*a^8*tan(1/2*x*e + 1/2*d)^4 - 300*a^7*b*tan(1/2*x*e + 1/2*d)^4 + 495*a^6*b^2*tan(1/2*x*e + 1/2*d)^4
- 480*a^5*b^3*tan(1/2*x*e + 1/2*d)^4 + 345*a^4*b^4*tan(1/2*x*e + 1/2*d)^4 - 180*a^3*b^5*tan(1/2*x*e + 1/2*d)^4
+ 57*a^2*b^6*tan(1/2*x*e + 1/2*d)^4 - 12*a*b^7*tan(1/2*x*e + 1/2*d)^4 + 150*a^8*tan(1/2*x*e + 1/2*d)^3 - 450*
a^7*b*tan(1/2*x*e + 1/2*d)^3 + 500*a^6*b^2*tan(1/2*x*e + 1/2*d)^3 - 300*a^5*b^3*tan(1/2*x*e + 1/2*d)^3 + 126*a
^4*b^4*tan(1/2*x*e + 1/2*d)^3 + 22*a^3*b^5*tan(1/2*x*e + 1/2*d)^3 - 48*a^2*b^6*tan(1/2*x*e + 1/2*d)^3 + 12*a*b
^7*tan(1/2*x*e + 1/2*d)^3 - 4*b^8*tan(1/2*x*e + 1/2*d)^3 + 150*a^8*tan(1/2*x*e + 1/2*d)^2 - 300*a^7*b*tan(1/2*
x*e + 1/2*d)^2 + 120*a^6*b^2*tan(1/2*x*e + 1/2*d)^2 + 60*a^5*b^3*tan(1/2*x*e + 1/2*d)^2 - 102*a^4*b^4*tan(1/2*
x*e + 1/2*d)^2 + 144*a^3*b^5*tan(1/2*x*e + 1/2*d)^2 - 60*a^2*b^6*tan(1/2*x*e + 1/2*d)^2 + 12*a*b^7*tan(1/2*x*e
+ 1/2*d)^2 + 75*a^8*tan(1/2*x*e + 1/2*d) - 75*a^7*b*tan(1/2*x*e + 1/2*d) - 75*a^6*b^2*tan(1/2*x*e + 1/2*d) +
75*a^5*b^3*tan(1/2*x*e + 1/2*d) - 39*a^4*b^4*tan(1/2*x*e + 1/2*d) + 39*a^3*b^5*tan(1/2*x*e + 1/2*d) + 33*a^2*b
^6*tan(1/2*x*e + 1/2*d) - 15*a*b^7*tan(1/2*x*e + 1/2*d) + 6*b^8*tan(1/2*x*e + 1/2*d) + 15*a^8 - 31*a^6*b^2 + 9
*a^4*b^4 + 15*a^2*b^6)/((a^3*b^6 - 3*a^2*b^7 + 3*a*b^8 - b^9)*(a*tan(1/2*x*e + 1/2*d)^2 - b*tan(1/2*x*e + 1/2*
d)^2 + 2*a*tan(1/2*x*e + 1/2*d) + a + b)^3) - 3*(5*a^3 + 3*a*b^2)*log(abs(tan(1/2*x*e + 1/2*d) + 1))/b^7)*e^(-
1)