\(\int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx\) [387]

Optimal result

Integrand size = 24, antiderivative size = 215 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{32 b^7 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a+b \cos (d+e x)+a \sin (d+e x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3208, 3235, 3232, 3202, 31} \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))^2}+\frac {a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{32 b^7 e}-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3} \]

[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 31

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

Rule 3202

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 3208

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 3232

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 3235

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 {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 ) \text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(632\) vs. \(2(215)=430\).

Time = 1.75 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {-12 a \left (5 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )+12 a \left (5 a^2+3 b^2\right ) \log \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )+\frac {b \left (150 a^6+130 a^4 b^2+24 a^2 b^4-3 a^2 \left (25 a^4-50 a^3 b+5 a^2 b^2-30 a b^3+4 b^4\right ) \cos (d+e x)-6 a^2 \left (15 a^4+20 a^3 b+9 a^2 b^2+2 a b^3-2 b^4\right ) \cos (2 (d+e x))+15 a^6 \cos (3 (d+e x))-30 a^5 b \cos (3 (d+e x))-41 a^4 b^2 \cos (3 (d+e x))-38 a^3 b^3 \cos (3 (d+e x))-12 a^2 b^4 \cos (3 (d+e x))-8 a b^5 \cos (3 (d+e x))+225 a^6 \sin (d+e x)+75 a^5 b \sin (d+e x)+180 a^4 b^2 \sin (d+e x)+15 a^3 b^3 \sin (d+e x)+27 a^2 b^4 \sin (d+e x)+12 a b^5 \sin (d+e x)+12 b^6 \sin (d+e x)-60 a^6 \sin (2 (d+e x))+120 a^5 b \sin (2 (d+e x))+54 a^4 b^2 \sin (2 (d+e x))+102 a^3 b^3 \sin (2 (d+e x))+6 a^2 b^4 \sin (2 (d+e x))+6 a b^5 \sin (2 (d+e x))-15 a^6 \sin (3 (d+e x))-45 a^5 b \sin (3 (d+e x))-4 a^4 b^2 \sin (3 (d+e x))+3 a^3 b^3 \sin (3 (d+e x))+15 a^2 b^4 \sin (3 (d+e x))+4 a b^5 \sin (3 (d+e x))+4 b^6 \sin (3 (d+e x))\right )}{(a+b) \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^3 \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )^3}}{384 b^7 e} \]

[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 [A] (verified)

Time = 3.18 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.85

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {-2 a -b}{2 b^{5} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {5 a^{2}+2 a b +2 b^{2}}{2 b^{6} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{7}}+\frac {\left (5 a^{3}-5 a^{2} b +3 a \,b^{2}-3 b^{3}\right ) a \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{7} \left (a -b \right )}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{4} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {2 a^{6}-3 a^{5} b +3 a^{4} b^{2}-6 a^{3} b^{3}-3 a \,b^{5}-b^{6}}{2 b^{5} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {5 a^{6}-12 a^{5} b +12 a^{4} b^{2}-12 a^{3} b^{3}+9 a^{2} b^{4}+2 b^{6}}{2 b^{6} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}}{16 e}\)	\(398\)
default	\(\frac {-\frac {1}{3 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {-2 a -b}{2 b^{5} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {5 a^{2}+2 a b +2 b^{2}}{2 b^{6} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{7}}+\frac {\left (5 a^{3}-5 a^{2} b +3 a \,b^{2}-3 b^{3}\right ) a \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{7} \left (a -b \right )}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{4} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {2 a^{6}-3 a^{5} b +3 a^{4} b^{2}-6 a^{3} b^{3}-3 a \,b^{5}-b^{6}}{2 b^{5} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {5 a^{6}-12 a^{5} b +12 a^{4} b^{2}-12 a^{3} b^{3}+9 a^{2} b^{4}+2 b^{6}}{2 b^{6} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}}{16 e}\)	\(398\)
risch	\(\frac {i \left (-15 a^{5} {\mathrm e}^{5 i \left (e x +d \right )}-75 a^{5} {\mathrm e}^{i \left (e x +d \right )}+150 a^{5} {\mathrm e}^{3 i \left (e x +d \right )}+12 b^{5} {\mathrm e}^{2 i \left (e x +d \right )}+4 b^{5}-45 a^{4} b +3 a^{2} b^{3}+6 a^{3} b^{2} {\mathrm e}^{5 i \left (e x +d \right )}+24 a \,b^{4} {\mathrm e}^{3 i \left (e x +d \right )}+150 a^{4} b \,{\mathrm e}^{2 i \left (e x +d \right )}+45 a^{2} b^{3} {\mathrm e}^{4 i \left (e x +d \right )}-15 i a^{5}-75 i a^{5} {\mathrm e}^{4 i \left (e x +d \right )}+150 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}+41 i a^{3} b^{2}+12 i a \,b^{4}+60 i a^{3} b^{2} {\mathrm e}^{2 i \left (e x +d \right )}+12 i a \,b^{4} {\mathrm e}^{2 i \left (e x +d \right )}-45 i a^{3} b^{2} {\mathrm e}^{4 i \left (e x +d \right )}+60 a^{3} b^{2} {\mathrm e}^{i \left (e x +d \right )}+15 a \,b^{4} {\mathrm e}^{i \left (e x +d \right )}+75 a^{4} b \,{\mathrm e}^{4 i \left (e x +d \right )}+9 a \,b^{4} {\mathrm e}^{5 i \left (e x +d \right )}+60 a^{2} b^{3} {\mathrm e}^{2 i \left (e x +d \right )}+130 a^{3} b^{2} {\mathrm e}^{3 i \left (e x +d \right )}-30 i a^{4} b \,{\mathrm e}^{5 i \left (e x +d \right )}-18 i a^{2} b^{3} {\mathrm e}^{5 i \left (e x +d \right )}+150 i a^{4} b \,{\mathrm e}^{i \left (e x +d \right )}+30 i a^{2} b^{3} {\mathrm e}^{i \left (e x +d \right )}\right )}{48 \left (-i a \,{\mathrm e}^{2 i \left (e x +d \right )}+b \,{\mathrm e}^{2 i \left (e x +d \right )}+i a +2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{3} b^{6} e}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{32 b^{7} e}+\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{32 b^{5} e}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{32 b^{7} e}-\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{32 b^{5} e}\)	\(581\)
norman	\(\frac {-\frac {\left (50 a^{7}+100 a^{6} b +75 a^{5} b^{2}+50 a^{4} b^{3}+22 a^{3} b^{4}-6 a^{2} b^{5}-a \,b^{6}+4 b^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{16 e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (50 a^{7}-100 a^{6} b +75 a^{5} b^{2}-50 a^{4} b^{3}+22 a^{3} b^{4}+6 a^{2} b^{5}-a \,b^{6}-4 b^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{16 e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}-\frac {75 a^{8}+225 a^{7} b +250 a^{6} b^{2}+150 a^{5} b^{3}+63 a^{4} b^{4}-11 a^{3} b^{5}-24 a^{2} b^{6}-6 b^{7} a -2 b^{8}}{96 a \,b^{6} e \left (5 a^{2}-3 b^{2}\right )}-\frac {\left (125 a^{8}+125 a^{7} b +50 a^{6} b^{2}+50 a^{5} b^{3}+5 a^{4} b^{4}+5 a^{3} b^{5}+32 a^{2} b^{6}+2 b^{7} a +2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{32 a \,b^{6} e \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (75 a^{8}-225 a^{7} b +250 a^{6} b^{2}-150 a^{5} b^{3}+63 a^{4} b^{4}+11 a^{3} b^{5}-24 a^{2} b^{6}+6 b^{7} a -2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{96 a \,b^{6} e \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (125 a^{8}-125 a^{7} b +50 a^{6} b^{2}-50 a^{5} b^{3}+5 a^{4} b^{4}-5 a^{3} b^{5}+32 a^{2} b^{6}-2 b^{7} a +2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{32 a \,b^{6} e \left (5 a^{2}-3 b^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 b^{7} e}+\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{32 b^{7} e}\)	\(677\)
parallelrisch	\(\frac {15 \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (\left (6 a^{3}-6 a \,b^{2}\right ) \cos \left (2 e x +2 d \right )+\left (3 a^{2} b -b^{3}\right ) \cos \left (3 e x +3 d \right )+\left (a^{3}-3 a \,b^{2}\right ) \sin \left (3 e x +3 d \right )-12 a^{2} b \sin \left (2 e x +2 d \right )+\left (-15 a^{2} b -3 b^{3}\right ) \cos \left (e x +d \right )+\left (-15 a^{3}-3 a \,b^{2}\right ) \sin \left (e x +d \right )-10 a^{3}-6 a \,b^{2}\right ) \left (a +b \right )^{2} a^{2} \ln \left (a +b +\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )-15 \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (\left (6 a^{3}-6 a \,b^{2}\right ) \cos \left (2 e x +2 d \right )+\left (3 a^{2} b -b^{3}\right ) \cos \left (3 e x +3 d \right )+\left (a^{3}-3 a \,b^{2}\right ) \sin \left (3 e x +3 d \right )-12 a^{2} b \sin \left (2 e x +2 d \right )+\left (-15 a^{2} b -3 b^{3}\right ) \cos \left (e x +d \right )+\left (-15 a^{3}-3 a \,b^{2}\right ) \sin \left (e x +d \right )-10 a^{3}-6 a \,b^{2}\right ) \left (a +b \right )^{2} a^{2} \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+60 b \left (\left (a +b \right ) \left (a^{6}+\frac {3}{2} a^{5} b +\frac {3}{5} a^{4} b^{2}+\frac {9}{10} a^{3} b^{3}+\frac {3}{10} a^{2} b^{4}-\frac {1}{10} a \,b^{5}+\frac {1}{5} b^{6}\right ) a \cos \left (2 e x +2 d \right )+\left (-\frac {1}{4} a^{8}+\frac {1}{30} b^{8}+\frac {41}{60} a^{6} b^{2}+\frac {3}{4} a^{5} b^{3}+\frac {7}{10} a^{4} b^{4}+\frac {1}{2} a^{3} b^{5}+\frac {2}{15} a^{2} b^{6}+\frac {1}{60} b^{7} a \right ) \cos \left (3 e x +3 d \right )+\frac {\left (a^{7}+5 a^{6} b +\frac {51}{10} a^{5} b^{2}+\frac {5}{2} a^{4} b^{3}+2 a^{3} b^{4}-\frac {3}{10} a^{2} b^{5}-\frac {1}{2} a \,b^{6}+\frac {1}{5} b^{7}\right ) a \sin \left (3 e x +3 d \right )}{6}+a^{2} \left (a^{6}-\frac {9}{10} a^{4} b^{2}-\frac {3}{5} a^{2} b^{4}+\frac {3}{10} b^{6}\right ) \sin \left (2 e x +2 d \right )+\left (\frac {5}{4} a^{8}+\frac {1}{10} b^{8}+\frac {1}{4} a^{6} b^{2}+\frac {9}{4} a^{5} b^{3}+\frac {7}{10} a^{4} b^{4}+\frac {11}{10} a^{3} b^{5}+\frac {4}{5} a^{2} b^{6}+\frac {1}{20} b^{7} a \right ) \cos \left (e x +d \right )-\frac {5 \left (\left (a^{7}+a^{6} b +\frac {3}{10} a^{5} b^{2}+\frac {1}{2} a^{4} b^{3}-\frac {2}{25} a^{3} b^{4}-\frac {3}{50} a^{2} b^{5}+\frac {7}{50} a \,b^{6}+\frac {1}{25} b^{7}\right ) \sin \left (e x +d \right )+\frac {2 \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (a^{5}+\frac {1}{2} a^{4} b -\frac {9}{10} a^{3} b^{2}-\frac {1}{5} a^{2} b^{3}-\frac {1}{10} a \,b^{4}-\frac {1}{5} b^{5}\right )}{3}\right ) a}{2}\right )}{96 \left (\left (6 a^{3}-6 a \,b^{2}\right ) \cos \left (2 e x +2 d \right )+\left (3 a^{2} b -b^{3}\right ) \cos \left (3 e x +3 d \right )+\left (a^{3}-3 a \,b^{2}\right ) \sin \left (3 e x +3 d \right )-12 a^{2} b \sin \left (2 e x +2 d \right )+\left (-15 a^{2} b -3 b^{3}\right ) \cos \left (e x +d \right )+\left (-15 a^{3}-3 a \,b^{2}\right ) \sin \left (e x +d \right )-10 a^{3}-6 a \,b^{2}\right ) b^{7} \left (a +b \right )^{2} e a}\)	\(884\)

[In]

int(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x,method=_RETURNVERBOSE)

[Out]

1/16/e*(-1/3/b^4/(1+tan(1/2*e*x+1/2*d))^3-1/2*(-2*a-b)/b^5/(1+tan(1/2*e*x+1/2*d))^2-1/2*(5*a^2+2*a*b+2*b^2)/b^
6/(1+tan(1/2*e*x+1/2*d))-1/2*a*(5*a^2+3*b^2)/b^7*ln(1+tan(1/2*e*x+1/2*d))+1/2*(5*a^3-5*a^2*b+3*a*b^2-3*b^3)*a/
b^7/(a-b)*ln(a*tan(1/2*e*x+1/2*d)-b*tan(1/2*e*x+1/2*d)+a+b)-1/3*(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/b^4/(a-b)^3/(a*t
an(1/2*e*x+1/2*d)-b*tan(1/2*e*x+1/2*d)+a+b)^3-1/2*(2*a^6-3*a^5*b+3*a^4*b^2-6*a^3*b^3-3*a*b^5-b^6)/b^5/(a-b)^3/
(a*tan(1/2*e*x+1/2*d)-b*tan(1/2*e*x+1/2*d)+a+b)^2-1/2*(5*a^6-12*a^5*b+12*a^4*b^2-12*a^3*b^3+9*a^2*b^4+2*b^6)/b
^6/(a-b)^3/(a*tan(1/2*e*x+1/2*d)-b*tan(1/2*e*x+1/2*d)+a+b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (201) = 402\).

Time = 0.30 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.39 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {60 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 2 \, {\left (15 \, a^{5} b - 41 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \cos \left (e x + d\right )^{3} - 12 \, {\left (10 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (10 \, a^{5} b - 9 \, a^{3} b^{3} - 2 \, a b^{5}\right )} \cos \left (e x + d\right ) + 3 \, {\left (20 \, a^{6} + 12 \, a^{4} b^{2} - {\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (5 \, a^{6} - 2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (e x + d\right ) + {\left (20 \, a^{6} + 12 \, a^{4} b^{2} - {\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) - 3 \, {\left (20 \, a^{6} + 12 \, a^{4} b^{2} - {\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (5 \, a^{6} - 2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (e x + d\right ) + {\left (20 \, a^{6} + 12 \, a^{4} b^{2} - {\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (e x + d\right )^{2} + 6 \, {\left (5 \, a^{5} b + 3 \, a^{3} b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) + 2 \, {\left (30 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + 2 \, b^{6} - {\left (45 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (10 \, a^{5} b - 9 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{192 \, {\left (6 \, a^{2} b^{8} e \cos \left (e x + d\right ) + 4 \, a^{3} b^{7} e - {\left (3 \, a^{2} b^{8} - b^{10}\right )} e \cos \left (e x + d\right )^{3} - 3 \, {\left (a^{3} b^{7} - a b^{9}\right )} e \cos \left (e x + d\right )^{2} + {\left (6 \, a^{2} b^{8} e \cos \left (e x + d\right ) + 4 \, a^{3} b^{7} e - {\left (a^{3} b^{7} - 3 \, a b^{9}\right )} e \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )}} \]

[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)]

Timed out. \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (201) = 402\).

Time = 0.27 (sec) , antiderivative size = 963, normalized size of antiderivative = 4.48 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Too large to display} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (201) = 402\).

Time = 0.33 (sec) , antiderivative size = 957, normalized size of antiderivative = 4.45 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/96*(2*(15*a^8*tan(1/2*e*x + 1/2*d)^5 - 75*a^7*b*tan(1/2*e*x + 1/2*d)^5 + 159*a^6*b^2*tan(1/2*e*x + 1/2*d)^5
- 195*a^5*b^3*tan(1/2*e*x + 1/2*d)^5 + 165*a^4*b^4*tan(1/2*e*x + 1/2*d)^5 - 105*a^3*b^5*tan(1/2*e*x + 1/2*d)^
5 + 51*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 + 75*a^
8*tan(1/2*e*x + 1/2*d)^4 - 300*a^7*b*tan(1/2*e*x + 1/2*d)^4 + 495*a^6*b^2*tan(1/2*e*x + 1/2*d)^4 - 480*a^5*b^3
*tan(1/2*e*x + 1/2*d)^4 + 345*a^4*b^4*tan(1/2*e*x + 1/2*d)^4 - 180*a^3*b^5*tan(1/2*e*x + 1/2*d)^4 + 57*a^2*b^6
*tan(1/2*e*x + 1/2*d)^4 - 12*a*b^7*tan(1/2*e*x + 1/2*d)^4 + 150*a^8*tan(1/2*e*x + 1/2*d)^3 - 450*a^7*b*tan(1/2
*e*x + 1/2*d)^3 + 500*a^6*b^2*tan(1/2*e*x + 1/2*d)^3 - 300*a^5*b^3*tan(1/2*e*x + 1/2*d)^3 + 126*a^4*b^4*tan(1/
2*e*x + 1/2*d)^3 + 22*a^3*b^5*tan(1/2*e*x + 1/2*d)^3 - 48*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 + 150*a^8*tan(1/2*e*x + 1/2*d)^2 - 300*a^7*b*tan(1/2*e*x + 1/2*d)^
2 + 120*a^6*b^2*tan(1/2*e*x + 1/2*d)^2 + 60*a^5*b^3*tan(1/2*e*x + 1/2*d)^2 - 102*a^4*b^4*tan(1/2*e*x + 1/2*d)^
2 + 144*a^3*b^5*tan(1/2*e*x + 1/2*d)^2 - 60*a^2*b^6*tan(1/2*e*x + 1/2*d)^2 + 12*a*b^7*tan(1/2*e*x + 1/2*d)^2 +
75*a^8*tan(1/2*e*x + 1/2*d) - 75*a^7*b*tan(1/2*e*x + 1/2*d) - 75*a^6*b^2*tan(1/2*e*x + 1/2*d) + 75*a^5*b^3*ta
n(1/2*e*x + 1/2*d) - 39*a^4*b^4*tan(1/2*e*x + 1/2*d) + 39*a^3*b^5*tan(1/2*e*x + 1/2*d) + 33*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) + 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*e*x + 1/2*d)^2 - b*tan(1/2*e*x + 1/2*d)^2 + 2*a*ta
n(1/2*e*x + 1/2*d) + a + b)^3) + 3*(5*a^3 + 3*a*b^2)*log(abs(2*a*tan(1/2*e*x + 1/2*d) - 2*b*tan(1/2*e*x + 1/2*
d) + 2*a - 2*abs(b))/abs(2*a*tan(1/2*e*x + 1/2*d) - 2*b*tan(1/2*e*x + 1/2*d) + 2*a + 2*abs(b)))/(b^6*abs(b)))/
e

Mupad [B] (verification not implemented)

Time = 32.74 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.40 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {a\,\left (2\,a+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )\right )\,\left (5\,a^2+3\,b^2\right )}{2\,b\,\left (5\,a^3+3\,a\,b^2\right )}\right )\,\left (5\,a^2+3\,b^2\right )}{16\,b^7\,e}-\frac {\frac {15\,a^8-31\,a^6\,b^2+9\,a^4\,b^4+15\,a^2\,b^6}{6\,b^6\,{\left (a-b\right )}^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (25\,a^8-50\,a^7\,b+20\,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\right )}{b^6\,{\left (a-b\right )}^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (25\,a^7-75\,a^6\,b+90\,a^5\,b^2-70\,a^4\,b^3+45\,a^3\,b^4-15\,a^2\,b^5+4\,a\,b^6\right )}{2\,b^6\,{\left (a-b\right )}^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (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\right )}{3\,b^6\,{\left (a-b\right )}^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (5\,a^6-15\,a^5\,b+18\,a^4\,b^2-14\,a^3\,b^3+9\,a^2\,b^4-3\,a\,b^5+2\,b^6\right )}{2\,b^6\,\left (a-b\right )}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (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\right )}{2\,b^6\,{\left (a-b\right )}^3}}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (48\,a^3-96\,a^2\,b+48\,a\,b^2\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (-120\,a^3-120\,a^2\,b+24\,a\,b^2+24\,b^3\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (-120\,a^3+120\,a^2\,b+24\,a\,b^2-24\,b^3\right )+24\,a\,b^2+24\,a^2\,b-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (96\,a\,b^2-160\,a^3\right )+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (48\,a^3+96\,a^2\,b+48\,a\,b^2\right )+8\,a^3+8\,b^3\right )} \]

[In]

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

[Out]

(a*atanh((a*(2*a + tan(d/2 + (e*x)/2)*(2*a - 2*b))*(5*a^2 + 3*b^2))/(2*b*(3*a*b^2 + 5*a^3)))*(5*a^2 + 3*b^2))/
(16*b^7*e) - ((15*a^8 + 15*a^2*b^6 + 9*a^4*b^4 - 31*a^6*b^2)/(6*b^6*(a - b)^3) + (tan(d/2 + (e*x)/2)^2*(2*a*b^
7 - 50*a^7*b + 25*a^8 - 10*a^2*b^6 + 24*a^3*b^5 - 17*a^4*b^4 + 10*a^5*b^3 + 20*a^6*b^2))/(b^6*(a - b)^3) + (ta
n(d/2 + (e*x)/2)^4*(4*a*b^6 - 75*a^6*b + 25*a^7 - 15*a^2*b^5 + 45*a^3*b^4 - 70*a^4*b^3 + 90*a^5*b^2))/(2*b^6*(
a - b)^2) + (tan(d/2 + (e*x)/2)^3*(6*a*b^7 - 225*a^7*b + 75*a^8 - 2*b^8 - 24*a^2*b^6 + 11*a^3*b^5 + 63*a^4*b^4
- 150*a^5*b^3 + 250*a^6*b^2))/(3*b^6*(a - b)^3) + (tan(d/2 + (e*x)/2)^5*(5*a^6 - 15*a^5*b - 3*a*b^5 + 2*b^6 +
9*a^2*b^4 - 14*a^3*b^3 + 18*a^4*b^2))/(2*b^6*(a - b)) + (tan(d/2 + (e*x)/2)*(25*a^8 - 25*a^7*b - 5*a*b^7 + 2*
b^8 + 11*a^2*b^6 + 13*a^3*b^5 - 13*a^4*b^4 + 25*a^5*b^3 - 25*a^6*b^2))/(2*b^6*(a - b)^3))/(e*(tan(d/2 + (e*x)/
2)^5*(48*a*b^2 - 96*a^2*b + 48*a^3) + tan(d/2 + (e*x)/2)^6*(24*a*b^2 - 24*a^2*b + 8*a^3 - 8*b^3) - tan(d/2 + (
e*x)/2)^2*(24*a*b^2 - 120*a^2*b - 120*a^3 + 24*b^3) - tan(d/2 + (e*x)/2)^4*(24*a*b^2 + 120*a^2*b - 120*a^3 - 2
4*b^3) + 24*a*b^2 + 24*a^2*b - tan(d/2 + (e*x)/2)^3*(96*a*b^2 - 160*a^3) + tan(d/2 + (e*x)/2)*(48*a*b^2 + 96*a
^2*b + 48*a^3) + 8*a^3 + 8*b^3))