Integrand size = 24, antiderivative size = 215 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {a \left (5 a^2+3 b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{32 b^7 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{96 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))^2}-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{96 b^6 e (a+b \cos (d+e x)+a \sin (d+e x))} \] Output:
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*si n(e*x+d))
Leaf count is larger than twice the leaf count of optimal. \(632\) vs. \(2(215)=430\).
Time = 1.61 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\frac {-12 a \left (5 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )+12 a \left (5 a^2+3 b^2\right ) \log \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )+\frac {b \left (150 a^6+130 a^4 b^2+24 a^2 b^4-3 a^2 \left (25 a^4-50 a^3 b+5 a^2 b^2-30 a b^3+4 b^4\right ) \cos (d+e x)-6 a^2 \left (15 a^4+20 a^3 b+9 a^2 b^2+2 a b^3-2 b^4\right ) \cos (2 (d+e x))+15 a^6 \cos (3 (d+e x))-30 a^5 b \cos (3 (d+e x))-41 a^4 b^2 \cos (3 (d+e x))-38 a^3 b^3 \cos (3 (d+e x))-12 a^2 b^4 \cos (3 (d+e x))-8 a b^5 \cos (3 (d+e x))+225 a^6 \sin (d+e x)+75 a^5 b \sin (d+e x)+180 a^4 b^2 \sin (d+e x)+15 a^3 b^3 \sin (d+e x)+27 a^2 b^4 \sin (d+e x)+12 a b^5 \sin (d+e x)+12 b^6 \sin (d+e x)-60 a^6 \sin (2 (d+e x))+120 a^5 b \sin (2 (d+e x))+54 a^4 b^2 \sin (2 (d+e x))+102 a^3 b^3 \sin (2 (d+e x))+6 a^2 b^4 \sin (2 (d+e x))+6 a b^5 \sin (2 (d+e x))-15 a^6 \sin (3 (d+e x))-45 a^5 b \sin (3 (d+e x))-4 a^4 b^2 \sin (3 (d+e x))+3 a^3 b^3 \sin (3 (d+e x))+15 a^2 b^4 \sin (3 (d+e x))+4 a b^5 \sin (3 (d+e x))+4 b^6 \sin (3 (d+e x))\right )}{(a+b) \left (\cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^3 \left ((a+b) \cos \left (\frac {1}{2} (d+e x)\right )+(a-b) \sin \left (\frac {1}{2} (d+e x)\right )\right )^3}}{384 b^7 e} \] Input:
Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]
Output:
(-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*C os[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*Si n[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)] + 1 5*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)
Time = 0.85 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 3608, 27, 3042, 3635, 25, 3042, 3632, 3042, 3602, 16}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(2 a \sin (d+e x)+2 a+2 b \cos (d+e x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(2 a \sin (d+e x)+2 a+2 b \cos (d+e x))^4}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {\int -\frac {-2 \sin (d+e x) a+3 a-2 b \cos (d+e x)}{4 (\sin (d+e x) a+a+b \cos (d+e x))^3}dx}{12 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-2 \sin (d+e x) a+3 a-2 b \cos (d+e x)}{(\sin (d+e x) a+a+b \cos (d+e x))^3}dx}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-2 \sin (d+e x) a+3 a-2 b \cos (d+e x)}{(\sin (d+e x) a+a+b \cos (d+e x))^3}dx}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle -\frac {\frac {\int -\frac {-5 \sin (d+e x) a^2-5 b \cos (d+e x) a+2 \left (5 a^2+2 b^2\right )}{(\sin (d+e x) a+a+b \cos (d+e x))^2}dx}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {-5 \sin (d+e x) a^2-5 b \cos (d+e x) a+2 \left (5 a^2+2 b^2\right )}{(\sin (d+e x) a+a+b \cos (d+e x))^2}dx}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {-5 \sin (d+e x) a^2-5 b \cos (d+e x) a+2 \left (5 a^2+2 b^2\right )}{(\sin (d+e x) a+a+b \cos (d+e x))^2}dx}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle -\frac {-\frac {-3 a \left (\frac {5 a^2}{b^2}+3\right ) \int \frac {1}{\sin (d+e x) a+a+b \cos (d+e x)}dx-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{b^2 e (a \sin (d+e x)+a+b \cos (d+e x))}}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-3 a \left (\frac {5 a^2}{b^2}+3\right ) \int \frac {1}{\sin (d+e x) a+a+b \cos (d+e x)}dx-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{b^2 e (a \sin (d+e x)+a+b \cos (d+e x))}}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 3602 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (\frac {5 a^2}{b^2}+3\right ) \int \frac {1}{a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )}d\cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )}{e}-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{b^2 e (a \sin (d+e x)+a+b \cos (d+e x))}}{2 b^2}-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {-\frac {5 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{2 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}-\frac {\frac {3 a \left (\frac {5 a^2}{b^2}+3\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{b e}-\frac {a \left (15 a^2+4 b^2\right ) \cos (d+e x)-b \left (15 a^2+4 b^2\right ) \sin (d+e x)}{b^2 e (a \sin (d+e x)+a+b \cos (d+e x))}}{2 b^2}}{48 b^2}-\frac {a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}\) |
Input:
Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]
Output:
-1/48*(a*Cos[d + e*x] - b*Sin[d + e*x])/(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]))/(2*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2) - ((3*a*(3 + (5*a^2)/b^2)*Log[a + b*Co t[d/2 + Pi/4 + (e*x)/2]])/(b*e) - (a*(15*a^2 + 4*b^2)*Cos[d + e*x] - b*(15 *a^2 + 4*b^2)*Sin[d + e*x])/(b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))) /(2*b^2))/(48*b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-1), x_Symbol] :> Module[{f = FreeFactors[Cot[(d + e*x)/2 + Pi/4], x]}, Si mp[-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]
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] + Simp[ 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*Cos[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]
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] + Simp[(a*A - b*B - c*C)/(a^2 - b^2 - c^2) Int[1/(a + b*Cos[d + e*x] + c*S in[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] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[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]
Time = 4.09 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.85
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (5 a^{3}-5 a^{2} b +3 a \,b^{2}-3 b^{3}\right ) a \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{7} \left (a -b \right )}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{4} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {2 a^{6}-3 a^{5} b +3 a^{4} b^{2}-6 a^{3} b^{3}-3 a \,b^{5}-b^{6}}{2 b^{5} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {5 a^{6}-12 a^{5} b +12 a^{4} b^{2}-12 a^{3} b^{3}+9 a^{2} b^{4}+2 b^{6}}{2 b^{6} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}-\frac {1}{3 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {-2 a -b}{2 b^{5} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {5 a^{2}+2 b a +2 b^{2}}{2 b^{6} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{7}}}{16 e}\) | \(398\) |
| default | \(\frac {\frac {\left (5 a^{3}-5 a^{2} b +3 a \,b^{2}-3 b^{3}\right ) a \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{2 b^{7} \left (a -b \right )}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{4} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {2 a^{6}-3 a^{5} b +3 a^{4} b^{2}-6 a^{3} b^{3}-3 a \,b^{5}-b^{6}}{2 b^{5} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{2}}-\frac {5 a^{6}-12 a^{5} b +12 a^{4} b^{2}-12 a^{3} b^{3}+9 a^{2} b^{4}+2 b^{6}}{2 b^{6} \left (a -b \right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}-\frac {1}{3 b^{4} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {-2 a -b}{2 b^{5} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {5 a^{2}+2 b a +2 b^{2}}{2 b^{6} \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 b^{7}}}{16 e}\) | \(398\) |
| risch | \(\frac {i \left (12 b^{5} {\mathrm e}^{2 i \left (e x +d \right )}-75 a^{5} {\mathrm e}^{i \left (e x +d \right )}+150 a^{5} {\mathrm e}^{3 i \left (e x +d \right )}-15 a^{5} {\mathrm e}^{5 i \left (e x +d \right )}+150 a^{4} b \,{\mathrm e}^{2 i \left (e x +d \right )}+75 a^{4} b \,{\mathrm e}^{4 i \left (e x +d \right )}+60 a^{2} b^{3} {\mathrm e}^{2 i \left (e x +d \right )}+12 i a \,b^{4}+41 i a^{3} b^{2}+130 a^{3} b^{2} {\mathrm e}^{3 i \left (e x +d \right )}+24 a \,b^{4} {\mathrm e}^{3 i \left (e x +d \right )}+6 a^{3} b^{2} {\mathrm e}^{5 i \left (e x +d \right )}+9 a \,b^{4} {\mathrm e}^{5 i \left (e x +d \right )}+45 a^{2} b^{3} {\mathrm e}^{4 i \left (e x +d \right )}+15 a \,b^{4} {\mathrm e}^{i \left (e x +d \right )}+60 a^{3} b^{2} {\mathrm e}^{i \left (e x +d \right )}+3 a^{2} b^{3}+4 b^{5}-45 a^{4} b -75 i a^{5} {\mathrm e}^{4 i \left (e x +d \right )}+150 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}-15 i a^{5}+60 i a^{3} b^{2} {\mathrm e}^{2 i \left (e x +d \right )}+30 i a^{2} b^{3} {\mathrm e}^{i \left (e x +d \right )}+150 i a^{4} b \,{\mathrm e}^{i \left (e x +d \right )}-18 i a^{2} b^{3} {\mathrm e}^{5 i \left (e x +d \right )}-45 i a^{3} b^{2} {\mathrm e}^{4 i \left (e x +d \right )}+12 i a \,b^{4} {\mathrm e}^{2 i \left (e x +d \right )}-30 i a^{4} b \,{\mathrm e}^{5 i \left (e x +d \right )}\right )}{48 \left (-i a \,{\mathrm e}^{2 i \left (e x +d \right )}+b \,{\mathrm e}^{2 i \left (e x +d \right )}+i a +2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{3} b^{6} e}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{32 b^{7} e}+\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a +b}{i b +a}\right )}{32 b^{5} e}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{32 b^{7} e}-\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+i\right )}{32 b^{5} e}\) | \(581\) |
| norman | \(\frac {-\frac {\left (50 a^{7}+100 a^{6} b +75 a^{5} b^{2}+50 a^{4} b^{3}+22 a^{3} b^{4}-6 a^{2} b^{5}-a \,b^{6}+4 b^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{16 b^{6} e \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (50 a^{7}-100 a^{6} b +75 a^{5} b^{2}-50 a^{4} b^{3}+22 a^{3} b^{4}+6 a^{2} b^{5}-a \,b^{6}-4 b^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{16 b^{6} e \left (5 a^{2}-3 b^{2}\right )}-\frac {75 a^{8}+225 a^{7} b +250 a^{6} b^{2}+150 a^{5} b^{3}+63 a^{4} b^{4}-11 a^{3} b^{5}-24 b^{6} a^{2}-6 a \,b^{7}-2 b^{8}}{96 a e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}-\frac {\left (125 a^{8}+125 a^{7} b +50 a^{6} b^{2}+50 a^{5} b^{3}+5 a^{4} b^{4}+5 a^{3} b^{5}+32 b^{6} a^{2}+2 a \,b^{7}+2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{32 a e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (75 a^{8}-225 a^{7} b +250 a^{6} b^{2}-150 a^{5} b^{3}+63 a^{4} b^{4}+11 a^{3} b^{5}-24 b^{6} a^{2}+6 a \,b^{7}-2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{96 a e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}+\frac {\left (125 a^{8}-125 a^{7} b +50 a^{6} b^{2}-50 a^{5} b^{3}+5 a^{4} b^{4}-5 a^{3} b^{5}+32 b^{6} a^{2}-2 a \,b^{7}+2 b^{8}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{32 a e \,b^{6} \left (5 a^{2}-3 b^{2}\right )}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )^{3}}-\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 b^{7} e}+\frac {a \left (5 a^{2}+3 b^{2}\right ) \ln \left (a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a +b \right )}{32 b^{7} e}\) | \(677\) |
| parallelrisch | \(\frac {15 a^{2} \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^{2}+\frac {3 b^{2}}{5}\right ) \left (a +b \right )^{2} \ln \left (a +b +\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )-15 a^{2} \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^{2}+\frac {3 b^{2}}{5}\right ) \left (a +b \right )^{2} \ln \left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+60 \left (a \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 ) \left (a +b \right ) \cos \left (2 e x +2 d \right )+\left (-\frac {1}{4} a^{8}+\frac {1}{30} b^{8}+\frac {2}{15} b^{6} a^{2}+\frac {1}{60} a \,b^{7}+\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}\right ) \cos \left (3 e x +3 d \right )+\frac {a \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 ) \sin \left (3 e x +3 d \right )}{6}+a^{2} \left (a^{6}+\frac {3}{10} b^{6}-\frac {9}{10} a^{4} b^{2}-\frac {3}{5} a^{2} b^{4}\right ) \sin \left (2 e x +2 d \right )+\left (\frac {5}{4} a^{8}+\frac {1}{10} b^{8}+\frac {4}{5} b^{6} a^{2}+\frac {1}{20} a \,b^{7}+\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}\right ) \cos \left (e x +d \right )-\frac {5 a \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^{5}+\frac {1}{2} a^{4} b -\frac {9}{10} b^{2} a^{3}-\frac {1}{5} a^{2} b^{3}-\frac {1}{10} b^{4} a -\frac {1}{5} b^{5}\right ) \left (a^{2}+\frac {3 b^{2}}{5}\right )}{3}\right )}{2}\right ) b}{96 b^{7} a \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 ) e \left (a +b \right )^{2}}\) | \(884\) |
Input:
int(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x,method=_RETURNVERBOSE)
Output:
1/16/e*(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*tan(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)-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)))
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (201) = 402\).
Time = 0.11 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.39 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="fricas")
Output:
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)*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(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))
Timed out. \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (201) = 402\).
Time = 0.08 (sec) , antiderivative size = 963, normalized size of antiderivative = 4.48 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="maxima")
Output:
-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 + 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)*sin (e*x + d)^2/(cos(e*x + d) + 1)^2 + 2*(75*a^8 - 225*a^7*b + 250*a^6*b^2 - 1 50*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)*sin(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)^...
Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (201) = 402\).
Time = 0.16 (sec) , antiderivative size = 957, normalized size of antiderivative = 4.45 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x, algorithm="giac")
Output:
-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*tan(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*...
Time = 20.63 (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 )} \] Input:
int(1/(2*a + 2*b*cos(d + e*x) + 2*a*sin(d + e*x))^4,x)
Output:
(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) + (tan(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 - 24*b^3) + 2 4*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))
Time = 0.24 (sec) , antiderivative size = 4415, normalized size of antiderivative = 20.53 \[ \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(2*a+2*b*cos(e*x+d)+2*a*sin(e*x+d))^4,x)
Output:
( - 45*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**8*b + 90* cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**7*b**2 - 57*cos( d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**6*b**3 + 24*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**5*b**4 - 3*cos(d + e*x)* log(tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**4*b**5 - 18*cos(d + e*x)*log( tan((d + e*x)/2) + 1)*sin(d + e*x)**2*a**3*b**6 + 9*cos(d + e*x)*log(tan(( d + e*x)/2) + 1)*sin(d + e*x)**2*a**2*b**7 - 90*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)*a**8*b + 180*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)*a**7*b**2 - 144*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*si n(d + e*x)*a**6*b**3 + 108*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)*a**5*b**4 - 54*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*sin(d + e*x)*a* *4*b**5 - 45*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*a**8*b + 90*cos(d + e* x)*log(tan((d + e*x)/2) + 1)*a**7*b**2 - 87*cos(d + e*x)*log(tan((d + e*x) /2) + 1)*a**6*b**3 + 84*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*a**5*b**4 - 51*cos(d + e*x)*log(tan((d + e*x)/2) + 1)*a**4*b**5 + 18*cos(d + e*x)*log (tan((d + e*x)/2) + 1)*a**3*b**6 - 9*cos(d + e*x)*log(tan((d + e*x)/2) + 1 )*a**2*b**7 + 45*cos(d + e*x)*log(tan((d + e*x)/2)*a - tan((d + e*x)/2)*b + a + b)*sin(d + e*x)**2*a**8*b - 90*cos(d + e*x)*log(tan((d + e*x)/2)*a - tan((d + e*x)/2)*b + a + b)*sin(d + e*x)**2*a**7*b**2 + 57*cos(d + e*x)*l og(tan((d + e*x)/2)*a - tan((d + e*x)/2)*b + a + b)*sin(d + e*x)**2*a**...