Integrand size = 24, antiderivative size = 207 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)+a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a-a \cos (d+e x)+c \sin (d+e x))} \] Output:
1/32*a*(5*a^2+3*c^2)*ln(a+c*cot(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 ))
Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(207)=414\).
Time = 7.84 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.39 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {\sin \left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (150 a^6+130 a^4 c^2+24 a^2 c^4-225 a^6 \cos (d+e x)-255 a^4 c^2 \cos (d+e x)-42 a^2 c^4 \cos (d+e x)-24 c^6 \cos (d+e x)+90 a^6 \cos (2 (d+e x))+174 a^4 c^2 \cos (2 (d+e x))-15 a^6 \cos (3 (d+e x))-49 a^4 c^2 \cos (3 (d+e x))+18 a^2 c^4 \cos (3 (d+e x))+8 c^6 \cos (3 (d+e x))-192 \left (5 a^3+3 a c^2\right ) \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )^3+192 \left (5 a^3+3 a c^2\right ) \log \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )^3+75 a^5 c \sin (d+e x)+75 a^3 c^3 \sin (d+e x)-12 a c^5 \sin (d+e x)-60 a^5 c \sin (2 (d+e x))-156 a^3 c^3 \sin (2 (d+e x))-12 a c^5 \sin (2 (d+e x))+15 a^5 c \sin (3 (d+e x))+79 a^3 c^3 \sin (3 (d+e x))+20 a c^5 \sin (3 (d+e x))\right )}{384 c^7 e (a-a \cos (d+e x)+c \sin (d+e x))^4} \] Input:
Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]
Output:
(Sin[(d + e*x)/2]*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])*(150*a^6 + 130 *a^4*c^2 + 24*a^2*c^4 - 225*a^6*Cos[d + e*x] - 255*a^4*c^2*Cos[d + e*x] - 42*a^2*c^4*Cos[d + e*x] - 24*c^6*Cos[d + e*x] + 90*a^6*Cos[2*(d + e*x)] + 174*a^4*c^2*Cos[2*(d + e*x)] - 15*a^6*Cos[3*(d + e*x)] - 49*a^4*c^2*Cos[3* (d + e*x)] + 18*a^2*c^4*Cos[3*(d + e*x)] + 8*c^6*Cos[3*(d + e*x)] - 192*(5 *a^3 + 3*a*c^2)*Log[Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^3*(c*Cos[(d + e*x)/ 2] + a*Sin[(d + e*x)/2])^3 + 192*(5*a^3 + 3*a*c^2)*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^3*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^3 + 75*a^5*c*Sin[d + e*x] + 75*a^3*c^3*Sin[d + e*x] - 12*a*c^5*Si n[d + e*x] - 60*a^5*c*Sin[2*(d + e*x)] - 156*a^3*c^3*Sin[2*(d + e*x)] - 12 *a*c^5*Sin[2*(d + e*x)] + 15*a^5*c*Sin[3*(d + e*x)] + 79*a^3*c^3*Sin[3*(d + e*x)] + 20*a*c^5*Sin[3*(d + e*x)]))/(384*c^7*e*(a - a*Cos[d + e*x] + c*S in[d + e*x])^4)
Time = 0.82 (sec) , antiderivative size = 218, 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, 3600, 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.
| \(\displaystyle \int \frac {1}{(-2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(-2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^4}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {\int -\frac {2 \cos (d+e x) a+3 a-2 c \sin (d+e x)}{4 (-\cos (d+e x) a+a+c \sin (d+e x))^3}dx}{12 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 \cos (d+e x) a+3 a-2 c \sin (d+e x)}{(-\cos (d+e x) a+a+c \sin (d+e x))^3}dx}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {2 \cos (d+e x) a+3 a-2 c \sin (d+e x)}{(-\cos (d+e x) a+a+c \sin (d+e x))^3}dx}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle -\frac {\frac {\int -\frac {5 \cos (d+e x) a^2-5 c \sin (d+e x) a+2 \left (5 a^2+2 c^2\right )}{(-\cos (d+e x) a+a+c \sin (d+e x))^2}dx}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {5 \cos (d+e x) a^2-5 c \sin (d+e x) a+2 \left (5 a^2+2 c^2\right )}{(-\cos (d+e x) a+a+c \sin (d+e x))^2}dx}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {5 \cos (d+e x) a^2-5 c \sin (d+e x) a+2 \left (5 a^2+2 c^2\right )}{(-\cos (d+e x) a+a+c \sin (d+e x))^2}dx}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle -\frac {-\frac {-3 a \left (\frac {5 a^2}{c^2}+3\right ) \int \frac {1}{-\cos (d+e x) a+a+c \sin (d+e x)}dx-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))}}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-3 a \left (\frac {5 a^2}{c^2}+3\right ) \int \frac {1}{-\cos (d+e x) a+a+c \sin (d+e x)}dx-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))}}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 3600 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (\frac {5 a^2}{c^2}+3\right ) \int \frac {1}{a+c \cot \left (\frac {1}{2} (d+e x)\right )}d\cot \left (\frac {1}{2} (d+e x)\right )}{e}-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))}}{2 c^2}-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {-\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{2 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}-\frac {\frac {3 a \left (\frac {5 a^2}{c^2}+3\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{c e}-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))}}{2 c^2}}{48 c^2}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}\) |
Input:
Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]
Output:
-1/48*(c*Cos[d + e*x] + a*Sin[d + e*x])/(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]))/(2*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2) - ((3*a*(3 + (5*a^2)/c^2)*Log[a + c*Co t[(d + e*x)/2]])/(c*e) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] + a*(15*a^2 + 4* c^2)*Sin[d + e*x])/(c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])))/(2*c^2)) /(48*c^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], x]}, Simp[-f/e Subst[Int[1/(a + c*f*x), x], x, Cot[(d + e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[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 = 1.49 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {-\frac {4 a^{6}+6 c^{2} a^{4}-2 c^{6}}{16 a^{3} c^{5} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {10 a^{6}+9 c^{2} a^{4}+c^{6}}{8 c^{6} a^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 c^{2} a^{4}+3 a^{2} c^{4}+c^{6}}{24 a^{3} c^{4} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {1}{24 c^{4} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}-\frac {10 a^{2}+3 c^{2}}{8 c^{6} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}+\frac {a}{4 c^{5} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}}{16 e}\) | \(253\) |
| default | \(\frac {-\frac {4 a^{6}+6 c^{2} a^{4}-2 c^{6}}{16 a^{3} c^{5} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {10 a^{6}+9 c^{2} a^{4}+c^{6}}{8 c^{6} a^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 c^{2} a^{4}+3 a^{2} c^{4}+c^{6}}{24 a^{3} c^{4} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {1}{24 c^{4} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}-\frac {10 a^{2}+3 c^{2}}{8 c^{6} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}+\frac {a}{4 c^{5} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}}{16 e}\) | \(253\) |
| norman | \(\frac {-\frac {1}{384 c e}+\frac {a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{128 c^{2} e}+\frac {\left (50 a^{6}+30 c^{2} a^{4}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{384 c^{7} e}-\frac {\left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{128 c^{3} e}+\frac {a \left (15 a^{4}+9 a^{2} c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{64 c^{6} e}-\frac {\left (60 a^{4}+36 a^{2} c^{2}+3 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{384 a \,c^{4} e}}{\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 c^{7} e}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 c^{7} e}\) | \(261\) |
| parallelrisch | \(\frac {60 \left (a^{2}+\frac {3 c^{2}}{5}\right ) a \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )-60 \left (a^{2}+\frac {3 c^{2}}{5}\right ) a \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\left (110 a^{6}+66 c^{2} a^{4}+3 a^{2} c^{4}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}+9 \left (30 a^{5} c +18 a^{3} c^{3}+a \,c^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+9 \left (20 c^{2} a^{4}+12 a^{2} c^{4}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-\cot \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} c^{6}+3 \cot \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a \,c^{5}+3 \left (-5 a^{2} c^{4}-3 c^{6}\right ) \cot \left (\frac {e x}{2}+\frac {d}{2}\right )}{384 c^{7} e \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}\) | \(262\) |
| risch | \(\frac {i \left (-4 c^{5}-3 a^{2} c^{3}-75 i a^{5} {\mathrm e}^{i \left (e x +d \right )}-41 i a^{3} c^{2}-12 i a \,c^{4}-15 i a^{5} {\mathrm e}^{5 i \left (e x +d \right )}+30 a^{4} c \,{\mathrm e}^{5 i \left (e x +d \right )}+18 a^{2} c^{3} {\mathrm e}^{5 i \left (e x +d \right )}-30 a^{2} c^{3} {\mathrm e}^{i \left (e x +d \right )}+75 i a^{5} {\mathrm e}^{4 i \left (e x +d \right )}+150 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}-45 a^{2} c^{3} {\mathrm e}^{4 i \left (e x +d \right )}+150 a^{4} c \,{\mathrm e}^{2 i \left (e x +d \right )}+45 a^{4} c -150 i a^{5} {\mathrm e}^{3 i \left (e x +d \right )}-150 a^{4} c \,{\mathrm e}^{i \left (e x +d \right )}-75 a^{4} c \,{\mathrm e}^{4 i \left (e x +d \right )}+60 a^{2} c^{3} {\mathrm e}^{2 i \left (e x +d \right )}+12 c^{5} {\mathrm e}^{2 i \left (e x +d \right )}+15 i a^{5}+15 i a \,c^{4} {\mathrm e}^{i \left (e x +d \right )}-130 i a^{3} c^{2} {\mathrm e}^{3 i \left (e x +d \right )}-24 i a \,c^{4} {\mathrm e}^{3 i \left (e x +d \right )}+9 i a \,c^{4} {\mathrm e}^{5 i \left (e x +d \right )}+60 i a^{3} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+6 i a^{3} c^{2} {\mathrm e}^{5 i \left (e x +d \right )}+45 i a^{3} c^{2} {\mathrm e}^{4 i \left (e x +d \right )}+60 i a^{3} c^{2} {\mathrm e}^{i \left (e x +d \right )}+12 i a \,c^{4} {\mathrm e}^{2 i \left (e x +d \right )}\right )}{48 \left (c \,{\mathrm e}^{2 i \left (e x +d \right )}-i a \,{\mathrm e}^{2 i \left (e x +d \right )}-c +2 i a \,{\mathrm e}^{i \left (e x +d \right )}-i a \right )^{3} c^{6} e}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}-1\right )}{32 c^{7} e}-\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}-1\right )}{32 c^{5} e}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i c -a}{i c +a}\right )}{32 c^{7} e}+\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i c -a}{i c +a}\right )}{32 c^{5} e}\) | \(591\) |
Input:
int(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x,method=_RETURNVERBOSE)
Output:
1/16/e*(-1/16*(4*a^6+6*a^4*c^2-2*c^6)/a^3/c^5/(c+a*tan(1/2*e*x+1/2*d))^2-1 /8*(10*a^6+9*a^4*c^2+c^6)/c^6/a^3/(c+a*tan(1/2*e*x+1/2*d))-1/24*(a^6+3*a^4 *c^2+3*a^2*c^4+c^6)/a^3/c^4/(c+a*tan(1/2*e*x+1/2*d))^3+1/2*a*(5*a^2+3*c^2) /c^7*ln(c+a*tan(1/2*e*x+1/2*d))-1/24/c^4/tan(1/2*e*x+1/2*d)^3-1/8*(10*a^2+ 3*c^2)/c^6/tan(1/2*e*x+1/2*d)+1/4/c^5*a/tan(1/2*e*x+1/2*d)^2-1/2*a*(5*a^2+ 3*c^2)/c^7*ln(tan(1/2*e*x+1/2*d)))
Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (204) = 408\).
Time = 0.11 (sec) , antiderivative size = 796, normalized size of antiderivative = 3.85 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="fricas")
Output:
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 - 1 2*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 - 12*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))
Timed out. \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))**4,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {a^{3} c^{5} - \frac {3 \, a^{4} c^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, {\left (5 \, a^{5} c^{3} + 3 \, a^{3} c^{5}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {{\left (110 \, a^{6} c^{2} + 66 \, a^{4} c^{4} + 3 \, a^{2} c^{6} + c^{8}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, {\left (50 \, a^{7} c + 30 \, a^{5} c^{3} + a c^{7}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, {\left (20 \, a^{8} + 12 \, a^{6} c^{2} + a^{2} c^{6}\right )} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}}}{\frac {a^{3} c^{9} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, a^{4} c^{8} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, a^{5} c^{7} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}} + \frac {a^{6} c^{6} \sin \left (e x + d\right )^{6}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{6}}} - \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (c + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \] Input:
integrate(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="maxima")
Output:
-1/384*((a^3*c^5 - 3*a^4*c^4*sin(e*x + d)/(cos(e*x + d) + 1) + 3*(5*a^5*c^ 3 + 3*a^3*c^5)*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 + (110*a^6*c^2 + 66*a^4 *c^4 + 3*a^2*c^6 + c^8)*sin(e*x + d)^3/(cos(e*x + d) + 1)^3 + 3*(50*a^7*c + 30*a^5*c^3 + a*c^7)*sin(e*x + d)^4/(cos(e*x + d) + 1)^4 + 3*(20*a^8 + 12 *a^6*c^2 + a^2*c^6)*sin(e*x + d)^5/(cos(e*x + d) + 1)^5)/(a^3*c^9*sin(e*x + d)^3/(cos(e*x + d) + 1)^3 + 3*a^4*c^8*sin(e*x + d)^4/(cos(e*x + d) + 1)^ 4 + 3*a^5*c^7*sin(e*x + d)^5/(cos(e*x + d) + 1)^5 + a^6*c^6*sin(e*x + d)^6 /(cos(e*x + d) + 1)^6) - 12*(5*a^3 + 3*a*c^2)*log(c + a*sin(e*x + d)/(cos( e*x + d) + 1))/c^7 + 12*(5*a^3 + 3*a*c^2)*log(sin(e*x + d)/(cos(e*x + d) + 1))/c^7)/e
Time = 0.17 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) \right |}\right )}{c^{7}} - \frac {12 \, {\left (5 \, a^{4} + 3 \, a^{2} c^{2}\right )} \log \left ({\left | a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + c \right |}\right )}{a c^{7}} + \frac {60 \, a^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 36 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 150 \, a^{7} c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 90 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 3 \, a c^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 110 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 66 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + c^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 15 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 9 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a^{3} c^{5}}{{\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )\right )}^{3} a^{3} c^{6}}}{384 \, e} \] Input:
integrate(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="giac")
Output:
-1/384*(12*(5*a^3 + 3*a*c^2)*log(abs(tan(1/2*e*x + 1/2*d)))/c^7 - 12*(5*a^ 4 + 3*a^2*c^2)*log(abs(a*tan(1/2*e*x + 1/2*d) + c))/(a*c^7) + (60*a^8*tan( 1/2*e*x + 1/2*d)^5 + 36*a^6*c^2*tan(1/2*e*x + 1/2*d)^5 + 3*a^2*c^6*tan(1/2 *e*x + 1/2*d)^5 + 150*a^7*c*tan(1/2*e*x + 1/2*d)^4 + 90*a^5*c^3*tan(1/2*e* x + 1/2*d)^4 + 3*a*c^7*tan(1/2*e*x + 1/2*d)^4 + 110*a^6*c^2*tan(1/2*e*x + 1/2*d)^3 + 66*a^4*c^4*tan(1/2*e*x + 1/2*d)^3 + 3*a^2*c^6*tan(1/2*e*x + 1/2 *d)^3 + c^8*tan(1/2*e*x + 1/2*d)^3 + 15*a^5*c^3*tan(1/2*e*x + 1/2*d)^2 + 9 *a^3*c^5*tan(1/2*e*x + 1/2*d)^2 - 3*a^4*c^4*tan(1/2*e*x + 1/2*d) + a^3*c^5 )/((a*tan(1/2*e*x + 1/2*d)^2 + c*tan(1/2*e*x + 1/2*d))^3*a^3*c^6))/e
Time = 18.91 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {a\,\left (c+2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (5\,a^2+3\,c^2\right )}{c\,\left (5\,a^3+3\,a\,c^2\right )}\right )\,\left (5\,a^2+3\,c^2\right )}{16\,c^7\,e}-\frac {\frac {1}{3\,c}-\frac {a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (5\,a^2+3\,c^2\right )}{c^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (110\,a^6+66\,a^4\,c^2+3\,a^2\,c^4+c^6\right )}{3\,a^3\,c^4}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (20\,a^6+12\,a^4\,c^2+c^6\right )}{a\,c^6}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (50\,a^6+30\,a^4\,c^2+c^6\right )}{a^2\,c^5}}{e\,\left (128\,a^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+384\,a^2\,c\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5+384\,a\,c^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+128\,c^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\right )} \] Input:
int(1/(2*a - 2*a*cos(d + e*x) + 2*c*sin(d + e*x))^4,x)
Output:
(a*atanh((a*(c + 2*a*tan(d/2 + (e*x)/2))*(5*a^2 + 3*c^2))/(c*(3*a*c^2 + 5* a^3)))*(5*a^2 + 3*c^2))/(16*c^7*e) - (1/(3*c) - (a*tan(d/2 + (e*x)/2))/c^2 + (tan(d/2 + (e*x)/2)^2*(5*a^2 + 3*c^2))/c^3 + (tan(d/2 + (e*x)/2)^3*(110 *a^6 + c^6 + 3*a^2*c^4 + 66*a^4*c^2))/(3*a^3*c^4) + (tan(d/2 + (e*x)/2)^5* (20*a^6 + c^6 + 12*a^4*c^2))/(a*c^6) + (tan(d/2 + (e*x)/2)^4*(50*a^6 + c^6 + 30*a^4*c^2))/(a^2*c^5))/(e*(128*a^3*tan(d/2 + (e*x)/2)^6 + 128*c^3*tan( d/2 + (e*x)/2)^3 + 384*a*c^2*tan(d/2 + (e*x)/2)^4 + 384*a^2*c*tan(d/2 + (e *x)/2)^5))
Time = 0.20 (sec) , antiderivative size = 764, normalized size of antiderivative = 3.69 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(2*a-2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x)
Output:
(60*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**6*a**7 + 36*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**6*a**5*c**2 + 180*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**5*a**6*c + 108*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**5*a**4*c**3 + 180*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)* *4*a**5*c**2 + 108*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**4*a**3*c* *4 + 60*log(tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**3*a**4*c**3 + 36*log (tan((d + e*x)/2)*a + c)*tan((d + e*x)/2)**3*a**2*c**5 - 60*log(tan((d + e *x)/2))*tan((d + e*x)/2)**6*a**7 - 36*log(tan((d + e*x)/2))*tan((d + e*x)/ 2)**6*a**5*c**2 - 180*log(tan((d + e*x)/2))*tan((d + e*x)/2)**5*a**6*c - 1 08*log(tan((d + e*x)/2))*tan((d + e*x)/2)**5*a**4*c**3 - 180*log(tan((d + e*x)/2))*tan((d + e*x)/2)**4*a**5*c**2 - 108*log(tan((d + e*x)/2))*tan((d + e*x)/2)**4*a**3*c**4 - 60*log(tan((d + e*x)/2))*tan((d + e*x)/2)**3*a**4 *c**3 - 36*log(tan((d + e*x)/2))*tan((d + e*x)/2)**3*a**2*c**5 + 20*tan((d + e*x)/2)**6*a**7 + 12*tan((d + e*x)/2)**6*a**5*c**2 + tan((d + e*x)/2)** 6*a*c**6 - 90*tan((d + e*x)/2)**4*a**5*c**2 - 54*tan((d + e*x)/2)**4*a**3* c**4 - 90*tan((d + e*x)/2)**3*a**4*c**3 - 54*tan((d + e*x)/2)**3*a**2*c**5 - 3*tan((d + e*x)/2)**3*c**7 - 15*tan((d + e*x)/2)**2*a**3*c**4 - 9*tan(( d + e*x)/2)**2*a*c**6 + 3*tan((d + e*x)/2)*a**2*c**5 - a*c**6)/(384*tan((d + e*x)/2)**3*a*c**7*e*(tan((d + e*x)/2)**3*a**3 + 3*tan((d + e*x)/2)**2*a **2*c + 3*tan((d + e*x)/2)*a*c**2 + c**3))