\(\int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx\) [380]

Optimal result

Integrand size = 24, antiderivative size = 207 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)+a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a-a \cos (d+e x)+c \sin (d+e x))} \]

[Out]

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))

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 207, 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, 3200, 31} \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]

[In]

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

[Out]

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

Rule 31

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

Rule 3200

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Cot[(d + e*x)/2], x]}, Dist[-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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(207)=414\).

Time = 7.97 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.39 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {\sin \left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (150 a^6+130 a^4 c^2+24 a^2 c^4-225 a^6 \cos (d+e x)-255 a^4 c^2 \cos (d+e x)-42 a^2 c^4 \cos (d+e x)-24 c^6 \cos (d+e x)+90 a^6 \cos (2 (d+e x))+174 a^4 c^2 \cos (2 (d+e x))-15 a^6 \cos (3 (d+e x))-49 a^4 c^2 \cos (3 (d+e x))+18 a^2 c^4 \cos (3 (d+e x))+8 c^6 \cos (3 (d+e x))-192 \left (5 a^3+3 a c^2\right ) \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )^3+192 \left (5 a^3+3 a c^2\right ) \log \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (c \cos \left (\frac {1}{2} (d+e x)\right )+a \sin \left (\frac {1}{2} (d+e x)\right )\right )^3+75 a^5 c \sin (d+e x)+75 a^3 c^3 \sin (d+e x)-12 a c^5 \sin (d+e x)-60 a^5 c \sin (2 (d+e x))-156 a^3 c^3 \sin (2 (d+e x))-12 a c^5 \sin (2 (d+e x))+15 a^5 c \sin (3 (d+e x))+79 a^3 c^3 \sin (3 (d+e x))+20 a c^5 \sin (3 (d+e x))\right )}{384 c^7 e (a-a \cos (d+e x)+c \sin (d+e x))^4} \]

[In]

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

[Out]

(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*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)]))/(384*c^7*e*(a - a*Cos[d + e*
x] + c*Sin[d + e*x])^4)

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {-\frac {1}{24 c^{4} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}-\frac {10 a^{2}+3 c^{2}}{8 c^{6} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}+\frac {a}{4 c^{5} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {4 a^{6}+6 a^{4} c^{2}-2 c^{6}}{16 a^{3} c^{5} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {10 a^{6}+9 a^{4} c^{2}+c^{6}}{8 c^{6} a^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{24 a^{3} c^{4} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}}{16 e}\)	\(253\)
default	\(\frac {-\frac {1}{24 c^{4} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}-\frac {10 a^{2}+3 c^{2}}{8 c^{6} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}+\frac {a}{4 c^{5} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}-\frac {4 a^{6}+6 a^{4} c^{2}-2 c^{6}}{16 a^{3} c^{5} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{2}}-\frac {10 a^{6}+9 a^{4} c^{2}+c^{6}}{8 c^{6} a^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}-\frac {a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{24 a^{3} c^{4} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 c^{7}}}{16 e}\)	\(253\)
norman	\(\frac {-\frac {1}{384 c e}+\frac {a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{128 c^{2} e}+\frac {\left (50 a^{6}+30 a^{4} c^{2}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{384 c^{7} e}-\frac {\left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{128 c^{3} e}+\frac {a \left (15 a^{4}+9 a^{2} c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{64 c^{6} e}-\frac {\left (60 a^{4}+36 a^{2} c^{2}+3 c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{384 a \,c^{4} e}}{\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}-\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 c^{7} e}+\frac {a \left (5 a^{2}+3 c^{2}\right ) \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{32 c^{7} e}\)	\(261\)
parallelrisch	\(\frac {60 \left (a^{2}+\frac {3 c^{2}}{5}\right ) \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} a \ln \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )-60 \left (a^{2}+\frac {3 c^{2}}{5}\right ) \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3} a \ln \left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )+\left (110 a^{6}+66 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}+9 \left (30 a^{5} c +18 a^{3} c^{3}+a \,c^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+9 \left (20 a^{4} c^{2}+12 a^{2} c^{4}+c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )-\cot \left (\frac {e x}{2}+\frac {d}{2}\right )^{3} c^{6}+3 \cot \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a \,c^{5}+3 \left (-5 a^{2} c^{4}-3 c^{6}\right ) \cot \left (\frac {e x}{2}+\frac {d}{2}\right )}{384 c^{7} e \left (c +a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )\right )^{3}}\)	\(262\)
risch	\(\frac {i \left (-4 c^{5}+45 a^{4} c +15 i a^{5}+15 i a \,c^{4} {\mathrm e}^{i \left (e x +d \right )}+12 i a \,c^{4} {\mathrm e}^{2 i \left (e x +d \right )}+60 i a^{3} c^{2} {\mathrm e}^{2 i \left (e x +d \right )}+45 i a^{3} c^{2} {\mathrm e}^{4 i \left (e x +d \right )}-3 a^{2} c^{3}-130 i a^{3} c^{2} {\mathrm e}^{3 i \left (e x +d \right )}-24 i a \,c^{4} {\mathrm e}^{3 i \left (e x +d \right )}-12 i a \,c^{4}+12 c^{5} {\mathrm e}^{2 i \left (e x +d \right )}+75 i a^{5} {\mathrm e}^{4 i \left (e x +d \right )}-150 i a^{5} {\mathrm e}^{3 i \left (e x +d \right )}-41 i a^{3} c^{2}-75 i a^{5} {\mathrm e}^{i \left (e x +d \right )}-15 i a^{5} {\mathrm e}^{5 i \left (e x +d \right )}-30 a^{2} c^{3} {\mathrm e}^{i \left (e x +d \right )}-150 a^{4} c \,{\mathrm e}^{i \left (e x +d \right )}+150 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}-75 a^{4} c \,{\mathrm e}^{4 i \left (e x +d \right )}+60 i a^{3} c^{2} {\mathrm e}^{i \left (e x +d \right )}+9 i a \,c^{4} {\mathrm e}^{5 i \left (e x +d \right )}+6 i a^{3} c^{2} {\mathrm e}^{5 i \left (e x +d \right )}+150 a^{4} c \,{\mathrm e}^{2 i \left (e x +d \right )}-45 a^{2} c^{3} {\mathrm e}^{4 i \left (e x +d \right )}+30 a^{4} c \,{\mathrm e}^{5 i \left (e x +d \right )}+18 a^{2} c^{3} {\mathrm e}^{5 i \left (e x +d \right )}+60 a^{2} c^{3} {\mathrm e}^{2 i \left (e x +d \right )}\right )}{48 \left (c \,{\mathrm e}^{2 i \left (e x +d \right )}-i a \,{\mathrm e}^{2 i \left (e x +d \right )}-c +2 i a \,{\mathrm e}^{i \left (e x +d \right )}-i a \right )^{3} c^{6} e}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i c -a}{i c +a}\right )}{32 c^{7} e}+\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i c -a}{i c +a}\right )}{32 c^{5} e}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (e x +d \right )}-1\right )}{32 c^{7} e}-\frac {3 a \ln \left ({\mathrm e}^{i \left (e x +d \right )}-1\right )}{32 c^{5} e}\)	\(591\)

[In]

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

[Out]

1/16/e*(-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))-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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (204) = 408\).

Time = 0.28 (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=\frac {60 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + 2 \, {\left (45 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - 4 \, c^{6}\right )} \cos \left (e x + d\right )^{3} - 12 \, {\left (10 \, a^{4} c^{2} + a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{4} c^{2} - 2 \, a^{2} c^{4} - 2 \, c^{6}\right )} \cos \left (e x + d\right ) - 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} - {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} - \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) + 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} - {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) + 2 \, {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 6 \, a c^{5} + {\left (15 \, a^{5} c - 41 \, a^{3} c^{3} - 12 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (10 \, a^{5} c - 9 \, a^{3} c^{3} - a c^{5}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{192 \, {\left ({\left (a^{3} c^{7} - 3 \, a c^{9}\right )} e \cos \left (e x + d\right )^{3} - 3 \, {\left (a^{3} c^{7} - a c^{9}\right )} e \cos \left (e x + d\right )^{2} + 3 \, {\left (a^{3} c^{7} + a c^{9}\right )} e \cos \left (e x + d\right ) - {\left (a^{3} c^{7} + 3 \, a c^{9}\right )} e + {\left (6 \, a^{2} c^{8} e \cos \left (e x + d\right ) - {\left (3 \, a^{2} c^{8} - c^{10}\right )} e \cos \left (e x + d\right )^{2} - {\left (3 \, a^{2} c^{8} + c^{10}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {a^{3} c^{5} - \frac {3 \, a^{4} c^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, {\left (5 \, a^{5} c^{3} + 3 \, a^{3} c^{5}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {{\left (110 \, a^{6} c^{2} + 66 \, a^{4} c^{4} + 3 \, a^{2} c^{6} + c^{8}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, {\left (50 \, a^{7} c + 30 \, a^{5} c^{3} + a c^{7}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, {\left (20 \, a^{8} + 12 \, a^{6} c^{2} + a^{2} c^{6}\right )} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}}}{\frac {a^{3} c^{9} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, a^{4} c^{8} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, a^{5} c^{7} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}} + \frac {a^{6} c^{6} \sin \left (e x + d\right )^{6}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{6}}} - \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (c + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \]

[In]

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

[Out]

-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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=-\frac {\frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) \right |}\right )}{c^{7}} - \frac {12 \, {\left (5 \, a^{4} + 3 \, a^{2} c^{2}\right )} \log \left ({\left | a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + c \right |}\right )}{a c^{7}} + \frac {60 \, a^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 36 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 150 \, a^{7} c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 90 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 3 \, a c^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 110 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 66 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + c^{8} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 15 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 9 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a^{3} c^{5}}{{\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + c \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )\right )}^{3} a^{3} c^{6}}}{384 \, e} \]

[In]

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

[Out]

-1/384*(12*(5*a^3 + 3*a*c^2)*log(abs(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

Mupad [B] (verification not implemented)

Time = 30.81 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {a\,\left (c+2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (5\,a^2+3\,c^2\right )}{c\,\left (5\,a^3+3\,a\,c^2\right )}\right )\,\left (5\,a^2+3\,c^2\right )}{16\,c^7\,e}-\frac {\frac {1}{3\,c}-\frac {a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (5\,a^2+3\,c^2\right )}{c^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (110\,a^6+66\,a^4\,c^2+3\,a^2\,c^4+c^6\right )}{3\,a^3\,c^4}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (20\,a^6+12\,a^4\,c^2+c^6\right )}{a\,c^6}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (50\,a^6+30\,a^4\,c^2+c^6\right )}{a^2\,c^5}}{e\,\left (128\,a^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+384\,a^2\,c\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5+384\,a\,c^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+128\,c^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\right )} \]

[In]

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

[Out]

(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*(1
10*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))