\(\int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\) [110]

Optimal result

Integrand size = 28, antiderivative size = 227 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d} \]

[Out]

a*b^4*x/(a^2+b^2)^3+1/2*a*b^2*x/(a^2+b^2)^2+3/8*a*x/(a^2+b^2)+1/2*b^3*cos(d*x+c)^2/(a^2+b^2)^2/d+1/4*b*cos(d*x
+c)^4/(a^2+b^2)/d+b^5*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^3/d+1/2*a*b^2*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)^2/
d+3/8*a*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)/d+1/4*a*cos(d*x+c)^3*sin(d*x+c)/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3179, 2715, 8, 3177, 3212} \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a b^2 \sin (c+d x) \cos (c+d x)}{2 d \left (a^2+b^2\right )^2}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d \left (a^2+b^2\right )}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {b^3 \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )^2} \]

[In]

Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(a*b^4*x)/(a^2 + b^2)^3 + (a*b^2*x)/(2*(a^2 + b^2)^2) + (3*a*x)/(8*(a^2 + b^2)) + (b^3*Cos[c + d*x]^2)/(2*(a^2
+ b^2)^2*d) + (b*Cos[c + d*x]^4)/(4*(a^2 + b^2)*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^
3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)^2*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*(a^2 + b^2)*
d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*(a^2 + b^2)*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3177

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[a*(x/(a^2 + b^2)), x] + Dist[b/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3179

Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[b*(Cos[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1), x]
, x] + Dist[b^2/(a^2 + b^2), Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rule 3212

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(b*B + c*C)*(x/(b^2 + c^2)), x] + Simp[(c*B - b*C)*(L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a \int \cos ^4(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2} \\ & = \frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {\left (a b^2\right ) \int \cos ^2(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {(3 a) \int \cos ^2(c+d x) \, dx}{4 \left (a^2+b^2\right )} \\ & = \frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a b^2\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}+\frac {(3 a) \int 1 \, dx}{8 \left (a^2+b^2\right )} \\ & = \frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {12 a^5 c+40 a^3 b^2 c+60 a b^4 c+12 a^5 d x+40 a^3 b^2 d x+60 a b^4 d x+4 b \left (a^4+4 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+8 a^5 \sin (2 (c+d x))+24 a^3 b^2 \sin (2 (c+d x))+16 a b^4 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+a b^4 \sin (4 (c+d x))}{32 \left (a^2+b^2\right )^3 d} \]

[In]

Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(12*a^5*c + 40*a^3*b^2*c + 60*a*b^4*c + 12*a^5*d*x + 40*a^3*b^2*d*x + 60*a*b^4*d*x + 4*b*(a^4 + 4*a^2*b^2 + 3*
b^4)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Cos[4*(c + d*x)] + 32*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 8*a^5
*Sin[2*(c + d*x)] + 24*a^3*b^2*Sin[2*(c + d*x)] + 16*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2
*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x)])/(32*(a^2 + b^2)^3*d)

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} a \,b^{4}\right ) \tan \left (d x +c \right )^{3}+\left (\frac {1}{2} a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (d x +c \right )^{2}+\left (\frac {7}{4} a^{3} b^{2}+\frac {9}{8} a \,b^{4}+\frac {5}{8} a^{5}\right ) \tan \left (d x +c \right )+\frac {a^{4} b}{4}+a^{2} b^{3}+\frac {3 b^{5}}{4}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}-\frac {b^{5} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (3 a^{5}+10 a^{3} b^{2}+15 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(197\)
default	\(\frac {\frac {\frac {\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} a \,b^{4}\right ) \tan \left (d x +c \right )^{3}+\left (\frac {1}{2} a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (d x +c \right )^{2}+\left (\frac {7}{4} a^{3} b^{2}+\frac {9}{8} a \,b^{4}+\frac {5}{8} a^{5}\right ) \tan \left (d x +c \right )+\frac {a^{4} b}{4}+a^{2} b^{3}+\frac {3 b^{5}}{4}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}-\frac {b^{5} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (3 a^{5}+10 a^{3} b^{2}+15 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(197\)
parallelrisch	\(\frac {32 b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-32 b^{5} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+4 \left (a^{4} b +4 a^{2} b^{3}+3 b^{5}\right ) \cos \left (2 d x +2 c \right )+b \left (a^{2}+b^{2}\right )^{2} \cos \left (4 d x +4 c \right )+8 \left (a^{5}+3 a^{3} b^{2}+2 a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+a \left (a^{2}+b^{2}\right )^{2} \sin \left (4 d x +4 c \right )+12 a^{5} d x +40 a^{3} b^{2} d x +60 a \,b^{4} d x -5 a^{4} b -18 a^{2} b^{3}-13 b^{5}}{32 d \left (a^{2}+b^{2}\right )^{3}}\)	\(211\)
risch	\(\frac {9 x a b}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}+\frac {3 i x \,a^{2}}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}-\frac {8 i x \,b^{2}}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )} b}{16 \left (-2 i b a +a^{2}-b^{2}\right ) d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a}{8 \left (-2 i b a +a^{2}-b^{2}\right ) d}-\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )} b}{16 \left (i b +a \right )^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} d}-\frac {2 i b^{5} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i b^{5} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \cos \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}-\frac {a \sin \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}\)	\(396\)
norman	\(\frac {\frac {\left (-2 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) x}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}+\frac {2 \left (-a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \left (-a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{2}+9 b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{2}+9 b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{5} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(823\)

[In]

int(cos(d*x+c)^5/(cos(d*x+c)*a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^3*(((3/8*a^5+5/4*a^3*b^2+7/8*a*b^4)*tan(d*x+c)^3+(1/2*a^2*b^3+1/2*b^5)*tan(d*x+c)^2+(7/4*a^3*
b^2+9/8*a*b^4+5/8*a^5)*tan(d*x+c)+1/4*a^4*b+a^2*b^3+3/4*b^5)/(1+tan(d*x+c)^2)^2-1/2*b^5*ln(1+tan(d*x+c)^2)+1/8
*(3*a^5+10*a^3*b^2+15*a*b^4)*arctan(tan(d*x+c)))+b^5/(a^2+b^2)^3*ln(a+b*tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {4 \, b^{5} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 4 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8*(4*b^5*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) + 2*(a^4*b + 2*a^2*b^3 + b^
5)*cos(d*x + c)^4 + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*d*x + 4*(a^2*b^3 + b^5)*cos(d*x + c)^2 + (2*(a^5 + 2*a^3*b
^2 + a*b^4)*cos(d*x + c)^3 + (3*a^5 + 10*a^3*b^2 + 7*a*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (213) = 426\).

Time = 0.34 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {4 \, b^{5} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, b^{5} \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {16 \, b^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}}{4 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(4*b^5*log(-a - 2*b*sin(d*x + c)/(cos(d*x + c) + 1) + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6) - 4*b^5*log(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
) - (16*b^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - (5*a^3 + 9*a*b^2)*sin(d*x + c)/(cos(d*x + c) + 1) + 8*(a^2*b
+ 2*b^3)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (3*a^3 - a*b^2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - (3*a^3 -
a*b^2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 8*(a^2*b + 2*b^3)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (5*a^3 +
9*a*b^2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 + 6*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*(a^4 + 2*a^2*b^2 +
b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {8 \, b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {6 \, b^{5} \tan \left (d x + c\right )^{4} + 3 \, a^{5} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 7 \, a b^{4} \tan \left (d x + c\right )^{3} + 4 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + 16 \, b^{5} \tan \left (d x + c\right )^{2} + 5 \, a^{5} \tan \left (d x + c\right ) + 14 \, a^{3} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{4} \tan \left (d x + c\right ) + 2 \, a^{4} b + 8 \, a^{2} b^{3} + 12 \, b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/8*(8*b^6*log(abs(b*tan(d*x + c) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 4*b^5*log(tan(d*x + c)^2 + 1)/
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6) + (6*b^5*tan(d*x + c)^4 + 3*a^5*tan(d*x + c)^3 + 10*a^3*b^2*tan(d*x + c)^3 + 7*a*b^4*tan(d*x + c)^3 + 4*
a^2*b^3*tan(d*x + c)^2 + 16*b^5*tan(d*x + c)^2 + 5*a^5*tan(d*x + c) + 14*a^3*b^2*tan(d*x + c) + 9*a*b^4*tan(d*
x + c) + 2*a^4*b + 8*a^2*b^3 + 12*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(d*x + c)^2 + 1)^2))/d

Mupad [B] (verification not implemented)

Time = 35.84 (sec) , antiderivative size = 6099, normalized size of antiderivative = 26.87 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^5/(a*cos(c + d*x) + b*sin(c + d*x)),x)

[Out]

(b^5*log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2))/(d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (64*b
^5*log(1/(cos(c + d*x) + 1)))/(d*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)) - ((4*b^3*tan(c/2 + (d*x)/2)^4
)/(a^4 + b^4 + 2*a^2*b^2) - (tan(c/2 + (d*x)/2)*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c/2 + (d
*x)/2)^3*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^5*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 +
2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^7*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*tan(c/2 + (d*x)/2)^
2*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2) + (2*b*tan(c/2 + (d*x)/2)^6*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2))/(
d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) - (a*
atan((tan(c/2 + (d*x)/2)*((((64*b^5*((a*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^11 + 3152*a^7*b^9
+ 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8
*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 403
2*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^
2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3
*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 +
6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4
+ 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((9*a^15 - 192*a*b^14 - 222*a^3*b^12
+ 475*a^5*b^10 + 1089*a^7*b^8 + 894*a^9*b^6 + 388*a^11*b^4 + 87*a^13*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a
^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^
11 + 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 2
0*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 +
6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^
4 + 192*a^4*b^2))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^3*(3*a^4 + 15*b^
4 + 10*a^2*b^2)^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 +
1344*a^13*b^4 + 192*a^15*b^2))/(1024*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4
*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(9*a^12 + 256*b^12 - 1441*a^2*b^10 - 715*a^4*b^8 - 82*a^6*b^6 +
130*a^8*b^4 + 51*a^10*b^2))/(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4 + 69*
a^10*b^2)^2 - (2*a*b*((64*a*b^13 + 210*a^3*b^11 + 181*a^5*b^9 + 60*a^7*b^7 + 9*a^9*b^5)/(2*(a^12 + b^12 + 6*a^
2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) + (64*b^5*((9*a^15 - 192*a*b^14 - 222*a^3*b^12 +
475*a^5*b^10 + 1089*a^7*b^8 + 894*a^9*b^6 + 388*a^11*b^4 + 87*a^13*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*
b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^11
+ 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a
^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 672
0*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^1
2 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 +
192*a^4*b^2)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) + (a*((a*((64*a*b^15 + 48*a^15*b + 624*a^3*b^13
+ 2016*a^5*b^11 + 3152*a^7*b^9 + 2688*a^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 1
5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 672
0*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 1
92*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 +
10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a*b^16 + 1
344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/(
(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2)) - (a^2*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)^2*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^
7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/(2*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192
*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b
^4 + 6*a^10*b^2)))*(9*a^10 - 496*b^10 + 305*a^2*b^8 + 412*a^4*b^6 + 238*a^6*b^4 + 60*a^8*b^2))/(9*a^12 + 256*b
^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4 + 69*a^10*b^2)^2)*(16*a^16 + 16*b^16 + 128*a^2*b^
14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/(3*a^6 + 15*a^2
*b^4 + 10*a^4*b^2) + (((64*b^5*((a*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^10 - 248*a^8*b^8 + 408*
a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 +
6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b
^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10
+ 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^
4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6
*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a
^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 +
6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((9*a^14*b + 48*a^2*b^13 + 193*a^4*b^11 + 3
33*a^6*b^9 + 330*a^8*b^7 + 202*a^10*b^5 + 69*a^12*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6
+ 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^10 - 248*a^8*b^8 + 4
08*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4
+ 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^1
0*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^
10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2))*(3*a
^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^3*(3*a^4 + 15*b^4 + 10*a^2*b^2)^3*(192
*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a
^14*b^3))/(1024*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15
*a^8*b^4 + 6*a^10*b^2)))*(9*a^12 + 256*b^12 - 1441*a^2*b^10 - 715*a^4*b^8 - 82*a^6*b^6 + 130*a^8*b^4 + 51*a^10
*b^2)*(16*a^16 + 16*b^16 + 128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12
*b^4 + 128*a^14*b^2))/((3*a^6 + 15*a^2*b^4 + 10*a^4*b^2)*(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490
*a^6*b^6 + 250*a^8*b^4 + 69*a^10*b^2)^2) - (2*a*b*((56*a^2*b^12 + 73*a^4*b^10 + 42*a^6*b^8 + 9*a^8*b^6)/(2*(a^
12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) + (64*b^5*((9*a^14*b + 48*a^2*b^1
3 + 193*a^4*b^11 + 333*a^6*b^9 + 330*a^8*b^7 + 202*a^10*b^5 + 69*a^12*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a
^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (64*b^5*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^12 - 888*a^6*b^
10 - 248*a^8*b^8 + 408*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*
a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 672
0*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^
12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4
+ 192*a^4*b^2)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) + (a*((a*((24*a^16 - 184*a^2*b^14 - 696*a^4*b^1
2 - 888*a^6*b^10 - 248*a^8*b^8 + 408*a^10*b^6 + 408*a^12*b^4 + 152*a^14*b^2)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15
*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*
a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 19
2*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 +
10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a^16*b + 19
2*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((
64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 1
5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2)) - (a^2*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)^2*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*
b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/(2*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*
a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^
4 + 6*a^10*b^2)))*(9*a^10 - 496*b^10 + 305*a^2*b^8 + 412*a^4*b^6 + 238*a^6*b^4 + 60*a^8*b^2)*(16*a^16 + 16*b^1
6 + 128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/(
(3*a^6 + 15*a^2*b^4 + 10*a^4*b^2)*(9*a^12 + 256*b^12 + 481*a^2*b^10 + 525*a^4*b^8 + 490*a^6*b^6 + 250*a^8*b^4
+ 69*a^10*b^2)^2))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(4*d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))