\(\int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx\) [141]

Optimal result

Integrand size = 28, antiderivative size = 165 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4+4*a*b*(a^2-b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^4/d-1/3*b/(a^2+b^2)/
d/(a+b*tan(d*x+c))^3-a*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-b*(3*a^2-b^2)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3165, 3564, 3610, 3612, 3611} \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4} \]

[In]

Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b
^2)^4*d) - b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(3
*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3165

Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b \tan (c+d x))^4} \, dx \\ & = -\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx}{a^2+b^2} \\ & = -\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {a^2-b^2-2 a b \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3} \\ & = \frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (4 a b \left (a^2-b^2\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^4} \\ & = \frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.10 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.54 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \left (a^2+2 a b-b^2\right ) (c+d x)}{(a-i b)^4 (a+i b)^4 d}+\frac {4 \left (i a^{10} b+a^9 b^2+2 i a^8 b^3+2 a^7 b^4-2 i a^4 b^7-2 a^3 b^8-i a^2 b^9-a b^{10}\right ) (c+d x)}{(a-i b)^8 (a+i b)^7 d}-\frac {4 i \left (a^3 b-a b^3\right ) \arctan (\tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {2 \left (a^3 b-a b^3\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )}{\left (a^2+b^2\right )^4 d}+\frac {b^4 \sin (c+d x)}{3 a (a-i b)^2 (a+i b)^2 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac {b^3 \left (6 a^2+b^2\right )}{3 a (a-i b)^3 (a+i b)^3 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {2 \left (9 a^2 b^2 \sin (c+d x)-2 b^4 \sin (c+d x)\right )}{3 a (a-i b)^3 (a+i b)^3 d (a \cos (c+d x)+b \sin (c+d x))} \]

[In]

Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

((a^2 - 2*a*b - b^2)*(a^2 + 2*a*b - b^2)*(c + d*x))/((a - I*b)^4*(a + I*b)^4*d) + (4*(I*a^10*b + a^9*b^2 + (2*
I)*a^8*b^3 + 2*a^7*b^4 - (2*I)*a^4*b^7 - 2*a^3*b^8 - I*a^2*b^9 - a*b^10)*(c + d*x))/((a - I*b)^8*(a + I*b)^7*d
) - ((4*I)*(a^3*b - a*b^3)*ArcTan[Tan[c + d*x]])/((a^2 + b^2)^4*d) + (2*(a^3*b - a*b^3)*Log[(a*Cos[c + d*x] +
b*Sin[c + d*x])^2])/((a^2 + b^2)^4*d) + (b^4*Sin[c + d*x])/(3*a*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*
Sin[c + d*x])^3) - (b^3*(6*a^2 + b^2))/(3*a*(a - I*b)^3*(a + I*b)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (
2*(9*a^2*b^2*Sin[c + d*x] - 2*b^4*Sin[c + d*x]))/(3*a*(a - I*b)^3*(a + I*b)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*
x]))

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(183\)
default	\(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(183\)
risch	\(-\frac {x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {8 i a^{3} b x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {8 i a \,b^{3} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {8 i a^{3} b c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {8 i a \,b^{3} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {4 i b^{2} \left (-12 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+18 i a^{3} b -4 i a \,b^{3}+9 a^{4}-11 a^{2} b^{2}+2 b^{4}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(565\)
parallelrisch	\(\frac {36 b \left (\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (a^{2}+b^{2}\right ) \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )\right ) a^{4} \left (a +b \right ) \left (a -b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-36 b \left (\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (a^{2}+b^{2}\right ) \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )\right ) a^{4} \left (a +b \right ) \left (a -b \right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+3 a \left (a^{9} d x -9 a^{7} b^{2} d x +19 a^{5} b^{4} d x -3 a^{3} b^{6} d x -10 a^{6} b^{3}-13 a^{4} b^{5}-4 a^{2} b^{7}-b^{9}\right ) \cos \left (3 d x +3 c \right )+\left (9 a^{9} b d x -57 a^{7} b^{3} d x +27 a^{5} b^{5} d x -3 a^{3} b^{7} d x +18 a^{8} b^{2}+14 a^{6} b^{4}-3 a^{4} b^{6}-b^{10}\right ) \sin \left (3 d x +3 c \right )+9 \left (b \left (x \,a^{7} d -6 x \,a^{5} b^{2} d +x \,a^{3} b^{4} d +2 a^{6} b +\frac {16}{3} a^{4} b^{3}+a^{2} b^{5}+\frac {1}{3} b^{7}\right ) \sin \left (d x +c \right )+a \cos \left (d x +c \right ) \left (x \,a^{7} d -6 x \,a^{5} b^{2} d +x \,a^{3} b^{4} d +\frac {10}{3} a^{4} b^{3}+a^{2} b^{5}+\frac {1}{3} b^{7}\right )\right ) \left (a^{2}+b^{2}\right )}{9 \left (\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (a^{2}+b^{2}\right ) \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )\right ) \left (a^{2}+b^{2}\right )^{4} d \,a^{3}}\)	\(578\)
norman	\(\text {Expression too large to display}\)	\(2137\)

[In]

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

[Out]

1/d*(1/(a^2+b^2)^4*(1/2*(-4*a^3*b+4*a*b^3)*ln(1+tan(d*x+c)^2)+(a^4-6*a^2*b^2+b^4)*arctan(tan(d*x+c)))-1/3*b/(a
^2+b^2)/(a+b*tan(d*x+c))^3-b*(3*a^2-b^2)/(a^2+b^2)^3/(a+b*tan(d*x+c))-a*b/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+4*a*b
*(a^2-b^2)/(a^2+b^2)^4*ln(a+b*tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (163) = 326\).

Time = 0.28 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.48 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {{\left (54 \, a^{4} b^{3} - 30 \, a^{2} b^{5} + 4 \, b^{7} - 3 \, {\left (a^{7} - 9 \, a^{5} b^{2} + 19 \, a^{3} b^{4} - 3 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (10 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + b^{7} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (a^{6} b - 4 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6} + {\left (3 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (13 \, a^{3} b^{4} - 9 \, a b^{6} + 3 \, {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} d x + {\left (18 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 12 \, a b^{6} + 3 \, {\left (3 \, a^{6} b - 19 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - b^{7}\right )} d x\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{11} + a^{9} b^{2} - 6 \, a^{7} b^{4} - 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} - 3 \, a b^{10}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{10} b + 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - a^{2} b^{9} - b^{11}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(cos(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*((54*a^4*b^3 - 30*a^2*b^5 + 4*b^7 - 3*(a^7 - 9*a^5*b^2 + 19*a^3*b^4 - 3*a*b^6)*d*x)*cos(d*x + c)^3 - 3*(1
0*a^4*b^3 - 11*a^2*b^5 + b^7 + 3*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*d*x)*cos(d*x + c) - 6*((a^6*b - 4*a^4*b^3 + 3*a
^2*b^5)*cos(d*x + c)^3 + 3*(a^4*b^3 - a^2*b^5)*cos(d*x + c) + (a^3*b^4 - a*b^6 + (3*a^5*b^2 - 4*a^3*b^4 + a*b^
6)*cos(d*x + c)^2)*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - (13
*a^3*b^4 - 9*a*b^6 + 3*(a^4*b^3 - 6*a^2*b^5 + b^7)*d*x + (18*a^5*b^2 - 58*a^3*b^4 + 12*a*b^6 + 3*(3*a^6*b - 19
*a^4*b^3 + 9*a^2*b^5 - b^7)*d*x)*cos(d*x + c)^2)*sin(d*x + c))/((a^11 + a^9*b^2 - 6*a^7*b^4 - 14*a^5*b^6 - 11*
a^3*b^8 - 3*a*b^10)*d*cos(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)
+ ((3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4*b^7 - a^2*b^9 - b^11)*d*cos(d*x + c)^2 + (a^8*b^3 + 4*a^6*b^5
+ 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4/(a*cos(d*x+c)+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (163) = 326\).

Time = 0.33 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {13 \, a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 3 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (7 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{3 \, d} \]

[In]

integrate(cos(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 12*(a^3*b - a*b^3)*
log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^3*b - a*b^3)*log(tan(d*x + c)^2
+ 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - (13*a^4*b + 2*a^2*b^3 + b^5 + 3*(3*a^2*b^3 - b^5)*tan(
d*x + c)^2 + 3*(7*a^3*b^2 - a*b^4)*tan(d*x + c))/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6 + (a^6*b^3 + 3*a^4*b^5
+ 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2 + 3*(a^8*b + 3
*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (163) = 326\).

Time = 0.38 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.24 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac {22 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} - 22 \, a b^{6} \tan \left (d x + c\right )^{3} + 75 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} - 60 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} - 3 \, b^{7} \tan \left (d x + c\right )^{2} + 87 \, a^{5} b^{2} \tan \left (d x + c\right ) - 48 \, a^{3} b^{4} \tan \left (d x + c\right ) - 3 \, a b^{6} \tan \left (d x + c\right ) + 35 \, a^{6} b - 7 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{3 \, d} \]

[In]

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

[Out]

1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^3*b - a*b^3)*l
og(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 12*(a^3*b^2 - a*b^4)*log(abs(b*tan(d*
x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) - (22*a^3*b^4*tan(d*x + c)^3 - 22*a*b^6*tan(d*x
+ c)^3 + 75*a^4*b^3*tan(d*x + c)^2 - 60*a^2*b^5*tan(d*x + c)^2 - 3*b^7*tan(d*x + c)^2 + 87*a^5*b^2*tan(d*x +
c) - 48*a^3*b^4*tan(d*x + c) - 3*a*b^6*tan(d*x + c) + 35*a^6*b - 7*a^4*b^3 + 3*a^2*b^5 + b^7)/((a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 37.65 (sec) , antiderivative size = 8586, normalized size of antiderivative = 52.04 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^4/(a*cos(c + d*x) + b*sin(c + d*x))^4,x)

[Out]

((4*tan(c/2 + (d*x)/2)^2*(b^7 + 3*a^2*b^5 + 10*a^4*b^3))/(a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*tan(c/
2 + (d*x)/2)^4*(b^7 + 3*a^2*b^5 + 10*a^4*b^3))/(a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*b*tan(c/2 + (d*x
)/2)^5*(6*a^4*b + b^5 + 3*a^2*b^3))/(a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (4*b*tan(c/2 + (d*x)/2)^3*(2*b^7
- 18*a^6*b + a^2*b^5 + 17*a^4*b^3))/(3*a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*b*tan(c/2 + (d*x)/2)*(6*
a^4*b + b^5 + 3*a^2*b^3))/(a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(12*a*b^2 - 3*a^3)
- a^3*tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^4*(12*a*b^2 - 3*a^3) - tan(c/2 + (d*x)/2)^3*(12*a^2*b - 8*b^3
) + a^3 + 6*a^2*b*tan(c/2 + (d*x)/2) + 6*a^2*b*tan(c/2 + (d*x)/2)^5)) - (log(a + 2*b*tan(c/2 + (d*x)/2) - a*ta
n(c/2 + (d*x)/2)^2)*(4*a*b^3 - 4*a^3*b))/(d*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (log((((-(a^4 +
b^4 - 6*a^2*b^2)^2/(a^2 + b^2)^8)^(1/2) - (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*(((-(a^4 + b^4 - 6*a^2*b^2)^2/(a
^2 + b^2)^8)^(1/2) - (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*((32*a*(a^6 - b^6 + 11*a^2*b^4 - 11*a^4*b^2))/(a^2 + b
^2)^3 + 96*a*b*((-(a^4 + b^4 - 6*a^2*b^2)^2/(a^2 + b^2)^8)^(1/2) - (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*(a + b*t
an(c/2 + (d*x)/2))*(a^2 + b^2) - (64*a^2*b*tan(c/2 + (d*x)/2)*(b^4 - 5*a^4 + 8*a^2*b^2))/(a^2 + b^2)^3) - (32*
a^2*b*(7*a^4 + 7*b^4 - 18*a^2*b^2))/(a^2 + b^2)^5 + (32*a*tan(c/2 + (d*x)/2)*(a^8 + 2*b^8 - 57*a^2*b^6 + 105*a
^4*b^4 - 27*a^6*b^2))/(a^2 + b^2)^6) + (128*a^3*b^2*(3*a^6 - 3*b^6 + 13*a^2*b^4 - 13*a^4*b^2))/(a^2 + b^2)^9 -
(128*a^2*b*tan(c/2 + (d*x)/2)*(a^8 - 2*b^8 + 5*a^2*b^6 - 15*a^4*b^4 + 11*a^6*b^2))/(a^2 + b^2)^9)*(((-(a^4 +
b^4 - 6*a^2*b^2)^2/(a^2 + b^2)^8)^(1/2) + (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*(((-(a^4 + b^4 - 6*a^2*b^2)^2/(a^
2 + b^2)^8)^(1/2) + (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*(96*a*b*((-(a^4 + b^4 - 6*a^2*b^2)^2/(a^2 + b^2)^8)^(1/
2) + (4*a*b*(a^2 - b^2))/(a^2 + b^2)^4)*(a + b*tan(c/2 + (d*x)/2))*(a^2 + b^2) - (32*a*(a^6 - b^6 + 11*a^2*b^4
- 11*a^4*b^2))/(a^2 + b^2)^3 + (64*a^2*b*tan(c/2 + (d*x)/2)*(b^4 - 5*a^4 + 8*a^2*b^2))/(a^2 + b^2)^3) - (32*a
^2*b*(7*a^4 + 7*b^4 - 18*a^2*b^2))/(a^2 + b^2)^5 + (32*a*tan(c/2 + (d*x)/2)*(a^8 + 2*b^8 - 57*a^2*b^6 + 105*a^
4*b^4 - 27*a^6*b^2))/(a^2 + b^2)^6) - (128*a^3*b^2*(3*a^6 - 3*b^6 + 13*a^2*b^4 - 13*a^4*b^2))/(a^2 + b^2)^9 +
(128*a^2*b*tan(c/2 + (d*x)/2)*(a^8 - 2*b^8 + 5*a^2*b^6 - 15*a^4*b^4 + 11*a^6*b^2))/(a^2 + b^2)^9))*(8*a*b^3 -
8*a^3*b))/(2*d*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - (2*atan((tan(c/2 + (d*x)/2)*((((32*(4*a^10*b
- 8*a^2*b^9 + 20*a^4*b^7 - 60*a^6*b^5 + 44*a^8*b^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 +
126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (((32*(2*a*b^14 + a^15 - 51*a^3*b^12 -
60*a^5*b^10 + 119*a^7*b^8 + 178*a^9*b^6 + 27*a^11*b^4 - 24*a^13*b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14
+ 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - ((8*a*b^3 - 8*a^3*b)*
((32*(2*a^2*b^17 - 10*a^18*b + 28*a^4*b^15 + 116*a^6*b^13 + 220*a^8*b^11 + 200*a^10*b^9 + 52*a^12*b^7 - 52*a^1
4*b^5 - 44*a^16*b^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84
*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a
^7*b^16 + 630*a^9*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^
2))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 1
26*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2))))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4
+ 4*a^6*b^2)))*(8*a*b^3 - 8*a^3*b))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - (((((32*(2*a^2*b^17
- 10*a^18*b + 28*a^4*b^15 + 116*a^6*b^13 + 220*a^8*b^11 + 200*a^10*b^9 + 52*a^12*b^7 - 52*a^14*b^5 - 44*a^16*b
^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^
14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^16 + 630*a^9
*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((a^8 + b^8 +
4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126
*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^8 + b^8 + 4*
a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2)*(3*a*b^22 +
30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^16 + 630*a^9*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135
*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^2*(a^18 + b^18 + 9*a^2
*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(2
*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) + (16*(8*a*b^3 - 8*a^3*
b)*(2*a*b - a^2 + b^2)^2*(2*a*b + a^2 - b^2)^2*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^16 + 630*a^9
*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((a^8 + b^8 +
4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^3*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 1
26*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(18*a*b^9 + 18*a^9*b - 280*a^3*b^7 + 556*a^5*b^5 - 280
*a^7*b^3))/(a^10 + b^10 + 53*a^2*b^8 - 38*a^4*b^6 - 38*a^6*b^4 + 53*a^8*b^2)^2 + ((((8*a*b^3 - 8*a^3*b)*((((32
*(2*a^2*b^17 - 10*a^18*b + 28*a^4*b^15 + 116*a^6*b^13 + 220*a^8*b^11 + 200*a^10*b^9 + 52*a^12*b^7 - 52*a^14*b^
5 - 44*a^16*b^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^1
2*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b
^16 + 630*a^9*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/
((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a
^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a
^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2
)*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^16 + 630*a^9*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a
^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^2*(a^18 +
b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a
^16*b^2))))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - (((32*(2*a*b^14 + a^15 - 51*a^3*b^12 - 60*a^
5*b^10 + 119*a^7*b^8 + 178*a^9*b^6 + 27*a^11*b^4 - 24*a^13*b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*
a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - ((8*a*b^3 - 8*a^3*b)*((32*(
2*a^2*b^17 - 10*a^18*b + 28*a^4*b^15 + 116*a^6*b^13 + 220*a^8*b^11 + 200*a^10*b^9 + 52*a^12*b^7 - 52*a^14*b^5
- 44*a^16*b^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*
b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^1
6 + 630*a^9*b^14 + 756*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((
a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8
*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2))))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^
6*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) + (32*(2*a*b
- a^2 + b^2)^3*(2*a*b + a^2 - b^2)^3*(3*a*b^22 + 30*a^3*b^20 + 135*a^5*b^18 + 360*a^7*b^16 + 630*a^9*b^14 + 7
56*a^11*b^12 + 630*a^13*b^10 + 360*a^15*b^8 + 135*a^17*b^6 + 30*a^19*b^4 + 3*a^21*b^2))/((a^8 + b^8 + 4*a^2*b^
6 + 6*a^4*b^4 + 4*a^6*b^2)^3*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b
^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(a^10 - b^10 + 109*a^2*b^8 - 466*a^4*b^6 + 466*a^6*b^4 - 109*a^
8*b^2))/(a^10 + b^10 + 53*a^2*b^8 - 38*a^4*b^6 - 38*a^6*b^4 + 53*a^8*b^2)^2)*(a^22 + b^22 + 11*a^2*b^20 + 55*a
^4*b^18 + 165*a^6*b^16 + 330*a^8*b^14 + 462*a^10*b^12 + 462*a^12*b^10 + 330*a^14*b^8 + 165*a^16*b^6 + 55*a^18*
b^4 + 11*a^20*b^2))/(32*a*b^4 + 32*a^5 - 192*a^3*b^2) + ((((8*a*b^3 - 8*a^3*b)*((((32*(a*b^18 - a^19 - 5*a^3*b
^16 - 40*a^5*b^14 - 80*a^7*b^12 - 46*a^9*b^10 + 46*a^11*b^8 + 80*a^13*b^6 + 40*a^15*b^4 + 5*a^17*b^2))/(a^18 +
b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a
^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^
10*b^13 + 756*a^12*b^11 + 630*a^14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 +
6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 +
84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*
a^4*b^4 + 4*a^6*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2)*(3*a^22*b + 3*a^2*b^21
+ 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*a^14*b^9 + 360*a^16*b^7 + 13
5*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^2*(a^18 + b^18 + 9*a^2*b^16 + 36*a
^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2))))/(2*(a^8 + b^8
+ 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (((32*(7*a^14*b + 7*a^2*b^13 + 10*a^4*b^11 - 23*a^6*b^9 - 52*a^8*b^7
- 23*a^10*b^5 + 10*a^12*b^3))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*
b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) + ((8*a*b^3 - 8*a^3*b)*((32*(a*b^18 - a^19 - 5*a^3*b^16 - 40*a^5
*b^14 - 80*a^7*b^12 - 46*a^9*b^10 + 46*a^11*b^8 + 80*a^13*b^6 + 40*a^15*b^4 + 5*a^17*b^2))/(a^18 + b^18 + 9*a^
2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (
16*(8*a*b^3 - 8*a^3*b)*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 75
6*a^12*b^11 + 630*a^14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 +
4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6
+ 36*a^14*b^4 + 9*a^16*b^2))))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b
+ a^2 - b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) + (32*(2*a*b - a^2 + b^2)^3*(2*a*b + a^2 - b^2)
^3*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*a^
14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^3*(a^18
+ b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*
a^16*b^2)))*(a^10 - b^10 + 109*a^2*b^8 - 466*a^4*b^6 + 466*a^6*b^4 - 109*a^8*b^2)*(a^22 + b^22 + 11*a^2*b^20 +
55*a^4*b^18 + 165*a^6*b^16 + 330*a^8*b^14 + 462*a^10*b^12 + 462*a^12*b^10 + 330*a^14*b^8 + 165*a^16*b^6 + 55*
a^18*b^4 + 11*a^20*b^2))/((32*a*b^4 + 32*a^5 - 192*a^3*b^2)*(a^10 + b^10 + 53*a^2*b^8 - 38*a^4*b^6 - 38*a^6*b^
4 + 53*a^8*b^2)^2) + (((32*(12*a^3*b^8 - 52*a^5*b^6 + 52*a^7*b^4 - 12*a^9*b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36
*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) + ((8*a*b^3 -
8*a^3*b)*((32*(7*a^14*b + 7*a^2*b^13 + 10*a^4*b^11 - 23*a^6*b^9 - 52*a^8*b^7 - 23*a^10*b^5 + 10*a^12*b^3))/(a^
18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 +
9*a^16*b^2) + ((8*a*b^3 - 8*a^3*b)*((32*(a*b^18 - a^19 - 5*a^3*b^16 - 40*a^5*b^14 - 80*a^7*b^12 - 46*a^9*b^10
+ 46*a^11*b^8 + 80*a^13*b^6 + 40*a^15*b^4 + 5*a^17*b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^1
2 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a^22*b
+ 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*a^14*b^9 + 360*
a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*
b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2))))/(2
*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2))))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - ((((
(32*(a*b^18 - a^19 - 5*a^3*b^16 - 40*a^5*b^14 - 80*a^7*b^12 - 46*a^9*b^10 + 46*a^11*b^8 + 80*a^13*b^6 + 40*a^1
5*b^4 + 5*a^17*b^2))/(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*
a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*
b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*a^14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3
))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 12
6*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))
/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) - (16*(8*a*b^3 - 8*a^3*b)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 -
b^2)*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*
a^14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^2*(a^1
8 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 + 126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 +
9*a^16*b^2)))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2) + (16*(
8*a*b^3 - 8*a^3*b)*(2*a*b - a^2 + b^2)^2*(2*a*b + a^2 - b^2)^2*(3*a^22*b + 3*a^2*b^21 + 30*a^4*b^19 + 135*a^6*
b^17 + 360*a^8*b^15 + 630*a^10*b^13 + 756*a^12*b^11 + 630*a^14*b^9 + 360*a^16*b^7 + 135*a^18*b^5 + 30*a^20*b^3
))/((a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)^3*(a^18 + b^18 + 9*a^2*b^16 + 36*a^4*b^14 + 84*a^6*b^12 +
126*a^8*b^10 + 126*a^10*b^8 + 84*a^12*b^6 + 36*a^14*b^4 + 9*a^16*b^2)))*(18*a*b^9 + 18*a^9*b - 280*a^3*b^7 + 5
56*a^5*b^5 - 280*a^7*b^3)*(a^22 + b^22 + 11*a^2*b^20 + 55*a^4*b^18 + 165*a^6*b^16 + 330*a^8*b^14 + 462*a^10*b^
12 + 462*a^12*b^10 + 330*a^14*b^8 + 165*a^16*b^6 + 55*a^18*b^4 + 11*a^20*b^2))/((32*a*b^4 + 32*a^5 - 192*a^3*b
^2)*(a^10 + b^10 + 53*a^2*b^8 - 38*a^4*b^6 - 38*a^6*b^4 + 53*a^8*b^2)^2))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b
^2))/(d*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2))