\(\int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx\) [278]

Optimal result

Integrand size = 18, antiderivative size = 93 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}+\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )} \]

[Out]

-a^2*b*x/(a^2+b^2)^2+1/2*b*x/(a^2+b^2)-a*b^2*ln(a*cos(x)+b*sin(x))/(a^2+b^2)^2+1/2*b*cos(x)*sin(x)/(a^2+b^2)+1
/2*a*sin(x)^2/(a^2+b^2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3188, 2715, 8, 2644, 30, 3177, 3212} \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )}-\frac {a b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2} \]

[In]

Int[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x]),x]

[Out]

-((a^2*b*x)/(a^2 + b^2)^2) + (b*x)/(2*(a^2 + b^2)) - (a*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 + (b*Cos[x
]*Sin[x])/(2*(a^2 + b^2)) + (a*Sin[x]^2)/(2*(a^2 + b^2))

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

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 3188

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

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 {a \int \cos (x) \sin (x) \, dx}{a^2+b^2}+\frac {b \int \cos ^2(x) \, dx}{a^2+b^2}-\frac {(a b) \int \frac {\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2} \\ & = -\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}-\frac {\left (a b^2\right ) \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {a \text {Subst}(\int x \, dx,x,\sin (x))}{a^2+b^2}+\frac {b \int 1 \, dx}{2 \left (a^2+b^2\right )} \\ & = -\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}+\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {4 i a b^2 \arctan (\tan (x))-a \left (a^2+b^2\right ) \cos (2 x)-2 b \left ((a+i b)^2 x+a b \log \left ((a \cos (x)+b \sin (x))^2\right )\right )+b \left (a^2+b^2\right ) \sin (2 x)}{4 \left (a^2+b^2\right )^2} \]

[In]

Integrate[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x]),x]

[Out]

((4*I)*a*b^2*ArcTan[Tan[x]] - a*(a^2 + b^2)*Cos[2*x] - 2*b*((a + I*b)^2*x + a*b*Log[(a*Cos[x] + b*Sin[x])^2])
+ b*(a^2 + b^2)*Sin[2*x])/(4*(a^2 + b^2)^2)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {a \,b^{2} \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tan \left (x \right )-\frac {a^{3}}{2}-\frac {a \,b^{2}}{2}}{1+\tan \left (x \right )^{2}}+\frac {b \left (a b \ln \left (1+\tan \left (x \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}\)	\(98\)
parallelrisch	\(\frac {-a^{3} \cos \left (2 x \right )-a \,b^{2} \cos \left (2 x \right )+a^{2} b \sin \left (2 x \right )+b^{3} \sin \left (2 x \right )-4 a \,b^{2} \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+4 a \,b^{2} \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-2 x \,a^{2} b +2 x \,b^{3}+a^{3}+a \,b^{2}}{4 \left (a^{2}+b^{2}\right )^{2}}\)	\(108\)
risch	\(\frac {x b}{4 i b a -2 a^{2}+2 b^{2}}-\frac {{\mathrm e}^{2 i x}}{8 \left (-i b +a \right )}-\frac {{\mathrm e}^{-2 i x}}{8 \left (i b +a \right )}+\frac {2 i a \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a \,b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(125\)
norman	\(\frac {\frac {b \tan \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {2 a \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}+\frac {2 a \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}-\frac {b \tan \left (\frac {x}{2}\right )^{5}}{a^{2}+b^{2}}-\frac {b \left (a^{2}-b^{2}\right ) x}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}+\frac {a \,b^{2} \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a \,b^{2} \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(295\)

[In]

int(cos(x)^2*sin(x)/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{2} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + {\left (a^{3} + a b^{2}\right )} \cos \left (x\right )^{2} - {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} b - b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]

[In]

integrate(cos(x)^2*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="fricas")

[Out]

-1/2*(a*b^2*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + (a^3 + a*b^2)*cos(x)^2 - (a^2*b + b^3)*cos
(x)*sin(x) + (a^2*b - b^3)*x)/(a^4 + 2*a^2*b^2 + b^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(x)**2*sin(x)/(a*cos(x)+b*sin(x)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (87) = 174\).

Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.28 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{2} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a b^{2} \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} b - b^{3}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {\frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]

[In]

integrate(cos(x)^2*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="maxima")

[Out]

-a*b^2*log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/(a^4 + 2*a^2*b^2 + b^4) + a*b^2*log(sin(x
)^2/(cos(x) + 1)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^2*b - b^3)*arctan(sin(x)/(cos(x) + 1))/(a^4 + 2*a^2*b^2 +
b^4) + (b*sin(x)/(cos(x) + 1) + 2*a*sin(x)^2/(cos(x) + 1)^2 - b*sin(x)^3/(cos(x) + 1)^3)/(a^2 + b^2 + 2*(a^2
+ b^2)*sin(x)^2/(cos(x) + 1)^2 + (a^2 + b^2)*sin(x)^4/(cos(x) + 1)^4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{3} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a b^{2} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a b^{2} \tan \left (x\right )^{2} - a^{2} b \tan \left (x\right ) - b^{3} \tan \left (x\right ) + a^{3} + 2 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}} \]

[In]

integrate(cos(x)^2*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="giac")

[Out]

-a*b^3*log(abs(b*tan(x) + a))/(a^4*b + 2*a^2*b^3 + b^5) + 1/2*a*b^2*log(tan(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4)
- 1/2*(a^2*b - b^3)*x/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a*b^2*tan(x)^2 - a^2*b*tan(x) - b^3*tan(x) + a^3 + 2*a*b^
2)/((a^4 + 2*a^2*b^2 + b^4)*(tan(x)^2 + 1))

Mupad [B] (verification not implemented)

Time = 29.67 (sec) , antiderivative size = 3419, normalized size of antiderivative = 36.76 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\text {Too large to display} \]

[In]

int((cos(x)^2*sin(x))/(a*cos(x) + b*sin(x)),x)

[Out]

((b*tan(x/2))/(a^2 + b^2) + (2*a*tan(x/2)^2)/(a^2 + b^2) - (b*tan(x/2)^3)/(a^2 + b^2))/(2*tan(x/2)^2 + tan(x/2
)^4 + 1) - (a*b^2*log(a + 2*b*tan(x/2) - a*tan(x/2)^2))/(a^4 + b^4 + 2*a^2*b^2) + (4*a*b^2*log(1/(cos(x) + 1))
)/(4*a^4 + 4*b^4 + 8*a^2*b^2) - (b*atan((tan(x/2)*((((4*a*b^2*((b*(a + b)*(a - b)*((8*(12*a^4*b^6 + 24*a^6*b^4
+ 12*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a*b^2*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*
b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b
^2)) - (16*a*b^3*(a + b)*(a - b)*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*
b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2) -
(b*(a + b)*((8*(2*a*b^8 - 7*a^3*b^6 - 8*a^5*b^4 + a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a*b^2*((8
*(12*a^4*b^6 + 24*a^6*b^4 + 12*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a*b^2*(12*a*b^10 + 48*a^3*b
^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))
)/(4*a^4 + 4*b^4 + 8*a^2*b^2))*(a - b))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (b^3*(a + b)^3*(a - b)^3*(12*a*b^10 + 48
*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^
2)))*(a^6 - b^6 + 35*a^2*b^4 - 35*a^4*b^2))/(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2 - (2*a*b*(5*a^4 + 5*b^4 -
26*a^2*b^2)*((8*(2*a^2*b^6 + a^4*b^4))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a*b^2*((8*(2*a*b^8 - 7*a^3*b^6
- 8*a^5*b^4 + a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a*b^2*((8*(12*a^4*b^6 + 24*a^6*b^4 + 12*a^8*
b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a*b^2*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a
^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2)))/(4*
a^4 + 4*b^4 + 8*a^2*b^2) + (b*((b*(a + b)*(a - b)*((8*(12*a^4*b^6 + 24*a^6*b^4 + 12*a^8*b^2))/(a^6 + b^6 + 3*a
^2*b^4 + 3*a^4*b^2) - (32*a*b^2*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b
^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (16*a*b^3*(a + b)*(a - b)
*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*
a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a - b))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (8*a*b^4*(a + b)
^2*(a - b)^2*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^
4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2)*(a^10 +
b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/(4*a*b^5 - 4*a^3*b^3) + (((b*(a + b)*(a - b)*((8*(3*
a^2*b^7 + 6*a^4*b^5 + 3*a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a*b^2*((8*(4*a^3*b^7 - 2*a^9*b - 2*
a*b^9 + 12*a^5*b^5 + 4*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a*b^2*(12*a^10*b + 12*a^2*b^9 + 48*
a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4
+ 4*b^4 + 8*a^2*b^2)))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (4*a*b^2*((b*(a + b)*(a - b)*((8*(4*a^3*b^7 - 2*a^9*b -
2*a*b^9 + 12*a^5*b^5 + 4*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a*b^2*(12*a^10*b + 12*a^2*b^9 + 4
8*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(
a^4 + b^4 + 2*a^2*b^2)) + (16*a*b^3*(a + b)*(a - b)*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8
*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*
b^4 + 8*a^2*b^2) + (b^3*(a + b)^3*(a - b)^3*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/(
(a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^6 - b^6 + 35*a^2*b^4 - 35*a^4*b^2)*(a^10 +
b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/((4*a*b^5 - 4*a^3*b^3)*(a^6 + b^6 + 15*a^2*b^4 + 15*a
^4*b^2)^2) + (2*a*b*(5*a^4 + 5*b^4 - 26*a^2*b^2)*((8*a^3*b^5)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a*b^2*(
(8*(3*a^2*b^7 + 6*a^4*b^5 + 3*a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a*b^2*((8*(4*a^3*b^7 - 2*a^9*
b - 2*a*b^9 + 12*a^5*b^5 + 4*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a*b^2*(12*a^10*b + 12*a^2*b^9
+ 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/
(4*a^4 + 4*b^4 + 8*a^2*b^2)))/(4*a^4 + 4*b^4 + 8*a^2*b^2) + (b*((b*(a + b)*(a - b)*((8*(4*a^3*b^7 - 2*a^9*b -
2*a*b^9 + 12*a^5*b^5 + 4*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a*b^2*(12*a^10*b + 12*a^2*b^9 + 4
8*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(
a^4 + b^4 + 2*a^2*b^2)) + (16*a*b^3*(a + b)*(a - b)*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8
*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a -
b))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (8*a*b^4*(a + b)^2*(a - b)^2*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*
b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))
)*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/((4*a*b^5 - 4*a^3*b^3)*(a^6 + b^6 + 15*a^2*
b^4 + 15*a^4*b^2)^2))*(a + b)*(a - b))/(a^4 + b^4 + 2*a^2*b^2)