\(\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx\) [8]
Optimal result
Integrand size = 16, antiderivative size = 91 \[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}-\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}
\]
[Out]
a^2*b*x/(a^2+b^2)^2+1/2*b*x/(a^2+b^2)-a^3*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 = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3178, 3176, 3212, 2715, 8}
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}-\frac {b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )}-\frac {a^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}
\]
[In]
Int[Sin[x]^3/(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^3*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 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 3176
Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[b*(x/(a^2 + b^2)), x] - Dist[a/(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 3178
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2
)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), 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 {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac {b \int \sin ^2(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 {a \sin ^2(x)}{2 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^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^3 \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 = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.03
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {-4 i a^3 x+6 a^2 b x+2 b^3 x+4 i a^3 \arctan (\tan (x))+a \left (a^2+b^2\right ) \cos (2 x)-2 a^3 \log \left ((a \cos (x)+b \sin (x))^2\right )-a^2 b \sin (2 x)-b^3 \sin (2 x)}{4 \left (a^2+b^2\right )^2}
\]
[In]
Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x]),x]
[Out]
((-4*I)*a^3*x + 6*a^2*b*x + 2*b^3*x + (4*I)*a^3*ArcTan[Tan[x]] + a*(a^2 + b^2)*Cos[2*x] - 2*a^3*Log[(a*Cos[x]
+ b*Sin[x])^2] - a^2*b*Sin[2*x] - b^3*Sin[2*x])/(4*(a^2 + b^2)^2)
Maple [A] (verified)
Time = 0.55 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.07
| | |
| method | result | size |
| | |
| default |
\(-\frac {a^{3} \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tan \left (x \right )+\frac {a^{3}}{2}+\frac {a \,b^{2}}{2}}{1+\tan \left (x \right )^{2}}+\frac {a^{3} \ln \left (1+\tan \left (x \right )^{2}\right )}{2}+\frac {\left (3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (x \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}\) |
\(97\) |
| parallelrisch |
\(\frac {-a^{2} b \sin \left (2 x \right )-b^{3} \sin \left (2 x \right )+a^{3} \cos \left (2 x \right )+a \,b^{2} \cos \left (2 x \right )-4 a^{3} \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+4 a^{3} \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+6 x \,a^{2} b +2 x \,b^{3}-a^{3}-a \,b^{2}}{4 \left (a^{2}+b^{2}\right )^{2}}\) |
\(109\) |
| risch |
\(\frac {x b}{4 i b a -2 a^{2}+2 b^{2}}+\frac {i x a}{2 i b a -a^{2}+b^{2}}+\frac {{\mathrm e}^{2 i x}}{-8 i b +8 a}+\frac {{\mathrm e}^{-2 i x}}{8 i b +8 a}+\frac {2 i a^{3} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) |
\(144\) |
| norman |
\(\frac {\frac {b \tan \left (\frac {x}{2}\right )^{5}}{a^{2}+b^{2}}-\frac {2 a \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}-\frac {2 a \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}-\frac {b \tan \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {b \left (3 a^{2}+b^{2}\right ) x}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 b \left (3 a^{2}+b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \left (3 a^{2}+b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (3 a^{2}+b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}+\frac {a^{3} \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a^{3} \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) |
\(293\) |
| | |
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[In]
int(sin(x)^3/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)
[Out]
-a^3/(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*a^3*ln(1+tan(x)^2)+1/2*(3*a^2*b+b^3)*arctan(tan(x)))
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.02
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{3} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left (a^{3} + a b^{2}\right )} \cos \left (x\right )^{2} + {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - {\left (3 \, a^{2} b + b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}}
\]
[In]
integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="fricas")
[Out]
-1/2*(a^3*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) - (3*a^2*b + b^3)*x)/(a^4 + 2*a^2*b^2 + b^4)
Sympy [F(-1)]
Timed out. \[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out}
\]
[In]
integrate(sin(x)**3/(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. 209 vs. \(2 (85) = 170\).
Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.30
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{3} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a^{3} \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {\frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}}
\]
[In]
integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="maxima")
[Out]
-a^3*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^3*log(sin(x)^2/
(cos(x) + 1)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (3*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 = 148, normalized size of antiderivative = 1.63
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{3} b \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a^{3} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{3} \tan \left (x\right )^{2} + a^{2} b \tan \left (x\right ) + b^{3} \tan \left (x\right ) - a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}}
\]
[In]
integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="giac")
[Out]
-a^3*b*log(abs(b*tan(x) + a))/(a^4*b + 2*a^2*b^3 + b^5) + 1/2*a^3*log(tan(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) +
1/2*(3*a^2*b + b^3)*x/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a^3*tan(x)^2 + a^2*b*tan(x) + b^3*tan(x) - a*b^2)/((a^4 +
2*a^2*b^2 + b^4)*(tan(x)^2 + 1))
Mupad [B] (verification not implemented)
Time = 29.18 (sec) , antiderivative size = 3512, normalized size of antiderivative = 38.59
\[
\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Too large to display}
\]
[In]
int(sin(x)^3/(a*cos(x) + b*sin(x)),x)
[Out]
(4*a^3*log(1/(cos(x) + 1)))/(4*a^4 + 4*b^4 + 8*a^2*b^2) - ((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^3*log(a + 2*b*tan(x/2) - a*tan(x/2)^2)
)/(a^4 + b^4 + 2*a^2*b^2) - (b*atan((tan(x/2)*((((4*a^3*((b*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4
- 8*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^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))/((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)))*(3*a^2 + b^2))/(2*(a^4 + b^4
+ 2*a^2*b^2)) + (16*a^3*b*(3*a^2 + 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)*(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^2 + b^2)*((8*(2*a*b^8 + 13*a^3*b^6 + 32*a^5*b^4 + 21*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2) + (4*a^3*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2) + (32*a^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))/((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)) - (b^3*(3*a^2 +
b^2)^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)))*(16*a^8 + b^8 + 5*a^2*b^6 - 13*a^4*b^4 - 73*a^6*b^2))/(16*a^8 + b^8 + 7*a^2*b^6 +
15*a^4*b^4 + 25*a^6*b^2)^2 + (2*a*b*(b^6 - 28*a^6 + 10*a^2*b^4 + 17*a^4*b^2)*((8*(8*a^8 + 2*a^4*b^4 + 9*a^6*b^
2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a^3*((8*(2*a*b^8 + 13*a^3*b^6 + 32*a^5*b^4 + 21*a^7*b^2))/(a^6 +
b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a^3*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b
^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^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))/((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*(3*a^2 + b^2)*((b*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b^6 + 3
*a^2*b^4 + 3*a^4*b^2) + (32*a^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))/((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)))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (16*a^3*b*(
3*a^2 + 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)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (8*a^3*b^2*(3*a^2 +
b^2)^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))))/(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^6*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^3 + 12*a^3*b) - (((4*a^3*((b*(3*a^2
+ b^2)*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^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))/((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^3*b*(3*a^2 + 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)*(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*((8*(a^2*b^7 + 2*a^4*b^5 + a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2) - (4*a^3*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^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))/((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))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (b^3*(3*a^2 +
b^2)^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)))*(16*a^8 + b^8 + 5*a^2*b^6 - 13*a^4*b^4 - 73*a^6*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^3 + 12*a^3*b)*(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^6*b^2)
^2) + (2*a*b*(b^6 - 28*a^6 + 10*a^2*b^4 + 17*a^4*b^2)*((8*(2*a^7*b + a^5*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2) + (4*a^3*((8*(a^2*b^7 + 2*a^4*b^5 + a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a^3*((8*(2*a*b^9 -
10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^3*(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))))/(4*a
^4 + 4*b^4 + 8*a^2*b^2)))/(4*a^4 + 4*b^4 + 8*a^2*b^2) + (b*((b*(3*a^2 + b^2)*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b
^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^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))/((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^3*b*(3*a^2 + 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)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*
a^2*b^2)) - (8*a^3*b^2*(3*a^2 + b^2)^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^3 + 12*a^3*b)*(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^
6*b^2)^2))*(3*a^2 + b^2))/(a^4 + b^4 + 2*a^2*b^2)