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

Optimal. Leaf size=131 \[ \frac{x \left (-6 a^2 b^2+a^4+b^4\right )}{2 \left (a^2+b^2\right )^3}+\frac{a b \sin ^2(x)}{\left (a^2+b^2\right )^2}+\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac{\left (b^2-a^2\right ) \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}+\frac{2 a b \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.544877, antiderivative size = 186, normalized size of antiderivative = 1.42, number of steps used = 21, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3111, 3109, 2635, 8, 2564, 30, 3098, 3133, 3097, 3075} \[ \frac{a^2 x}{2 \left (a^2+b^2\right )^2}-\frac{4 a^2 b^2 x}{\left (a^2+b^2\right )^3}+\frac{b^2 x}{2 \left (a^2+b^2\right )^2}+\frac{a b \sin ^2(x)}{\left (a^2+b^2\right )^2}-\frac{a^2 \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}+\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac{b^2 \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}+\frac{2 a^3 b \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac{2 a b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3111

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

Rule 3109

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2564

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 30

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

Rule 3098

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 3133

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]

Rule 3097

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 3075

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac{a \int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac{b \int \frac{\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2}\\ &=\frac{a^2 \int \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}+2 \frac{(a b) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac{\left (a^2 b\right ) \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{b^2 \int \cos ^2(x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac{\left (a b^2\right ) \int \frac{\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{1}{(a \cos (x)+b \sin (x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac{a^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}-2 \left (\frac{a^2 b^2 x}{\left (a^2+b^2\right )^3}-\frac{\left (a^3 b\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )-2 \left (\frac{a^2 b^2 x}{\left (a^2+b^2\right )^3}+\frac{\left (a b^3\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )+\frac{a^2 \int 1 \, dx}{2 \left (a^2+b^2\right )^2}+2 \frac{(a b) \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^2}+\frac{b^2 \int 1 \, dx}{2 \left (a^2+b^2\right )^2}\\ &=\frac{a^2 x}{2 \left (a^2+b^2\right )^2}+\frac{b^2 x}{2 \left (a^2+b^2\right )^2}-2 \left (\frac{a^2 b^2 x}{\left (a^2+b^2\right )^3}-\frac{a^3 b \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}\right )-2 \left (\frac{a^2 b^2 x}{\left (a^2+b^2\right )^3}+\frac{a b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}\right )-\frac{a^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{a b \sin ^2(x)}{\left (a^2+b^2\right )^2}+\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}\\ \end{align*}

Mathematica [A]  time = 1.58527, size = 145, normalized size = 1.11 \[ \frac{\sin (x)}{8 a (a \cos (x)+b \sin (x))}-\frac{-4 x \left (-6 a^2 b^2+a^4+b^4\right )+2 \left (a^4-b^4\right ) \sin (2 x)+4 a b \left (a^2+b^2\right ) \cos (2 x)+\frac{\left (a^2+b^2\right ) \left (-6 a^2 b^2+a^4+b^4\right ) \sin (x)}{a (a \cos (x)+b \sin (x))}-16 a b \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{8 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/(8*a*(a*Cos[x] + b*Sin[x])) - (-4*(a^4 - 6*a^2*b^2 + b^4)*x + 4*a*b*(a^2 + b^2)*Cos[2*x] - 16*a*b*(a^2
- b^2)*Log[a*Cos[x] + b*Sin[x]] + ((a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Sin[x])/(a*(a*Cos[x] + b*Sin[x])) + 2*(
a^4 - b^4)*Sin[2*x])/(8*(a^2 + b^2)^3)

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Maple [B]  time = 0.105, size = 260, normalized size = 2. \begin{align*} -{\frac{{a}^{2}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+2\,{\frac{{a}^{3}b\ln \left ( a+b\tan \left ( x \right ) \right ) }{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-2\,{\frac{a{b}^{3}\ln \left ( a+b\tan \left ( x \right ) \right ) }{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\tan \left ( x \right ){a}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{\tan \left ( x \right ){b}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{{a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{a{b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) a{b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{\arctan \left ( \tan \left ( x \right ) \right ){a}^{2}{b}^{2}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ){b}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ){a}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.69715, size = 347, normalized size = 2.65 \begin{align*} \frac{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} x}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac{2 \,{\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{4 \, a^{2} b +{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (x\right )^{2} +{\left (a^{3} + a b^{2}\right )} \tan \left (x\right )}{2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^4 - 6*a^2*b^2 + b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(a^3*b - a*b^3)*log(b*tan(x) + a)/(a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^3*b - a*b^3)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/2*(4
*a^2*b + (3*a^2*b - b^3)*tan(x)^2 + (a^3 + a*b^2)*tan(x))/(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)
*tan(x)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*tan(x)^2 + (a^4*b + 2*a^2*b^3 + b^5)*tan(x))

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Fricas [A]  time = 0.566495, size = 540, normalized size = 4.12 \begin{align*} -\frac{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{3} +{\left (a^{2} b^{3} - b^{5} -{\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} x\right )} \cos \left (x\right ) - 2 \,{\left ({\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) +{\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (x\right )\right )} \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 (3 \, a^{3} b^{2} + a b^{4} -{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2} +{\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} x\right )} \sin \left (x\right )}{2 \,{\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.09116, size = 296, normalized size = 2.26 \begin{align*} \frac{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} x}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac{3 \, a^{2} b \tan \left (x\right )^{2} - b^{3} \tan \left (x\right )^{2} + a^{3} \tan \left (x\right ) + a b^{2} \tan \left (x\right ) + 4 \, a^{2} b}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^4 - 6*a^2*b^2 + b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^3*b - a*b^3)*log(tan(x)^2 + 1)/(a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(a^3*b^2 - a*b^4)*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)
- 1/2*(3*a^2*b*tan(x)^2 - b^3*tan(x)^2 + a^3*tan(x) + a*b^2*tan(x) + 4*a^2*b)/((a^4 + 2*a^2*b^2 + b^4)*(b*tan
(x)^3 + a*tan(x)^2 + b*tan(x) + a))