3.292 \(\int \frac{\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{3 a b x \left (-6 a^2 b^2+a^4+b^4\right )}{4 \left (a^2+b^2\right )^4}+\frac{a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{2 a^2 b^2 \sin ^2(x)}{\left (a^2+b^2\right )^3}-\frac{b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{a b \sin (x) \cos ^3(x)}{2 \left (a^2+b^2\right )^2}+\frac{a b \left (5 a^2-3 b^2\right ) \sin (x) \cos (x)}{4 \left (a^2+b^2\right )^3}-\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}-\frac{3 a^2 b^2 \left (a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4} \]

[Out]

(-3*a*b*(a^4 - 6*a^2*b^2 + b^4)*x)/(4*(a^2 + b^2)^4) - (b^2*Cos[x]^4)/(4*(a^2 + b^2)^2) - (3*a^2*b^2*(a^2 - b^
2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (a*b*(5*a^2 - 3*b^2)*Cos[x]*Sin[x])/(4*(a^2 + b^2)^3) - (a*b*Cos[
x]^3*Sin[x])/(2*(a^2 + b^2)^2) - (2*a^2*b^2*Sin[x]^2)/(a^2 + b^2)^3 + (a^2*Sin[x]^4)/(4*(a^2 + b^2)^2) - (a^2*
b^3*Sin[x])/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))

________________________________________________________________________________________

Rubi [A]  time = 1.25073, antiderivative size = 289, normalized size of antiderivative = 1.38, number of steps used = 48, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3111, 3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133, 3097, 3075} \[ -\frac{a^3 b x}{\left (a^2+b^2\right )^3}+\frac{6 a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac{a b x}{4 \left (a^2+b^2\right )^2}-\frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{2 a^2 b^2 \sin ^2(x)}{\left (a^2+b^2\right )^3}-\frac{b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{a b \sin (x) \cos ^3(x)}{2 \left (a^2+b^2\right )^2}+\frac{a^3 b \sin (x) \cos (x)}{\left (a^2+b^2\right )^3}-\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}+\frac{a b \sin (x) \cos (x)}{4 \left (a^2+b^2\right )^2}-\frac{a b^3 \sin (x) \cos (x)}{\left (a^2+b^2\right )^3}-\frac{3 a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac{3 a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

(6*a^3*b^3*x)/(a^2 + b^2)^4 - (a^3*b*x)/(a^2 + b^2)^3 - (a*b^3*x)/(a^2 + b^2)^3 + (a*b*x)/(4*(a^2 + b^2)^2) -
(b^2*Cos[x]^4)/(4*(a^2 + b^2)^2) - (3*a^4*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (3*a^2*b^4*Log[a*Cos[x
] + b*Sin[x]])/(a^2 + b^2)^4 + (a^3*b*Cos[x]*Sin[x])/(a^2 + b^2)^3 - (a*b^3*Cos[x]*Sin[x])/(a^2 + b^2)^3 + (a*
b*Cos[x]*Sin[x])/(4*(a^2 + b^2)^2) - (a*b*Cos[x]^3*Sin[x])/(2*(a^2 + b^2)^2) - (2*a^2*b^2*Sin[x]^2)/(a^2 + b^2
)^3 + (a^2*Sin[x]^4)/(4*(a^2 + b^2)^2) - (a^2*b^3*Sin[x])/((a^2 + b^2)^3*(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 2565

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

Rule 30

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

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

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 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 ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac{a \int \frac{\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac{b \int \frac{\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2}\\ &=\frac{a^2 \int \cos (x) \sin ^3(x) \, dx}{\left (a^2+b^2\right )^2}+2 \frac{(a b) \int \cos ^2(x) \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac{\left (a^2 b\right ) \int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{b^2 \int \cos ^3(x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}-2 \frac{\left (a b^2\right ) \int \frac{\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx}{\left (a^2+b^2\right )^2}\\ &=-2 \left (\frac{\left (a^3 b\right ) \int \sin ^2(x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a^2 b^2\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^3 b^2\right ) \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )+\frac{\left (a^3 b^2\right ) \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-2 \left (\frac{\left (a^2 b^2\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a b^3\right ) \int \cos ^2(x) \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^2 b^3\right ) \int \frac{\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}\right )+\frac{\left (a^2 b^3\right ) \int \frac{\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^3 b^3\right ) \int \frac{1}{(a \cos (x)+b \sin (x))^2} \, dx}{\left (a^2+b^2\right )^3}+\frac{a^2 \operatorname{Subst}\left (\int x^3 \, dx,x,\sin (x)\right )}{\left (a^2+b^2\right )^2}+2 \left (-\frac{a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}+\frac{(a b) \int \cos ^2(x) \, dx}{4 \left (a^2+b^2\right )^2}\right )-\frac{b^2 \operatorname{Subst}\left (\int x^3 \, dx,x,\cos (x)\right )}{\left (a^2+b^2\right )^2}\\ &=\frac{2 a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac{b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}+\frac{a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}-\frac{\left (a^4 b^2\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac{\left (a^2 b^4\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}-2 \left (-\frac{a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac{a^3 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac{\left (a^4 b^2\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac{\left (a^3 b\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^3}+\frac{\left (a^2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^3}\right )-2 \left (-\frac{a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac{a b^3 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}-\frac{\left (a^2 b^4\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^4}+\frac{\left (a^2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^3}+\frac{\left (a b^3\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^3}\right )+2 \left (\frac{a b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )^2}-\frac{a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}+\frac{(a b) \int 1 \, dx}{8 \left (a^2+b^2\right )^2}\right )\\ &=\frac{2 a^3 b^3 x}{\left (a^2+b^2\right )^4}-\frac{b^2 \cos ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac{a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac{a^2 \sin ^4(x)}{4 \left (a^2+b^2\right )^2}-\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))}+2 \left (\frac{a b x}{8 \left (a^2+b^2\right )^2}+\frac{a b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )^2}-\frac{a b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )^2}\right )-2 \left (-\frac{a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac{a^3 b x}{2 \left (a^2+b^2\right )^3}+\frac{a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}-\frac{a^3 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac{a^2 b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^3}\right )-2 \left (-\frac{a^3 b^3 x}{\left (a^2+b^2\right )^4}+\frac{a b^3 x}{2 \left (a^2+b^2\right )^3}-\frac{a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^4}+\frac{a b^3 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^3}+\frac{a^2 b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^3}\right )\\ \end{align*}

Mathematica [C]  time = 2.75806, size = 409, normalized size = 1.95 \[ \frac{-12 a b x \left (a^2-3 b^2\right ) \left (3 a^2-b^2\right )+6 i x \left (-15 a^4 b^2+15 a^2 b^4+a^6-b^6\right )-2 a b \left (a^2+b^2\right )^2 \sin (4 x)+16 a b \left (a^4-b^4\right ) \sin (2 x)+\left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \cos (4 x)-4 \left (-6 a^2 b^2+a^4+b^4\right ) \left (a^2+b^2\right ) \cos (2 x)-6 i \left (-15 a^4 b^2+15 a^2 b^4+a^6-b^6\right ) \tan ^{-1}(\tan (x))+\frac{2 b \left (-10 a^2 b^2+3 a^4+3 b^4\right ) \left (a^2+b^2\right ) \sin (x)}{a \cos (x)+b \sin (x)}+3 \left (-15 a^4 b^2+15 a^2 b^4+a^6-b^6\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+\frac{3 \left (a^2+b^2\right )^2 \left (a \cos (x) \left (\left (b^2-a^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )-2 i x (a+i b)^2\right )+b \sin (x) \left (\left (b^2-a^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 (a+i b) (a (-1-i x)+b (x+i))\right )+2 i \left (a^2-b^2\right ) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))\right )}{a \cos (x)+b \sin (x)}}{32 \left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

(-12*a*b*(a^2 - 3*b^2)*(3*a^2 - b^2)*x + (6*I)*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*x - (6*I)*(a^6 - 15*a^4*b
^2 + 15*a^2*b^4 - b^6)*ArcTan[Tan[x]] - 4*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Cos[2*x] + (a^2 - b^2)*(a^2 + b^
2)^2*Cos[4*x] + 3*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Log[(a*Cos[x] + b*Sin[x])^2] + (2*b*(a^2 + b^2)*(3*a^4
- 10*a^2*b^2 + 3*b^4)*Sin[x])/(a*Cos[x] + b*Sin[x]) + (3*(a^2 + b^2)^2*(a*Cos[x]*((-2*I)*(a + I*b)^2*x + (-a^
2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(2*(a + I*b)*(a*(-1 - I*x) + b*(I + x)) + (-a^2 + b^2)*Log[(a*Cos[x
] + b*Sin[x])^2])*Sin[x] + (2*I)*(a^2 - b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x])))/(a*Cos[x] + b*Sin[x]) + 16
*a*b*(a^4 - b^4)*Sin[2*x] - 2*a*b*(a^2 + b^2)^2*Sin[4*x])/(32*(a^2 + b^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.108, size = 515, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x))^2,x)

[Out]

a^3*b^2/(a^2+b^2)^3/(a+b*tan(x))-3*a^4*b^2/(a^2+b^2)^4*ln(a+b*tan(x))+3*a^2*b^4/(a^2+b^2)^4*ln(a+b*tan(x))+1/2
/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)^3*a^3*b^3-3/4/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)^3*a*b^5+5/4/(a^2+b^2)^4/(ta
n(x)^2+1)^2*tan(x)^3*a^5*b-1/2/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)^2*a^6+1/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)^2*a
^4*b^2+3/2/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)^2*a^2*b^4+3/4/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)*a^5*b-1/2/(a^2+b^
2)^4/(tan(x)^2+1)^2*tan(x)*a^3*b^3-5/4/(a^2+b^2)^4/(tan(x)^2+1)^2*tan(x)*a*b^5-1/4/(a^2+b^2)^4/(tan(x)^2+1)^2*
a^6+5/4/(a^2+b^2)^4/(tan(x)^2+1)^2*a^4*b^2+5/4/(a^2+b^2)^4/(tan(x)^2+1)^2*a^2*b^4-1/4/(a^2+b^2)^4/(tan(x)^2+1)
^2*b^6+3/2/(a^2+b^2)^4*ln(tan(x)^2+1)*a^4*b^2-3/2/(a^2+b^2)^4*ln(tan(x)^2+1)*a^2*b^4-3/4/(a^2+b^2)^4*arctan(ta
n(x))*a^5*b+9/2/(a^2+b^2)^4*arctan(tan(x))*a^3*b^3-3/4/(a^2+b^2)^4*arctan(tan(x))*a*b^5

________________________________________________________________________________________

Maxima [B]  time = 1.65306, size = 616, normalized size = 2.93 \begin{align*} -\frac{3 \,{\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac{3 \,{\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \,{\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac{a^{5} - 10 \, a^{3} b^{2} + a b^{4} - 3 \,{\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (x\right )^{4} - 3 \,{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (x\right )^{3} +{\left (2 \, a^{5} - 17 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \tan \left (x\right )^{2} -{\left (2 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \tan \left (x\right )}{4 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{5} +{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{4} + 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )^{3} + 2 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (x\right )^{2} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="maxima")

[Out]

-3/4*(a^5*b - 6*a^3*b^3 + a*b^5)*x/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 3*(a^4*b^2 - a^2*b^4)*log
(b*tan(x) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3/2*(a^4*b^2 - a^2*b^4)*log(tan(x)^2 + 1)/(a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 1/4*(a^5 - 10*a^3*b^2 + a*b^4 - 3*(3*a^3*b^2 - a*b^4)*tan(x)^4
- 3*(a^4*b + a^2*b^3)*tan(x)^3 + (2*a^5 - 17*a^3*b^2 + 5*a*b^4)*tan(x)^2 - (2*a^4*b + a^2*b^3 - b^5)*tan(x))/(
a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*tan(x)^5 + (a^7 + 3*a^5*b^2 + 3*a^
3*b^4 + a*b^6)*tan(x)^4 + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*tan(x)^3 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 +
a*b^6)*tan(x)^2 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*tan(x))

________________________________________________________________________________________

Fricas [A]  time = 0.650174, size = 822, normalized size = 3.91 \begin{align*} \frac{8 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{5} - 8 \,{\left (2 \, a^{7} + 3 \, a^{5} b^{2} - a b^{6}\right )} \cos \left (x\right )^{3} +{\left (5 \, a^{7} + 21 \, a^{5} b^{2} + 27 \, a^{3} b^{4} - 21 \, a b^{6} - 24 \,{\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} x\right )} \cos \left (x\right ) - 48 \,{\left ({\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (x\right ) +{\left (a^{4} b^{3} - a^{2} b^{5}\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 (5 \, a^{6} b - 51 \, a^{4} b^{3} - 21 \, a^{2} b^{5} + 3 \, b^{7} - 8 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} + 24 \,{\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (x\right )^{2} - 24 \,{\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} x\right )} \sin \left (x\right )}{32 \,{\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (x\right ) +{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="fricas")

[Out]

1/32*(8*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x)^5 - 8*(2*a^7 + 3*a^5*b^2 - a*b^6)*cos(x)^3 + (5*a^7 + 21*
a^5*b^2 + 27*a^3*b^4 - 21*a*b^6 - 24*(a^6*b - 6*a^4*b^3 + a^2*b^5)*x)*cos(x) - 48*((a^5*b^2 - a^3*b^4)*cos(x)
+ (a^4*b^3 - a^2*b^5)*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + (5*a^6*b - 51*a^4*b^3 -
21*a^2*b^5 + 3*b^7 - 8*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(x)^4 + 24*(a^6*b + 2*a^4*b^3 + a^2*b^5)*cos(x
)^2 - 24*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*x)*sin(x))/((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cos(x) +
(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*sin(x))

________________________________________________________________________________________

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)**3*sin(x)**3/(a*cos(x)+b*sin(x))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.11637, size = 587, normalized size = 2.8 \begin{align*} -\frac{3 \,{\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} x}{4 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} + \frac{3 \,{\left (a^{4} b^{2} - a^{2} b^{4}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} - \frac{3 \,{\left (a^{4} b^{3} - a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (x\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{3 \, a^{4} b^{3} \tan \left (x\right ) - 3 \, a^{2} b^{5} \tan \left (x\right ) + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4}}{{\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 (x\right ) + a\right )}} - \frac{9 \, a^{4} b^{2} \tan \left (x\right )^{4} - 9 \, a^{2} b^{4} \tan \left (x\right )^{4} - 5 \, a^{5} b \tan \left (x\right )^{3} - 2 \, a^{3} b^{3} \tan \left (x\right )^{3} + 3 \, a b^{5} \tan \left (x\right )^{3} + 2 \, a^{6} \tan \left (x\right )^{2} + 14 \, a^{4} b^{2} \tan \left (x\right )^{2} - 24 \, a^{2} b^{4} \tan \left (x\right )^{2} - 3 \, a^{5} b \tan \left (x\right ) + 2 \, a^{3} b^{3} \tan \left (x\right ) + 5 \, a b^{5} \tan \left (x\right ) + a^{6} + 4 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + b^{6}}{4 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="giac")

[Out]

-3/4*(a^5*b - 6*a^3*b^3 + a*b^5)*x/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3/2*(a^4*b^2 - a^2*b^4)*l
og(tan(x)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 3*(a^4*b^3 - a^2*b^5)*log(abs(b*tan(x) + a)
)/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (3*a^4*b^3*tan(x) - 3*a^2*b^5*tan(x) + 4*a^5*b^2 - 2*a^3
*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(x) + a)) - 1/4*(9*a^4*b^2*tan(x)^4 - 9*a^2*b^4*t
an(x)^4 - 5*a^5*b*tan(x)^3 - 2*a^3*b^3*tan(x)^3 + 3*a*b^5*tan(x)^3 + 2*a^6*tan(x)^2 + 14*a^4*b^2*tan(x)^2 - 24
*a^2*b^4*tan(x)^2 - 3*a^5*b*tan(x) + 2*a^3*b^3*tan(x) + 5*a*b^5*tan(x) + a^6 + 4*a^4*b^2 - 14*a^2*b^4 + b^6)/(
(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(tan(x)^2 + 1)^2)