Integrand size = 20, antiderivative size = 193 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^3 b^3 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac {a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac {a \cos ^5(x)}{5 \left (a^2+b^2\right )}+\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}-\frac {a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \sin ^5(x)}{5 \left (a^2+b^2\right )} \] Output:
a^3*b^3*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)-a^3*b ^2*cos(x)/(a^2+b^2)^3+1/3*a*b^2*cos(x)^3/(a^2+b^2)^2-a*cos(x)^3/(3*a^2+3*b ^2)+a*cos(x)^5/(5*a^2+5*b^2)+a^2*b^3*sin(x)/(a^2+b^2)^3-1/3*a^2*b*sin(x)^3 /(a^2+b^2)^2+b*sin(x)^3/(3*a^2+3*b^2)-b*sin(x)^5/(5*a^2+5*b^2)
Time = 1.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {2 a^3 b^3 \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {-30 a \left (a^4+8 a^2 b^2-b^4\right ) \cos (x)-5 a \left (a^4-2 a^2 b^2-3 b^4\right ) \cos (3 x)+3 a^5 \cos (5 x)+6 a^3 b^2 \cos (5 x)+3 a b^4 \cos (5 x)-30 a^4 b \sin (x)+240 a^2 b^3 \sin (x)+30 b^5 \sin (x)+15 a^4 b \sin (3 x)+10 a^2 b^3 \sin (3 x)-5 b^5 \sin (3 x)-3 a^4 b \sin (5 x)-6 a^2 b^3 \sin (5 x)-3 b^5 \sin (5 x)}{240 \left (a^2+b^2\right )^3} \] Input:
Integrate[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]
Output:
(-2*a^3*b^3*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (-30*a*(a^4 + 8*a^2*b^2 - b^4)*Cos[x] - 5*a*(a^4 - 2*a^2*b^2 - 3*b^4)*Co s[3*x] + 3*a^5*Cos[5*x] + 6*a^3*b^2*Cos[5*x] + 3*a*b^4*Cos[5*x] - 30*a^4*b *Sin[x] + 240*a^2*b^3*Sin[x] + 30*b^5*Sin[x] + 15*a^4*b*Sin[3*x] + 10*a^2* b^3*Sin[3*x] - 5*b^5*Sin[3*x] - 3*a^4*b*Sin[5*x] - 6*a^2*b^3*Sin[5*x] - 3* b^5*Sin[5*x])/(240*(a^2 + b^2)^3)
Time = 1.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.050, Rules used = {3042, 3588, 3042, 3044, 244, 2009, 3045, 244, 2009, 3588, 3042, 3044, 15, 3045, 15, 3588, 3042, 3117, 3118, 3553, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin ^3(x) \cos ^3(x)}{a \cos (x)+b \sin (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^3 \cos (x)^3}{a \cos (x)+b \sin (x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {b \int \cos ^3(x) \sin ^2(x)dx}{a^2+b^2}+\frac {a \int \cos ^2(x) \sin ^3(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \int \cos (x)^3 \sin (x)^2dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {b \int \sin ^2(x) \left (1-\sin ^2(x)\right )d\sin (x)}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {b \int \left (\sin ^2(x)-\sin ^4(x)\right )d\sin (x)}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \int \cos (x)^2 \sin (x)^3dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -\frac {a \int \cos ^2(x) \left (1-\cos ^2(x)\right )d\cos (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {a \int \left (\cos ^2(x)-\cos ^4(x)\right )d\cos (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)^2}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \cos ^2(x) \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin ^2(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {a b \left (\frac {a \int \sin ^2(x)d\sin (x)}{a^2+b^2}+\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \cos (x)^2 \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -\frac {a b \left (-\frac {b \int \cos ^2(x)d\cos (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \cos (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \sin \left (x+\frac {\pi }{2}\right )dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {a \int \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {a b \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \left (\frac {\sin ^3(x)}{3}-\frac {\sin ^5(x)}{5}\right )}{a^2+b^2}-\frac {a \left (\frac {\cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\right )}{a^2+b^2}\) |
Input:
Int[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]
Output:
-((a*(Cos[x]^3/3 - Cos[x]^5/5))/(a^2 + b^2)) + (b*(Sin[x]^3/3 - Sin[x]^5/5 ))/(a^2 + b^2) - (a*b*(-1/3*(b*Cos[x]^3)/(a^2 + b^2) + (a*Sin[x]^3)/(3*(a^ 2 + b^2)) - (a*b*((a*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^ 2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) + (b*Sin[x])/(a^2 + b^2)))/(a^2 + b^2)))/(a^2 + b^2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[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] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(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])
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[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]
Time = 0.78 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\frac {2 a^{2} b^{3} \tan \left (\frac {x}{2}\right )^{9}+2 b^{4} a \tan \left (\frac {x}{2}\right )^{8}+2 \left (\frac {16}{3} a^{2} b^{3}+\frac {4}{3} b^{5}\right ) \tan \left (\frac {x}{2}\right )^{7}+2 \left (-2 a^{5}-6 a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{6}+2 \left (-\frac {16}{5} a^{4} b +\frac {34}{15} a^{2} b^{3}-\frac {8}{15} b^{5}\right ) \tan \left (\frac {x}{2}\right )^{5}+2 \left (\frac {2}{3} a^{5}-\frac {10}{3} a^{3} b^{2}+2 b^{4} a \right ) \tan \left (\frac {x}{2}\right )^{4}+2 \left (\frac {16}{3} a^{2} b^{3}+\frac {4}{3} b^{5}\right ) \tan \left (\frac {x}{2}\right )^{3}+2 \left (-\frac {2}{3} a^{5}-\frac {14}{3} a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}+2 \tan \left (\frac {x}{2}\right ) a^{2} b^{3}-\frac {4 a^{5}}{15}-\frac {28 a^{3} b^{2}}{15}+\frac {2 b^{4} a}{5}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{5}}-\frac {16 a^{3} b^{3} \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}\right ) \sqrt {a^{2}+b^{2}}}\) | \(302\) |
| risch | \(-\frac {i {\mathrm e}^{3 i x} b}{96 \left (-2 i a b +a^{2}-b^{2}\right )}-\frac {{\mathrm e}^{3 i x} a}{96 \left (-2 i a b +a^{2}-b^{2}\right )}+\frac {i {\mathrm e}^{i x} a b}{-12 i a^{2} b +4 i b^{3}+4 a^{3}-12 a \,b^{2}}-\frac {{\mathrm e}^{i x} a^{2}}{16 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right )}+\frac {{\mathrm e}^{i x} b^{2}}{-48 i a^{2} b +16 i b^{3}+16 a^{3}-48 a \,b^{2}}-\frac {i {\mathrm e}^{-i x} a b}{4 \left (i b +a \right )^{3}}-\frac {{\mathrm e}^{-i x} a^{2}}{16 \left (i b +a \right )^{3}}+\frac {{\mathrm e}^{-i x} b^{2}}{16 \left (i b +a \right )^{3}}+\frac {i {\mathrm e}^{-3 i x} b}{96 \left (i b +a \right )^{2}}-\frac {{\mathrm e}^{-3 i x} a}{96 \left (i b +a \right )^{2}}-\frac {i b^{3} a^{3} \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{3}}+\frac {i b^{3} a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{3}}-\frac {a \cos \left (5 x \right )}{80 \left (-a^{2}-b^{2}\right )}+\frac {b \sin \left (5 x \right )}{-80 a^{2}-80 b^{2}}\) | \(400\) |
Input:
int(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)
Output:
2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*(a^2*b^3*tan(1/2*x)^9+b^4*a*tan(1/2*x)^8+( 16/3*a^2*b^3+4/3*b^5)*tan(1/2*x)^7+(-2*a^5-6*a^3*b^2)*tan(1/2*x)^6+(-16/5* a^4*b+34/15*a^2*b^3-8/15*b^5)*tan(1/2*x)^5+(2/3*a^5-10/3*a^3*b^2+2*b^4*a)* tan(1/2*x)^4+(16/3*a^2*b^3+4/3*b^5)*tan(1/2*x)^3+(-2/3*a^5-14/3*a^3*b^2)*t an(1/2*x)^2+tan(1/2*x)*a^2*b^3-2/15*a^5-14/15*a^3*b^2+1/5*b^4*a)/(1+tan(1/ 2*x)^2)^5-16*a^3*b^3/(8*a^6+24*a^4*b^2+24*a^2*b^4+8*b^6)/(a^2+b^2)^(1/2)*a rctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+b^2)^(1/2))
Time = 0.10 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.59 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {15 \, \sqrt {a^{2} + b^{2}} a^{3} b^{3} \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{5} - 10 \, {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (x\right )^{3} - 30 \, {\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (x\right ) - 2 \, {\left (3 \, a^{6} b - 11 \, a^{4} b^{3} - 16 \, a^{2} b^{5} - 2 \, b^{7} + 3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} - {\left (6 \, a^{6} b + 13 \, a^{4} b^{3} + 8 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} \] Input:
integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="fricas")
Output:
1/30*(15*sqrt(a^2 + b^2)*a^3*b^3*log((2*a*b*cos(x)*sin(x) + (a^2 - b^2)*co s(x)^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos (x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)) + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x)^5 - 10*(a^7 + 2*a^5*b^2 + a^3*b^4)*cos(x)^3 - 30*(a^5*b^2 + a^3*b^4)*cos(x) - 2*(3*a^6*b - 11*a^4*b^3 - 16*a^2*b^5 - 2*b^7 + 3*(a^6 *b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(x)^4 - (6*a^6*b + 13*a^4*b^3 + 8*a^2 *b^5 + b^7)*cos(x)^2)*sin(x))/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b ^8)
Timed out. \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out} \] Input:
integrate(cos(x)**3*sin(x)**3/(a*cos(x)+b*sin(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (177) = 354\).
Time = 0.13 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="maxima")
Output:
a^3*b^3*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(c os(x) + 1) - sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a ^2 + b^2)) - 2/15*(2*a^5 + 14*a^3*b^2 - 3*a*b^4 - 15*a^2*b^3*sin(x)/(cos(x ) + 1) - 15*a*b^4*sin(x)^8/(cos(x) + 1)^8 - 15*a^2*b^3*sin(x)^9/(cos(x) + 1)^9 + 10*(a^5 + 7*a^3*b^2)*sin(x)^2/(cos(x) + 1)^2 - 20*(4*a^2*b^3 + b^5) *sin(x)^3/(cos(x) + 1)^3 - 10*(a^5 - 5*a^3*b^2 + 3*a*b^4)*sin(x)^4/(cos(x) + 1)^4 + 2*(24*a^4*b - 17*a^2*b^3 + 4*b^5)*sin(x)^5/(cos(x) + 1)^5 + 30*( a^5 + 3*a^3*b^2)*sin(x)^6/(cos(x) + 1)^6 - 20*(4*a^2*b^3 + b^5)*sin(x)^7/( cos(x) + 1)^7)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + 5*(a^6 + 3*a^4*b^2 + 3 *a^2*b^4 + b^6)*sin(x)^2/(cos(x) + 1)^2 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sin(x)^4/(cos(x) + 1)^4 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*si n(x)^6/(cos(x) + 1)^6 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sin(x)^8/(co s(x) + 1)^8 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sin(x)^10/(cos(x) + 1)^1 0)
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (177) = 354\).
Time = 0.19 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.87 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{3} b^{3} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{9} + 15 \, a b^{4} \tan \left (\frac {1}{2} \, x\right )^{8} + 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + 20 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{7} - 30 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{6} - 90 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 48 \, a^{4} b \tan \left (\frac {1}{2} \, x\right )^{5} + 34 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} - 8 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 10 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{4} - 50 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 30 \, a b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} + 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 20 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 10 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{2} - 70 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 15 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 2 \, a^{5} - 14 \, a^{3} b^{2} + 3 \, a b^{4}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{5}} \] Input:
integrate(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="giac")
Output:
a^3*b^3*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2* x) - 2*b + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a ^2 + b^2)) + 2/15*(15*a^2*b^3*tan(1/2*x)^9 + 15*a*b^4*tan(1/2*x)^8 + 80*a^ 2*b^3*tan(1/2*x)^7 + 20*b^5*tan(1/2*x)^7 - 30*a^5*tan(1/2*x)^6 - 90*a^3*b^ 2*tan(1/2*x)^6 - 48*a^4*b*tan(1/2*x)^5 + 34*a^2*b^3*tan(1/2*x)^5 - 8*b^5*t an(1/2*x)^5 + 10*a^5*tan(1/2*x)^4 - 50*a^3*b^2*tan(1/2*x)^4 + 30*a*b^4*tan (1/2*x)^4 + 80*a^2*b^3*tan(1/2*x)^3 + 20*b^5*tan(1/2*x)^3 - 10*a^5*tan(1/2 *x)^2 - 70*a^3*b^2*tan(1/2*x)^2 + 15*a^2*b^3*tan(1/2*x) - 2*a^5 - 14*a^3*b ^2 + 3*a*b^4)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(1/2*x)^2 + 1)^5)
Time = 17.37 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.11 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\frac {8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (4\,a^2\,b^3+b^5\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^5+7\,a^3\,b^2\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6\,\left (a^5+3\,a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {2\,\left (2\,a^5+14\,a^3\,b^2-3\,a\,b^4\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (a^5-5\,a^3\,b^2+3\,a\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {8\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7\,\left (4\,a^2+b^2\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,a^2\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,a\,b^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,a^2\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {4\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\left (24\,a^4-17\,a^2\,b^2+4\,b^4\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}+\frac {2\,a^3\,b^3\,\mathrm {atanh}\left (\frac {2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{2\,{\left (a^2+b^2\right )}^{7/2}}\right )}{{\left (a^2+b^2\right )}^{7/2}} \] Input:
int((cos(x)^3*sin(x)^3)/(a*cos(x) + b*sin(x)),x)
Output:
((8*tan(x/2)^3*(b^5 + 4*a^2*b^3))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*tan(x/2)^2*(a^5 + 7*a^3*b^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*tan(x/2)^6*(a^5 + 3*a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (2*(2*a^5 - 3*a*b^4 + 14*a^3*b^2))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) ) + (4*tan(x/2)^4*(3*a*b^4 + a^5 - 5*a^3*b^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (8*b^3*tan(x/2)^7*(4*a^2 + b^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*a^2*b^3*tan(x/2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*a*b^4*tan(x/2)^8)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*a^2*b^3*tan (x/2)^9)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*b*tan(x/2)^5*(24*a^4 + 4 *b^4 - 17*a^2*b^2))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(5*tan(x/2)^ 2 + 10*tan(x/2)^4 + 10*tan(x/2)^6 + 5*tan(x/2)^8 + tan(x/2)^10 + 1) + (2*a ^3*b^3*atanh((2*a^6*b + 2*b^7 + 6*a^2*b^5 + 6*a^4*b^3 - 2*a*tan(x/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(2*(a^2 + b^2)^(7/2))))/(a^2 + b^2)^(7/2)
Time = 0.17 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {-2 a^{7}+9 \cos \left (x \right ) \sin \left (x \right )^{4} a^{3} b^{4}+3 \cos \left (x \right ) \sin \left (x \right )^{4} a \,b^{6}-8 \cos \left (x \right ) \sin \left (x \right )^{2} a^{5} b^{2}-13 \cos \left (x \right ) \sin \left (x \right )^{2} a^{3} b^{4}-6 \cos \left (x \right ) \sin \left (x \right )^{2} a \,b^{6}-2 \cos \left (x \right ) a^{7}-3 \sin \left (x \right )^{5} b^{7}+5 \sin \left (x \right )^{3} b^{7}+3 \cos \left (x \right ) \sin \left (x \right )^{4} a^{7}-16 \cos \left (x \right ) a^{5} b^{2}-11 \cos \left (x \right ) a^{3} b^{4}+3 \cos \left (x \right ) a \,b^{6}-3 \sin \left (x \right )^{5} a^{6} b -9 \sin \left (x \right )^{5} a^{4} b^{3}-9 \sin \left (x \right )^{5} a^{2} b^{5}+5 \sin \left (x \right )^{3} a^{4} b^{3}+10 \sin \left (x \right )^{3} a^{2} b^{5}+15 \sin \left (x \right ) a^{4} b^{3}+15 \sin \left (x \right ) a^{2} b^{5}-16 a^{5} b^{2}-3 a \,b^{6}+30 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a i -b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b^{3} i -\cos \left (x \right ) \sin \left (x \right )^{2} a^{7}+9 \cos \left (x \right ) \sin \left (x \right )^{4} a^{5} b^{2}-17 a^{3} b^{4}}{15 a^{8}+60 a^{6} b^{2}+90 a^{4} b^{4}+60 a^{2} b^{6}+15 b^{8}} \] Input:
int(cos(x)^3*sin(x)^3/(a*cos(x)+b*sin(x)),x)
Output:
(30*sqrt(a**2 + b**2)*atan((tan(x/2)*a*i - b*i)/sqrt(a**2 + b**2))*a**3*b* *3*i + 3*cos(x)*sin(x)**4*a**7 + 9*cos(x)*sin(x)**4*a**5*b**2 + 9*cos(x)*s in(x)**4*a**3*b**4 + 3*cos(x)*sin(x)**4*a*b**6 - cos(x)*sin(x)**2*a**7 - 8 *cos(x)*sin(x)**2*a**5*b**2 - 13*cos(x)*sin(x)**2*a**3*b**4 - 6*cos(x)*sin (x)**2*a*b**6 - 2*cos(x)*a**7 - 16*cos(x)*a**5*b**2 - 11*cos(x)*a**3*b**4 + 3*cos(x)*a*b**6 - 3*sin(x)**5*a**6*b - 9*sin(x)**5*a**4*b**3 - 9*sin(x)* *5*a**2*b**5 - 3*sin(x)**5*b**7 + 5*sin(x)**3*a**4*b**3 + 10*sin(x)**3*a** 2*b**5 + 5*sin(x)**3*b**7 + 15*sin(x)*a**4*b**3 + 15*sin(x)*a**2*b**5 - 2* a**7 - 16*a**5*b**2 - 17*a**3*b**4 - 3*a*b**6)/(15*(a**8 + 4*a**6*b**2 + 6 *a**4*b**4 + 4*a**2*b**6 + b**8))