Integrand size = 20, antiderivative size = 175 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^3 b^2 x}{\left (a^2+b^2\right )^3}-\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {a x}{8 \left (a^2+b^2\right )}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac {a b^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac {a \cos (x) \sin (x)}{8 \left (a^2+b^2\right )}-\frac {a \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}-\frac {a^2 b \sin ^2(x)}{2 \left (a^2+b^2\right )^2} \]
a^3*b^2*x/(a^2+b^2)^3-1/2*a*b^2*x/(a^2+b^2)^2+1/8*a*x/(a^2+b^2)-1/4*b*cos( x)^4/(a^2+b^2)+a^2*b^3*ln(a*cos(x)+b*sin(x))/(a^2+b^2)^3-1/2*a*b^2*cos(x)* sin(x)/(a^2+b^2)^2+1/8*a*cos(x)*sin(x)/(a^2+b^2)-1/4*a*cos(x)^3*sin(x)/(a^ 2+b^2)-1/2*a^2*b*sin(x)^2/(a^2+b^2)^2
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {-4 a^5 x+4 i a^4 b x-24 a^3 b^2 x-24 i a^2 b^3 x+12 a b^4 x+4 i b^5 x-4 i b \left (a^4-6 a^2 b^2+b^4\right ) \arctan (\tan (x))+4 b \left (-a^4+b^4\right ) \cos (2 x)+a^4 b \cos (4 x)+2 a^2 b^3 \cos (4 x)+b^5 \cos (4 x)-4 a^4 b \log (a \cos (x)+b \sin (x))-8 a^2 b^3 \log (a \cos (x)+b \sin (x))-4 b^5 \log (a \cos (x)+b \sin (x))+2 a^4 b \log \left ((a \cos (x)+b \sin (x))^2\right )-12 a^2 b^3 \log \left ((a \cos (x)+b \sin (x))^2\right )+2 b^5 \log \left ((a \cos (x)+b \sin (x))^2\right )+8 a^3 b^2 \sin (2 x)+8 a b^4 \sin (2 x)+a^5 \sin (4 x)+2 a^3 b^2 \sin (4 x)+a b^4 \sin (4 x)}{32 \left (a^2+b^2\right )^3} \]
-1/32*(-4*a^5*x + (4*I)*a^4*b*x - 24*a^3*b^2*x - (24*I)*a^2*b^3*x + 12*a*b ^4*x + (4*I)*b^5*x - (4*I)*b*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[x]] + 4*b* (-a^4 + b^4)*Cos[2*x] + a^4*b*Cos[4*x] + 2*a^2*b^3*Cos[4*x] + b^5*Cos[4*x] - 4*a^4*b*Log[a*Cos[x] + b*Sin[x]] - 8*a^2*b^3*Log[a*Cos[x] + b*Sin[x]] - 4*b^5*Log[a*Cos[x] + b*Sin[x]] + 2*a^4*b*Log[(a*Cos[x] + b*Sin[x])^2] - 1 2*a^2*b^3*Log[(a*Cos[x] + b*Sin[x])^2] + 2*b^5*Log[(a*Cos[x] + b*Sin[x])^2 ] + 8*a^3*b^2*Sin[2*x] + 8*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4 *x] + a*b^4*Sin[4*x])/(a^2 + b^2)^3
Time = 1.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3588, 3042, 3045, 15, 3048, 3042, 3115, 24, 3588, 3042, 3044, 15, 3115, 24, 3577, 3042, 3612}
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.
| \(\displaystyle \int \frac {\sin ^2(x) \cos ^3(x)}{a \cos (x)+b \sin (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^2 \cos (x)^3}{a \cos (x)+b \sin (x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {b \int \cos ^3(x) \sin (x)dx}{a^2+b^2}+\frac {a \int \cos ^2(x) \sin ^2(x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \int \cos (x)^3 \sin (x)dx}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -\frac {b \int \cos ^3(x)d\cos (x)}{a^2+b^2}+\frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \int \cos (x)^2 \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {a \left (\frac {1}{4} \int \cos ^2(x)dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {1}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)^2 \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {a b \left (\frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (-\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle -\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin (x) \cos ^3(x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
-1/4*(b*Cos[x]^4)/(a^2 + b^2) + (a*(-1/4*(Cos[x]^3*Sin[x]) + (x/2 + (Cos[x ]*Sin[x])/2)/4))/(a^2 + b^2) - (a*b*(-((a*b*((a*x)/(a^2 + b^2) + (b*Log[a* Cos[x] + b*Sin[x]])/(a^2 + b^2)))/(a^2 + b^2)) + (a*Sin[x]^2)/(2*(a^2 + b^ 2)) + (b*(x/2 + (Cos[x]*Sin[x])/2))/(a^2 + b^2)))/(a^2 + b^2)
3.3.82.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[(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] + Simp[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]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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] + Simp[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]
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]
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)*(Log[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]
Time = 0.60 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\frac {\left (\frac {1}{8} a^{5}-\frac {1}{4} a^{3} b^{2}-\frac {3}{8} a \,b^{4}\right ) \tan \left (x \right )^{3}+\left (\frac {1}{2} a^{4} b +\frac {1}{2} a^{2} b^{3}\right ) \tan \left (x \right )^{2}+\left (-\frac {3}{4} a^{3} b^{2}-\frac {5}{8} a \,b^{4}-\frac {1}{8} a^{5}\right ) \tan \left (x \right )+\frac {a^{4} b}{4}-\frac {b^{5}}{4}}{\left (1+\tan \left (x \right )^{2}\right )^{2}}+\frac {a \left (-4 a \,b^{3} \ln \left (1+\tan \left (x \right )^{2}\right )+\left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{2} b^{3} \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}\) | \(163\) |
| parallelrisch | \(\frac {32 a^{2} b^{3} \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )-32 a^{2} b^{3} \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-b \left (a^{2}+b^{2}\right )^{2} \cos \left (4 x \right )-a \left (a^{2}+b^{2}\right )^{2} \sin \left (4 x \right )+\left (4 a^{4} b -4 b^{5}\right ) \cos \left (2 x \right )+\left (-8 a^{3} b^{2}-8 a \,b^{4}\right ) \sin \left (2 x \right )+4 a^{5} x +24 a^{3} b^{2} x -12 a \,b^{4} x -3 a^{4} b +2 a^{2} b^{3}+5 b^{5}}{32 \left (a^{2}+b^{2}\right )^{3}}\) | \(165\) |
| risch | \(\frac {3 i x a b}{4 \left (6 i b \,a^{2}-2 i b^{3}-2 a^{3}+6 a \,b^{2}\right )}-\frac {x \,a^{2}}{4 \left (6 i b \,a^{2}-2 i b^{3}-2 a^{3}+6 a \,b^{2}\right )}-\frac {b \,{\mathrm e}^{2 i x}}{16 \left (2 i b a -a^{2}+b^{2}\right )}-\frac {b \,{\mathrm e}^{-2 i x}}{16 \left (-i a +b \right )^{2}}-\frac {2 i a^{2} b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {a^{2} b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b \cos \left (4 x \right )}{-32 a^{2}-32 b^{2}}+\frac {a \sin \left (4 x \right )}{-32 a^{2}-32 b^{2}}\) | \(240\) |
| norman | \(\frac {\frac {2 b^{3} \tan \left (\frac {x}{2}\right )^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 b^{3} \tan \left (\frac {x}{2}\right )^{8}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \left (-2 a^{2} b +b^{3}\right ) \tan \left (\frac {x}{2}\right )^{4}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \left (-2 a^{2} b +b^{3}\right ) \tan \left (\frac {x}{2}\right )^{6}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a x \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right )}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}-\frac {\left (a^{2}+5 b^{2}\right ) a \tan \left (\frac {x}{2}\right )}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{2}+5 b^{2}\right ) a \tan \left (\frac {x}{2}\right )^{9}}{4 a^{4}+8 a^{2} b^{2}+4 b^{4}}+\frac {\left (3 a^{2}-b^{2}\right ) a \tan \left (\frac {x}{2}\right )^{3}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}-\frac {\left (3 a^{2}-b^{2}\right ) a \tan \left (\frac {x}{2}\right )^{7}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {5 a \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 a \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 a \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 a \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{8}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a \left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) x \tan \left (\frac {x}{2}\right )^{10}}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{5}}+\frac {a^{2} b^{3} \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {a^{2} b^{3} \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(682\) |
1/(a^2+b^2)^3*(((1/8*a^5-1/4*a^3*b^2-3/8*a*b^4)*tan(x)^3+(1/2*a^4*b+1/2*a^ 2*b^3)*tan(x)^2+(-3/4*a^3*b^2-5/8*a*b^4-1/8*a^5)*tan(x)+1/4*a^4*b-1/4*b^5) /(1+tan(x)^2)^2+1/8*a*(-4*a*b^3*ln(1+tan(x)^2)+(a^4+6*a^2*b^2-3*b^4)*arcta n(tan(x))))+a^2*b^3/(a^2+b^2)^3*ln(a+b*tan(x))
Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {4 \, a^{2} b^{3} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{4} + 4 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right )^{2} + {\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} x - {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} - {\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} \]
1/8*(4*a^2*b^3*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - 2*( a^4*b + 2*a^2*b^3 + b^5)*cos(x)^4 + 4*(a^4*b + a^2*b^3)*cos(x)^2 + (a^5 + 6*a^3*b^2 - 3*a*b^4)*x - (2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^3 - (a^5 - 2* a^3*b^2 - 3*a*b^4)*cos(x))*sin(x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)
Timed out. \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (161) = 322\).
Time = 0.31 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{2} b^{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^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {a^{2} b^{3} \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {\frac {8 \, b^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {16 \, a^{2} b \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {8 \, b^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {{\left (a^{3} + 5 \, a b^{2}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {{\left (a^{3} + 5 \, a b^{2}\right )} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} \]
a^2*b^3*log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - a^2*b^3*log(sin(x)^2/(cos(x) + 1)^2 + 1) /(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(a^5 + 6*a^3*b^2 - 3*a*b^4)*arc tan(sin(x)/(cos(x) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(8*b^3* sin(x)^2/(cos(x) + 1)^2 - 16*a^2*b*sin(x)^4/(cos(x) + 1)^4 + 8*b^3*sin(x)^ 6/(cos(x) + 1)^6 - (a^3 + 5*a*b^2)*sin(x)/(cos(x) + 1) + (7*a^3 + 3*a*b^2) *sin(x)^3/(cos(x) + 1)^3 - (7*a^3 + 3*a*b^2)*sin(x)^5/(cos(x) + 1)^5 + (a^ 3 + 5*a*b^2)*sin(x)^7/(cos(x) + 1)^7)/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2* a^2*b^2 + b^4)*sin(x)^2/(cos(x) + 1)^2 + 6*(a^4 + 2*a^2*b^2 + b^4)*sin(x)^ 4/(cos(x) + 1)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(x)^6/(cos(x) + 1)^6 + (a^ 4 + 2*a^2*b^2 + b^4)*sin(x)^8/(cos(x) + 1)^8)
Time = 0.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{2} b^{4} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {a^{2} b^{3} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} x}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {6 \, a^{2} b^{3} \tan \left (x\right )^{4} + a^{5} \tan \left (x\right )^{3} - 2 \, a^{3} b^{2} \tan \left (x\right )^{3} - 3 \, a b^{4} \tan \left (x\right )^{3} + 4 \, a^{4} b \tan \left (x\right )^{2} + 16 \, a^{2} b^{3} \tan \left (x\right )^{2} - a^{5} \tan \left (x\right ) - 6 \, a^{3} b^{2} \tan \left (x\right ) - 5 \, a b^{4} \tan \left (x\right ) + 2 \, a^{4} b + 6 \, a^{2} b^{3} - 2 \, b^{5}}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \]
a^2*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 *a^2*b^3*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/8*(a^5 + 6*a^3*b^2 - 3*a*b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/8*(6*a^2* b^3*tan(x)^4 + a^5*tan(x)^3 - 2*a^3*b^2*tan(x)^3 - 3*a*b^4*tan(x)^3 + 4*a^ 4*b*tan(x)^2 + 16*a^2*b^3*tan(x)^2 - a^5*tan(x) - 6*a^3*b^2*tan(x) - 5*a*b ^4*tan(x) + 2*a^4*b + 6*a^2*b^3 - 2*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b ^6)*(tan(x)^2 + 1)^2)
Time = 36.11 (sec) , antiderivative size = 5870, normalized size of antiderivative = 33.54 \[ \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Too large to display} \]
((tan(x/2)^3*(3*a*b^2 + 7*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(x/2)^5* (3*a*b^2 + 7*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(x/2)*(5*a*b^2 + a^3) )/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(x/2)^7*(5*a*b^2 + a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (2*b^3*tan(x/2)^2)/(a^4 + b^4 + 2*a^2*b^2) + (2*b^3*tan(x /2)^6)/(a^4 + b^4 + 2*a^2*b^2) - (4*a^2*b*tan(x/2)^4)/(a^4 + b^4 + 2*a^2*b ^2))/(4*tan(x/2)^2 + 6*tan(x/2)^4 + 4*tan(x/2)^6 + tan(x/2)^8 + 1) - (a*at an((tan(x/2)*((((64*a^2*b^3*((a*((16*a^15*b + 16*a^3*b^13 + 288*a^5*b^11 + 1008*a^7*b^9 + 1472*a^9*b^7 + 1008*a^11*b^5 + 288*a^13*b^3)/(2*(a^12 + b^ 12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (3 2*a^2*b^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 67 20*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64* b^6 + 192*a^2*b^4 + 192*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(a^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a^3*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)*(1 92*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^ 2*b^4 + 192*a^4*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6* a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((a^15 + 18*a^3*b^12 - 141*a^5*b ^10 - 327*a^7*b^8 - 146*a^9*b^6 + 36*a^11*b^4 + 15*a^13*b^2)/(2*(a^12 +...