Integrand size = 18, antiderivative size = 92 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^2 x}{\left (a^2+b^2\right )^2}+\frac {a x}{2 \left (a^2+b^2\right )}+\frac {a^2 b \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}-\frac {a \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )} \]
-a*b^2*x/(a^2+b^2)^2+1/2*a*x/(a^2+b^2)+a^2*b*ln(a*cos(x)+b*sin(x))/(a^2+b^ 2)^2-1/2*a*cos(x)*sin(x)/(a^2+b^2)+1/2*b*sin(x)^2/(a^2+b^2)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {-2 a^3 x-6 i a^2 b x+6 a b^2 x+2 i b^3 x-2 i b \left (-3 a^2+b^2\right ) \arctan (\tan (x))+2 b \left (a^2+b^2\right ) \cos (2 x)-2 \left (a^2+b^2\right ) (a x+b \log (a \cos (x)+b \sin (x)))-3 a^2 b \log \left ((a \cos (x)+b \sin (x))^2\right )+b^3 \log \left ((a \cos (x)+b \sin (x))^2\right )+2 a^3 \sin (2 x)+2 a b^2 \sin (2 x)}{8 \left (a^2+b^2\right )^2} \]
-1/8*(-2*a^3*x - (6*I)*a^2*b*x + 6*a*b^2*x + (2*I)*b^3*x - (2*I)*b*(-3*a^2 + b^2)*ArcTan[Tan[x]] + 2*b*(a^2 + b^2)*Cos[2*x] - 2*(a^2 + b^2)*(a*x + b *Log[a*Cos[x] + b*Sin[x]]) - 3*a^2*b*Log[(a*Cos[x] + b*Sin[x])^2] + b^3*Lo g[(a*Cos[x] + b*Sin[x])^2] + 2*a^3*Sin[2*x] + 2*a*b^2*Sin[2*x])/(a^2 + b^2 )^2
Time = 0.56 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3588, 3042, 3044, 15, 3115, 24, 3576, 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.
| \(\displaystyle \int \frac {\sin ^2(x) \cos (x)}{a \cos (x)+b \sin (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^2 \cos (x)}{a \cos (x)+b \sin (x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {a \int \sin ^2(x)dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \int \sin (x)^2dx}{a^2+b^2}+\frac {b \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {a \int \sin (x)^2dx}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {\sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle -\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle \frac {b \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\) |
-((a*b*((b*x)/(a^2 + b^2) - (a*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)))/(a^ 2 + b^2)) + (b*Sin[x]^2)/(2*(a^2 + b^2)) + (a*(x/2 - (Cos[x]*Sin[x])/2))/( a^2 + b^2)
3.3.76.3.1 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[((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[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_. ) + (d_.)*(x_)]), x_Symbol] :> Simp[b*(x/(a^2 + b^2)), x] - Simp[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]
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 = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {a^{2} b \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tan \left (x \right )-\frac {a^{2} b}{2}-\frac {b^{3}}{2}}{1+\tan \left (x \right )^{2}}+\frac {a \left (-a b \ln \left (1+\tan \left (x \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}\) | \(98\) |
| parallelrisch | \(\frac {-a^{2} b \cos \left (2 x \right )-b^{3} \cos \left (2 x \right )-a^{3} \sin \left (2 x \right )-a \,b^{2} \sin \left (2 x \right )-4 a^{2} b \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+4 a^{2} b \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+2 a^{3} x -2 a \,b^{2} x +a^{2} b +b^{3}}{4 \left (a^{2}+b^{2}\right )^{2}}\) | \(110\) |
| risch | \(-\frac {a x}{2 \left (2 i b a -a^{2}+b^{2}\right )}+\frac {i {\mathrm e}^{2 i x}}{-8 i b +8 a}-\frac {i {\mathrm e}^{-2 i x}}{8 \left (i b +a \right )}-\frac {2 i a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a^{2} b \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) | \(126\) |
| norman | \(\frac {\frac {a \tan \left (\frac {x}{2}\right )^{5}}{a^{2}+b^{2}}+\frac {2 b \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}+\frac {2 b \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}-\frac {a \tan \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {a \left (a^{2}-b^{2}\right ) x}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 a \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 a \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}+\frac {a^{2} b \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a^{2} b \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) | \(295\) |
a^2*b/(a^2+b^2)^2*ln(a+b*tan(x))+1/(a^2+b^2)^2*(((-1/2*a^3-1/2*a*b^2)*tan( x)-1/2*a^2*b-1/2*b^3)/(1+tan(x)^2)+1/2*a*(-a*b*ln(1+tan(x)^2)+(a^2-b^2)*ar ctan(tan(x))))
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{2} b \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{2} - {\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{3} - a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
1/2*(a^2*b*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (a^2*b + b^3)*cos(x)^2 - (a^3 + a*b^2)*cos(x)*sin(x) + (a^3 - a*b^2)*x)/(a^4 + 2* a^2*b^2 + b^4)
Timed out. \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (86) = 172\).
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.29 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{2} b \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a^{2} b \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{3} - a b^{2}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {\frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]
a^2*b*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^2*b*log(sin(x)^2/(cos(x) + 1)^2 + 1)/(a^4 + 2*a^2*b^ 2 + b^4) + (a^3 - a*b^2)*arctan(sin(x)/(cos(x) + 1))/(a^4 + 2*a^2*b^2 + b^ 4) - (a*sin(x)/(cos(x) + 1) - 2*b*sin(x)^2/(cos(x) + 1)^2 - a*sin(x)^3/(co s(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)
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a^{2} b^{2} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {a^{2} b \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} - a b^{2}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {a^{2} b \tan \left (x\right )^{2} - a^{3} \tan \left (x\right ) - a b^{2} \tan \left (x\right ) - b^{3}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}} \]
a^2*b^2*log(abs(b*tan(x) + a))/(a^4*b + 2*a^2*b^3 + b^5) - 1/2*a^2*b*log(t an(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 1/2*(a^3 - a*b^2)*x/(a^4 + 2*a^2*b^ 2 + b^4) + 1/2*(a^2*b*tan(x)^2 - a^3*tan(x) - a*b^2*tan(x) - b^3)/((a^4 + 2*a^2*b^2 + b^4)*(tan(x)^2 + 1))
Time = 30.05 (sec) , antiderivative size = 3401, normalized size of antiderivative = 36.97 \[ \int \frac {\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\text {Too large to display} \]
((a*tan(x/2)^3)/(a^2 + b^2) - (a*tan(x/2))/(a^2 + b^2) + (2*b*tan(x/2)^2)/ (a^2 + b^2))/(2*tan(x/2)^2 + tan(x/2)^4 + 1) + (a^2*b*log(a + 2*b*tan(x/2) - a*tan(x/2)^2))/(a^4 + b^4 + 2*a^2*b^2) - (4*a^2*b*log(1/(cos(x) + 1)))/ (4*a^4 + 4*b^4 + 8*a^2*b^2) - (a*atan((tan(x/2)*((((4*a^2*b*((a*(a + b)*(( 8*(12*a^9*b + 12*a^5*b^5 + 24*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2 ) - (32*a^2*b*(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)))*(a - b))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (16*a^3*b*(a + b)*(a - b)*(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) - (a*((8*(a^9 + 2*a^3*b^6 - 7*a^5*b^4 - 8*a^7*b^ 2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a^2*b*((8*(12*a^9*b + 12*a^5* b^5 + 24*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^2*b*(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))*(a + b)*(a - b))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (a^3*(a + b)^3*(a - b)^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)))*(a^6 - b^6 + 35*a^2*b^4 - 35*a^4*b^2))/(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2 - (2*a*b *(5*a^4 + 5*b^4 - 26*a^2*b^2)*((8*(a^7*b + 2*a^5*b^3))/(a^6 + b^6 + 3*a...