Integrand size = 18, antiderivative size = 93 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}+\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )} \]
-a^2*b*x/(a^2+b^2)^2+1/2*b*x/(a^2+b^2)-a*b^2*ln(a*cos(x)+b*sin(x))/(a^2+b^ 2)^2+1/2*b*cos(x)*sin(x)/(a^2+b^2)+1/2*a*sin(x)^2/(a^2+b^2)
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {4 i a b^2 \arctan (\tan (x))-a \left (a^2+b^2\right ) \cos (2 x)-2 b \left ((a+i b)^2 x+a b \log \left ((a \cos (x)+b \sin (x))^2\right )\right )+b \left (a^2+b^2\right ) \sin (2 x)}{4 \left (a^2+b^2\right )^2} \]
((4*I)*a*b^2*ArcTan[Tan[x]] - a*(a^2 + b^2)*Cos[2*x] - 2*b*((a + I*b)^2*x + a*b*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(a^2 + b^2)*Sin[2*x])/(4*(a^2 + b^ 2)^2)
Time = 0.52 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98, 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, 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.
| \(\displaystyle \int \frac {\sin (x) \cos ^2(x)}{a \cos (x)+b \sin (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x) \cos (x)^2}{a \cos (x)+b \sin (x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {b \int \cos ^2(x)dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \cos (x) \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {a \int \sin (x)d\sin (x)}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {b \int \sin \left (x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a b \int \frac {\cos (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a b \left (\frac {b \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle \frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )}{a^2+b^2}-\frac {a b \left (\frac {a x}{a^2+b^2}+\frac {b \log (a \cos (x)+b \sin (x))}{a^2+b^2}\right )}{a^2+b^2}\) |
-((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)
3.3.78.3.1 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[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 = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(-\frac {a \,b^{2} \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tan \left (x \right )-\frac {a^{3}}{2}-\frac {a \,b^{2}}{2}}{1+\tan \left (x \right )^{2}}+\frac {b \left (a b \ln \left (1+\tan \left (x \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (x \right )\right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}\) | \(98\) |
| parallelrisch | \(\frac {-a^{3} \cos \left (2 x \right )-a \,b^{2} \cos \left (2 x \right )+a^{2} b \sin \left (2 x \right )+b^{3} \sin \left (2 x \right )-4 a \,b^{2} \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )+4 a \,b^{2} \ln \left (\frac {1}{\cos \left (x \right )+1}\right )-2 x \,a^{2} b +2 x \,b^{3}+a^{3}+a \,b^{2}}{4 \left (a^{2}+b^{2}\right )^{2}}\) | \(108\) |
| risch | \(\frac {x b}{4 i b a -2 a^{2}+2 b^{2}}-\frac {{\mathrm e}^{2 i x}}{8 \left (-i b +a \right )}-\frac {{\mathrm e}^{-2 i x}}{8 \left (i b +a \right )}+\frac {2 i a \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a \,b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) | \(125\) |
| norman | \(\frac {\frac {b \tan \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {2 a \tan \left (\frac {x}{2}\right )^{2}}{a^{2}+b^{2}}+\frac {2 a \tan \left (\frac {x}{2}\right )^{4}}{a^{2}+b^{2}}-\frac {b \tan \left (\frac {x}{2}\right )^{5}}{a^{2}+b^{2}}-\frac {b \left (a^{2}-b^{2}\right ) x}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{4}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{6}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}+\frac {a \,b^{2} \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a \,b^{2} \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\) | \(295\) |
-a*b^2/(a^2+b^2)^2*ln(a+b*tan(x))+1/(a^2+b^2)^2*(((1/2*a^2*b+1/2*b^3)*tan( x)-1/2*a^3-1/2*a*b^2)/(1+tan(x)^2)+1/2*b*(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 = 1.01 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{2} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) + {\left (a^{3} + a b^{2}\right )} \cos \left (x\right )^{2} - {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} b - b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
-1/2*(a*b^2*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) + (a^3 + a*b^2)*cos(x)^2 - (a^2*b + b^3)*cos(x)*sin(x) + (a^2*b - b^3)*x)/(a^4 + 2 *a^2*b^2 + b^4)
Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (87) = 174\).
Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.28 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{2} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a b^{2} \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} b - b^{3}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {\frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]
-a*b^2*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*b^2*log(sin(x)^2/(cos(x) + 1)^2 + 1)/(a^4 + 2*a^2*b ^2 + b^4) - (a^2*b - b^3)*arctan(sin(x)/(cos(x) + 1))/(a^4 + 2*a^2*b^2 + b ^4) + (b*sin(x)/(cos(x) + 1) + 2*a*sin(x)^2/(cos(x) + 1)^2 - b*sin(x)^3/(c os(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 = 156, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a b^{3} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a b^{2} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a b^{2} \tan \left (x\right )^{2} - a^{2} b \tan \left (x\right ) - b^{3} \tan \left (x\right ) + a^{3} + 2 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (x\right )^{2} + 1\right )}} \]
-a*b^3*log(abs(b*tan(x) + a))/(a^4*b + 2*a^2*b^3 + b^5) + 1/2*a*b^2*log(ta n(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a^2*b - b^3)*x/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a*b^2*tan(x)^2 - a^2*b*tan(x) - b^3*tan(x) + a^3 + 2*a*b^2) /((a^4 + 2*a^2*b^2 + b^4)*(tan(x)^2 + 1))
Time = 29.67 (sec) , antiderivative size = 3419, normalized size of antiderivative = 36.76 \[ \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\text {Too large to display} \]
((b*tan(x/2))/(a^2 + b^2) + (2*a*tan(x/2)^2)/(a^2 + b^2) - (b*tan(x/2)^3)/ (a^2 + b^2))/(2*tan(x/2)^2 + tan(x/2)^4 + 1) - (a*b^2*log(a + 2*b*tan(x/2) - a*tan(x/2)^2))/(a^4 + b^4 + 2*a^2*b^2) + (4*a*b^2*log(1/(cos(x) + 1)))/ (4*a^4 + 4*b^4 + 8*a^2*b^2) - (b*atan((tan(x/2)*((((4*a*b^2*((b*(a + b)*(a - b)*((8*(12*a^4*b^6 + 24*a^6*b^4 + 12*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a*b^2*(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))))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (16*a*b^3*(a + b)*(a - b)*(12*a*b^1 0 + 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) - (b*(a + b)*((8*(2*a*b^8 - 7*a^3*b^6 - 8*a^5* b^4 + a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a*b^2*((8*(12*a^4 *b^6 + 24*a^6*b^4 + 12*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32 *a*b^2*(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))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (b^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*(2*a^2*b^6 + a^4*b^4))/(a^6 + b...