Integrand size = 16, antiderivative size = 65 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \cos (x)}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2} \]
a*b*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(3/2)-a*cos(x)/ (a^2+b^2)+b*sin(x)/(a^2+b^2)
Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {2 a b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {-a \cos (x)+b \sin (x)}{a^2+b^2} \]
(-2*a*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) + (- (a*Cos[x]) + b*Sin[x])/(a^2 + b^2)
Time = 0.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 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.
\(\displaystyle \int \frac {\sin (x) \cos (x)}{a \cos (x)+b \sin (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x) \cos (x)}{a \cos (x)+b \sin (x)}dx\) |
\(\Big \downarrow \) 3588 |
\(\displaystyle \frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \cos (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \int \sin (x)dx}{a^2+b^2}+\frac {b \int \sin \left (x+\frac {\pi }{2}\right )dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {a \int \sin (x)dx}{a^2+b^2}-\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {a b \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle \frac {a b \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2}\) |
(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)
3.3.75.3.1 Defintions of rubi rules used
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[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 = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {4 a b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {2 b \tan \left (\frac {x}{2}\right )-2 a}{\left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\) | \(82\) |
risch | \(-\frac {{\mathrm e}^{i x}}{2 \left (-i b +a \right )}-\frac {{\mathrm e}^{-i x}}{2 \left (i b +a \right )}+\frac {i b a \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )}-\frac {i b a \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )}\) | \(141\) |
-4*a*b/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2 +b^2)^(1/2))+2/(a^2+b^2)*(b*tan(1/2*x)-a)/(1+tan(1/2*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (61) = 122\).
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.18 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a b \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) + 2 \, {\left (a^{2} b + b^{3}\right )} \sin \left (x\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
1/2*(sqrt(a^2 + b^2)*a*b*log((2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(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)) - 2*(a^3 + a*b^2)*cos(x) + 2*(a^2*b + b^ 3)*sin(x))/(a^4 + 2*a^2*b^2 + b^4)
Result contains complex when optimal does not.
Time = 67.47 (sec) , antiderivative size = 699, normalized size of antiderivative = 10.75 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\begin {cases} \tilde {\infty } \sin {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\sin {\left (x \right )}}{b} & \text {for}\: a = 0 \\\frac {i \sin ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} + \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {i \cos ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = - i b \\- \frac {i \sin ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} + \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} + \frac {i \cos ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = i b \\\frac {a b \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {a b \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {a b \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {a b \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {2 a \sqrt {a^{2} + b^{2}}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {2 b \sqrt {a^{2} + b^{2}} \tan {\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*sin(x), Eq(a, 0) & Eq(b, 0)), (sin(x)/b, Eq(a, 0)), (I*sin( x)**2/(3*I*b*sin(x) + 3*b*cos(x)) + sin(x)*cos(x)/(3*I*b*sin(x) + 3*b*cos( x)) - I*cos(x)**2/(3*I*b*sin(x) + 3*b*cos(x)), Eq(a, -I*b)), (-I*sin(x)**2 /(-3*I*b*sin(x) + 3*b*cos(x)) + sin(x)*cos(x)/(-3*I*b*sin(x) + 3*b*cos(x)) + I*cos(x)**2/(-3*I*b*sin(x) + 3*b*cos(x)), Eq(a, I*b)), (a*b*log(tan(x/2 ) - b/a - sqrt(a**2 + b**2)/a)*tan(x/2)**2/(a**2*sqrt(a**2 + b**2)*tan(x/2 )**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2* sqrt(a**2 + b**2)) + a*b*log(tan(x/2) - b/a - sqrt(a**2 + b**2)/a)/(a**2*s qrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b **2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - a*b*log(tan(x/2) - b/a + sqrt (a**2 + b**2)/a)*tan(x/2)**2/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sq rt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b* *2)) - a*b*log(tan(x/2) - b/a + sqrt(a**2 + b**2)/a)/(a**2*sqrt(a**2 + b** 2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)* *2 + b**2*sqrt(a**2 + b**2)) - 2*a*sqrt(a**2 + b**2)/(a**2*sqrt(a**2 + b** 2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)* *2 + b**2*sqrt(a**2 + b**2)) + 2*b*sqrt(a**2 + b**2)*tan(x/2)/(a**2*sqrt(a **2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)* tan(x/2)**2 + b**2*sqrt(a**2 + b**2)), True))
Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.62 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}}{a^{2} + b^{2} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} \]
a*b*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(cos(x ) + 1) - sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) - 2*(a - b*sin(x)/(cos(x) + 1 ))/(a^2 + b^2 + (a^2 + b^2)*sin(x)^2/(cos(x) + 1)^2)
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.45 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, x\right ) - a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]
a*b*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^2 + b^2)^(3/2) + 2*(b*tan(1/2*x) - a)/((a^2 + b^2)*(tan(1/2*x)^2 + 1))
Time = 22.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {2\,a\,b\,\mathrm {atanh}\left (\frac {2\,a^2\,b+2\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+b^2\right )}{2\,{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {\frac {2\,a}{a^2+b^2}-\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]