\(\int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx\) [275]

Optimal result

Integrand size = 16, antiderivative size = 65 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \cos (x)}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3188, 2717, 2718, 3153, 212} \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {b \sin (x)}{a^2+b^2}-\frac {a \cos (x)}{a^2+b^2} \]

[In]

Int[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x]),x]

[Out]

(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)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3153

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Dist[-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]

Rule 3188

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \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} \\ & = -\frac {a \cos (x)}{a^2+b^2}+\frac {b \sin (x)}{a^2+b^2}+\frac {(a b) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{a^2+b^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {2 a b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {-a \cos (x)+b \sin (x)}{a^2+b^2} \]

[In]

Integrate[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x]),x]

[Out]

(-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)

Maple [A] (verified)

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	\(-\frac {{\mathrm e}^{i x}}{2 \left (-i b +a \right )}-\frac {{\mathrm e}^{-i x}}{2 \left (i b +a \right )}+\frac {i b a \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )}-\frac {i b a \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )}\)	\(141\)

[In]

int(cos(x)*sin(x)/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [B] (verification not implemented)

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 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a b \log \left (\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) + 2 \, {\left (a^{2} b + b^{3}\right )} \sin \left (x\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]

[In]

integrate(cos(x)*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="fricas")

[Out]

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)

Sympy [C] (verification not implemented)

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} \]

[In]

integrate(cos(x)*sin(x)/(a*cos(x)+b*sin(x)),x)

[Out]

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*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)*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*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))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.62 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}}{a^{2} + b^{2} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} \]

[In]

integrate(cos(x)*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="maxima")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.45 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {a b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, x\right ) - a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]

[In]

integrate(cos(x)*sin(x)/(a*cos(x)+b*sin(x)),x, algorithm="giac")

[Out]

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))

Mupad [B] (verification not implemented)

Time = 22.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43 \[ \int \frac {\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx=\frac {2\,a\,b\,\mathrm {atanh}\left (\frac {2\,a^2\,b+2\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+b^2\right )}{2\,{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {\frac {2\,a}{a^2+b^2}-\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]

[In]

int((cos(x)*sin(x))/(a*cos(x) + b*sin(x)),x)

[Out]

(2*a*b*atanh((2*a^2*b + 2*b^3 - 2*a*tan(x/2)*(a^2 + b^2))/(2*(a^2 + b^2)^(3/2))))/(a^2 + b^2)^(3/2) - ((2*a)/(
a^2 + b^2) - (2*b*tan(x/2))/(a^2 + b^2))/(tan(x/2)^2 + 1)