\(\int \frac {x}{a+b \cos (x) \sin (x)} \, dx\) [581]

Optimal result

Integrand size = 12, antiderivative size = 225 \[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=-\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}} \]

[Out]

-I*x*ln(1-I*b*exp(2*I*x)/(2*a-(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)+I*x*ln(1-I*b*exp(2*I*x)/(2*a+(4*a^2-b^2)^(
1/2)))/(4*a^2-b^2)^(1/2)-1/2*polylog(2,I*b*exp(2*I*x)/(2*a-(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)+1/2*polylog(2
,I*b*exp(2*I*x)/(2*a+(4*a^2-b^2)^(1/2)))/(4*a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4680, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}-\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x \log \left (1-\frac {i b e^{2 i x}}{\sqrt {4 a^2-b^2}+2 a}\right )}{\sqrt {4 a^2-b^2}} \]

[In]

Int[x/(a + b*Cos[x]*Sin[x]),x]

[Out]

((-I)*x*Log[1 - (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (I*x*Log[1 - (I*b*E^((2*I)*x
))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - PolyLog[2, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])]/(2*
Sqrt[4*a^2 - b^2]) + PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])]/(2*Sqrt[4*a^2 - b^2])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4680

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cos[(c_.) + (d_.)*(x_)]*(b_.)*Sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol]
:> Int[(e + f*x)^m*(a + b*(Sin[2*c + 2*d*x]/2))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{a+\frac {1}{2} b \sin (2 x)} \, dx \\ & = 2 \int \frac {e^{2 i x} x}{\frac {i b}{2}+2 a e^{2 i x}-\frac {1}{2} i b e^{4 i x}} \, dx \\ & = -\frac {(2 i b) \int \frac {e^{2 i x} x}{2 a-\sqrt {4 a^2-b^2}-i b e^{2 i x}} \, dx}{\sqrt {4 a^2-b^2}}+\frac {(2 i b) \int \frac {e^{2 i x} x}{2 a+\sqrt {4 a^2-b^2}-i b e^{2 i x}} \, dx}{\sqrt {4 a^2-b^2}} \\ & = -\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i \int \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right ) \, dx}{\sqrt {4 a^2-b^2}}-\frac {i \int \log \left (1-\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right ) \, dx}{\sqrt {4 a^2-b^2}} \\ & = -\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{2 a-\sqrt {4 a^2-b^2}}\right )}{x} \, dx,x,e^{2 i x}\right )}{2 \sqrt {4 a^2-b^2}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{2 a+\sqrt {4 a^2-b^2}}\right )}{x} \, dx,x,e^{2 i x}\right )}{2 \sqrt {4 a^2-b^2}} \\ & = -\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {i x \log \left (1-\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a-\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{2 i x}}{2 a+\sqrt {4 a^2-b^2}}\right )}{2 \sqrt {4 a^2-b^2}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(788\) vs. \(2(225)=450\).

Time = 1.25 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.50 \[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=\frac {1}{2} \left (\frac {\pi \arctan \left (\frac {b+2 a \tan (x)}{\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2}}+\frac {2 \arccos \left (-\frac {2 a}{b}\right ) \text {arctanh}\left (\frac {(2 a-b) \cot \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )+(\pi -4 x) \text {arctanh}\left (\frac {(2 a+b) \tan \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )-\left (\arccos \left (-\frac {2 a}{b}\right )+2 i \text {arctanh}\left (\frac {(2 a-b) \cot \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )\right ) \log \left (\frac {(2 a+b) \left (-2 a+b-i \sqrt {-4 a^2+b^2}\right ) \left (1+i \cot \left (\frac {\pi }{4}+x\right )\right )}{b \left (2 a+b+\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}\right )-\left (\arccos \left (-\frac {2 a}{b}\right )-2 i \text {arctanh}\left (\frac {(2 a-b) \cot \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )\right ) \log \left (\frac {(2 a+b) \left (2 i a-i b+\sqrt {-4 a^2+b^2}\right ) \left (i+\cot \left (\frac {\pi }{4}+x\right )\right )}{b \left (2 a+b+\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}\right )+\left (\arccos \left (-\frac {2 a}{b}\right )+2 i \left (\text {arctanh}\left (\frac {(2 a-b) \cot \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )+\text {arctanh}\left (\frac {(2 a+b) \tan \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {-4 a^2+b^2} e^{-i x}}{2 \sqrt {b} \sqrt {a+b \cos (x) \sin (x)}}\right )+\left (\arccos \left (-\frac {2 a}{b}\right )-2 i \text {arctanh}\left (\frac {(2 a-b) \cot \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )-2 i \text {arctanh}\left (\frac {(2 a+b) \tan \left (\frac {\pi }{4}+x\right )}{\sqrt {-4 a^2+b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {-4 a^2+b^2} e^{i x}}{2 \sqrt {b} \sqrt {a+b \cos (x) \sin (x)}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (2 a-i \sqrt {-4 a^2+b^2}\right ) \left (2 a+b-\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}{b \left (2 a+b+\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (2 a+i \sqrt {-4 a^2+b^2}\right ) \left (2 a+b-\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}{b \left (2 a+b+\sqrt {-4 a^2+b^2} \cot \left (\frac {\pi }{4}+x\right )\right )}\right )\right )}{\sqrt {-4 a^2+b^2}}\right ) \]

[In]

Integrate[x/(a + b*Cos[x]*Sin[x]),x]

[Out]

((Pi*ArcTan[(b + 2*a*Tan[x])/Sqrt[4*a^2 - b^2]])/Sqrt[4*a^2 - b^2] + (2*ArcCos[(-2*a)/b]*ArcTanh[((2*a - b)*Co
t[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] + (Pi - 4*x)*ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] - (ArcCos[
(-2*a)/b] + (2*I)*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*Log[((2*a + b)*(-2*a + b - I*Sqrt[-4*
a^2 + b^2])*(1 + I*Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))] - (ArcCos[(-2*a)/b] - (2*
I)*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*Log[((2*a + b)*((2*I)*a - I*b + Sqrt[-4*a^2 + b^2])*
(I + Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))] + (ArcCos[(-2*a)/b] + (2*I)*(ArcTanh[((
2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] + ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]]))*Log[((-1
)^(1/4)*Sqrt[-4*a^2 + b^2])/(2*Sqrt[b]*E^(I*x)*Sqrt[a + b*Cos[x]*Sin[x]])] + (ArcCos[(-2*a)/b] - (2*I)*ArcTanh
[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] - (2*I)*ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*
Log[-1/2*((-1)^(3/4)*Sqrt[-4*a^2 + b^2]*E^(I*x))/(Sqrt[b]*Sqrt[a + b*Cos[x]*Sin[x]])] + I*(PolyLog[2, ((2*a -
I*Sqrt[-4*a^2 + b^2])*(2*a + b - Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4
+ x]))] - PolyLog[2, ((2*a + I*Sqrt[-4*a^2 + b^2])*(2*a + b - Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))/(b*(2*a + b +
Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))]))/Sqrt[-4*a^2 + b^2])/2

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (191 ) = 382\).

Time = 0.91 (sec) , antiderivative size = 1284, normalized size of antiderivative = 5.71

method	result	size
risch	\(\text {Expression too large to display}\)	\(1284\)

[In]

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

[Out]

4*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2)))*(-(2*
a-b)*(2*a+b))^(1/2)*a*x+4/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*(-(2*a-b)*(2*a+b))^(1/2)*a*x^2-2/(8*
a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2)))*b^2*x+2/(8*a
^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2)))*(-(2*a-b
)*(2*a+b))^(1/2)*a+8/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+
b))^(1/2)))*a^2*x-4*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a-b)
*(2*a+b))^(1/2)))*a^2+I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a+(-(2*a-
b)*(2*a+b))^(1/2)))*b^2+2*I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*x^2*b^2-4*I/(8*a^2-2*b^2)/(-2*I*a-
(-(2*a-b)*(2*a+b))^(1/2))*(-(2*a-b)*(2*a+b))^(1/2)*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2)))*a*x-8*
I/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*a^2*x^2+8/(8*a^2-2*b^2)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2))*ln
(1-b*exp(2*I*x)/(-2*I*a+(-(2*a-b)*(2*a+b))^(1/2)))*a^2*x-4/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*(-(
2*a-b)*(2*a+b))^(1/2)*a*x^2-2/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*ln(1-b*exp(2*I*x)/(-2*I*a-(-(2*a
-b)*(2*a+b))^(1/2)))*b^2*x-2/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*(-(2*a-b)*(2*a+b))^(1/2)*polylog(
2,b*exp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2)))*a-8*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*a^2*x^
2+2*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2))*x^2*b^2-4*I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/
2))*polylog(2,b*exp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2)))*a^2+I/(8*a^2-2*b^2)/(-2*I*a-(-(2*a-b)*(2*a+b))^(
1/2))*polylog(2,b*exp(2*I*x)/(-2*I*a-(-(2*a-b)*(2*a+b))^(1/2)))*b^2

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (179) = 358\).

Time = 1.19 (sec) , antiderivative size = 1676, normalized size of antiderivative = 7.45 \[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-((2*I*a*cos(x) + 2*a*sin(x) - (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b
^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) - b)/b) + b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-((-2*I*a*co
s(x) - 2*a*sin(x) + (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a
)/b) - b)/b) - b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-((2*I*a*cos(x) - 2*a*sin(x) - (b*cos(x) + I*b*sin(x))*sqrt(-(
4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) - b)/b) - b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-(
(-2*I*a*cos(x) + 2*a*sin(x) + (b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b
^2) + 2*I*a)/b) - b)/b) + b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-((2*I*a*cos(x) - 2*a*sin(x) + (b*cos(x) + I*b*sin(
x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) - b)/b) + b*x*sqrt(-(4*a^2 - b^2)/b
^2)*log(-((-2*I*a*cos(x) + 2*a*sin(x) - (b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2
- b^2)/b^2) - 2*I*a)/b) - b)/b) - b*x*sqrt(-(4*a^2 - b^2)/b^2)*log(-((2*I*a*cos(x) + 2*a*sin(x) + (b*cos(x) -
I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) - b)/b) - b*x*sqrt(-(4*a^
2 - b^2)/b^2)*log(-((-2*I*a*cos(x) - 2*a*sin(x) - (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*s
qrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) - b)/b) + I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((2*I*a*cos(x) + 2*a*sin(x) -
(b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) - b)/b + 1) +
I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x) - 2*a*sin(x) + (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/
b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) - b)/b + 1) + I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((2*I*a*co
s(x) - 2*a*sin(x) - (b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*
a)/b) - b)/b + 1) + I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x) + 2*a*sin(x) + (b*cos(x) + I*b*sin(x))*
sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) + 2*I*a)/b) - b)/b + 1) - I*b*sqrt(-(4*a^2 - b^2)/
b^2)*dilog(((2*I*a*cos(x) - 2*a*sin(x) + (b*cos(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^
2 - b^2)/b^2) - 2*I*a)/b) - b)/b + 1) - I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x) + 2*a*sin(x) - (b*c
os(x) + I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt((b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) - b)/b + 1) - I*b*s
qrt(-(4*a^2 - b^2)/b^2)*dilog(((2*I*a*cos(x) + 2*a*sin(x) + (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*
sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b) - b)/b + 1) - I*b*sqrt(-(4*a^2 - b^2)/b^2)*dilog(((-2*I*a*cos(x)
- 2*a*sin(x) - (b*cos(x) - I*b*sin(x))*sqrt(-(4*a^2 - b^2)/b^2))*sqrt(-(b*sqrt(-(4*a^2 - b^2)/b^2) - 2*I*a)/b
) - b)/b + 1))/(4*a^2 - b^2)

Sympy [F]

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

[In]

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

[Out]

Integral(x/(a + b*sin(x)*cos(x)), x)

Maxima [F]

\[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=\int { \frac {x}{b \cos \left (x\right ) \sin \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(x/(b*cos(x)*sin(x) + a), x)

Giac [F]

\[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=\int { \frac {x}{b \cos \left (x\right ) \sin \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(x/(b*cos(x)*sin(x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{a+b \cos (x) \sin (x)} \, dx=\int \frac {x}{a+b\,\cos \left (x\right )\,\sin \left (x\right )} \,d x \]

[In]

int(x/(a + b*cos(x)*sin(x)),x)

[Out]

int(x/(a + b*cos(x)*sin(x)), x)