Integrand size = 12, antiderivative size = 186 \[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\frac {x \log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}-\frac {x \log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{\sqrt {4 a^2+b^2}}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}} \]
x*ln(1+b*exp(2*x)/(2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-x*ln(1+b*exp( 2*x)/(2*a+(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)+1/2*polylog(2,-b*exp(2*x)/ (2*a-(4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)-1/2*polylog(2,-b*exp(2*x)/(2*a+ (4*a^2+b^2)^(1/2)))/(4*a^2+b^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.77 \[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\frac {2 x \left (\log \left (1+\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )-\log \left (1+\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {b e^{2 x}}{-2 a+\sqrt {4 a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{2 \sqrt {4 a^2+b^2}} \]
(2*x*(Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])] - Log[1 + (b*E^(2*x)) /(2*a + Sqrt[4*a^2 + b^2])]) + PolyLog[2, (b*E^(2*x))/(-2*a + Sqrt[4*a^2 + b^2])] - PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4* a^2 + b^2])
Time = 0.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6162, 3042, 3803, 27, 2694, 27, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {x}{a+b \sinh (x) \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 6162 |
| \(\displaystyle \int \frac {x}{a+\frac {1}{2} b \sinh (2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {x}{a-\frac {1}{2} i b \sin (2 i x)}dx\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle 2 \int -\frac {2 e^{2 x} x}{-4 e^{2 x} a-b e^{4 x}+b}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \int \frac {e^{2 x} x}{-4 e^{2 x} a-b e^{4 x}+b}dx\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -4 \left (\frac {b \int -\frac {e^{2 x} x}{2 \left (2 a+b e^{2 x}-\sqrt {4 a^2+b^2}\right )}dx}{\sqrt {4 a^2+b^2}}-\frac {b \int -\frac {e^{2 x} x}{2 \left (2 a+b e^{2 x}+\sqrt {4 a^2+b^2}\right )}dx}{\sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \left (\frac {b \int \frac {e^{2 x} x}{2 a+b e^{2 x}+\sqrt {4 a^2+b^2}}dx}{2 \sqrt {4 a^2+b^2}}-\frac {b \int \frac {e^{2 x} x}{2 a+b e^{2 x}-\sqrt {4 a^2+b^2}}dx}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {\int \log \left (\frac {e^{2 x} b}{2 a+\sqrt {4 a^2+b^2}}+1\right )dx}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {\int \log \left (\frac {e^{2 x} b}{2 a-\sqrt {4 a^2+b^2}}+1\right )dx}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {x \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}-\frac {\int e^{-2 x} \log \left (\frac {e^{2 x} b}{2 a+\sqrt {4 a^2+b^2}}+1\right )de^{2 x}}{4 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {x \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}-\frac {\int e^{-2 x} \log \left (\frac {e^{2 x} b}{2 a-\sqrt {4 a^2+b^2}}+1\right )de^{2 x}}{4 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -4 \left (\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a+\sqrt {4 a^2+b^2}}\right )}{4 b}+\frac {x \log \left (\frac {b e^{2 x}}{\sqrt {4 a^2+b^2}+2 a}+1\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}\right )}{4 b}+\frac {x \log \left (\frac {b e^{2 x}}{2 a-\sqrt {4 a^2+b^2}}+1\right )}{2 b}\right )}{2 \sqrt {4 a^2+b^2}}\right )\) |
-4*(-1/2*(b*((x*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/(2*b) + Po lyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/(4*b)))/Sqrt[4*a^2 + b^ 2] + (b*((x*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/(2*b) + PolyLo g[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(4*b)))/(2*Sqrt[4*a^2 + b^2 ]))
3.9.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Si mp[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]
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]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + Cosh[(c_.) + (d_.)*(x_)]*(b_.)*Sinh[ (c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[(e + f*x)^m*(a + b*(Sinh[2*c + 2*d*x]/2))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(162)=324\).
Time = 0.91 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.02
| method | result | size |
| risch | \(\frac {\ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) x}{-2 a -\sqrt {4 a^{2}+b^{2}}}-\frac {x^{2}}{-2 a -\sqrt {4 a^{2}+b^{2}}}+\frac {2 \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a x}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}-\frac {2 a \,x^{2}}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right )}{-4 a -2 \sqrt {4 a^{2}+b^{2}}}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{-2 a -\sqrt {4 a^{2}+b^{2}}}\right ) a}{\sqrt {4 a^{2}+b^{2}}\, \left (-2 a -\sqrt {4 a^{2}+b^{2}}\right )}+\frac {x \ln \left (1-\frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{\sqrt {4 a^{2}+b^{2}}}-\frac {x^{2}}{\sqrt {4 a^{2}+b^{2}}}+\frac {\operatorname {polylog}\left (2, \frac {b \,{\mathrm e}^{2 x}}{\sqrt {4 a^{2}+b^{2}}-2 a}\right )}{2 \sqrt {4 a^{2}+b^{2}}}\) | \(376\) |
1/(-2*a-(4*a^2+b^2)^(1/2))*ln(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))*x-1/( -2*a-(4*a^2+b^2)^(1/2))*x^2+2/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*l n(1-b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))*a*x-2/(4*a^2+b^2)^(1/2)/(-2*a-(4* a^2+b^2)^(1/2))*a*x^2+1/2/(-2*a-(4*a^2+b^2)^(1/2))*polylog(2,b*exp(2*x)/(- 2*a-(4*a^2+b^2)^(1/2)))+1/(4*a^2+b^2)^(1/2)/(-2*a-(4*a^2+b^2)^(1/2))*polyl og(2,b*exp(2*x)/(-2*a-(4*a^2+b^2)^(1/2)))*a+1/(4*a^2+b^2)^(1/2)*x*ln(1-b*e xp(2*x)/((4*a^2+b^2)^(1/2)-2*a))-1/(4*a^2+b^2)^(1/2)*x^2+1/2/(4*a^2+b^2)^( 1/2)*polylog(2,b*exp(2*x)/((4*a^2+b^2)^(1/2)-2*a))
Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (160) = 320\).
Time = 0.28 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.05 \[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=-\frac {b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} + b}{b}\right ) + b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} - b}{b}\right ) - b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} + b}{b}\right ) - b x \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} - b}{b}\right ) + b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} + b}{b} + 1\right ) + b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) - {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {-\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} + 2 \, a}{b}} - b}{b} + 1\right ) - b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} + b}{b} + 1\right ) - b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {{\left (2 \, a \cosh \left (x\right ) + 2 \, a \sinh \left (x\right ) + {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}}\right )} \sqrt {\frac {b \sqrt {\frac {4 \, a^{2} + b^{2}}{b^{2}}} - 2 \, a}{b}} - b}{b} + 1\right )}{4 \, a^{2} + b^{2}} \]
-(b*x*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b) + b*x*sqrt((4*a^2 + b^2)/b^2)*log(-((2*a*cosh(x) + 2*a*sin h(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a ^2 + b^2)/b^2) + 2*a)/b) - b)/b) - b*x*sqrt((4*a^2 + b^2)/b^2)*log(((2*a*c osh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sq rt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b) - b*x*sqrt((4*a^2 + b^2)/b ^2)*log(-((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) - b)/b) + b*sqrt(( 4*a^2 + b^2)/b^2)*dilog(-((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh (x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b^2)/b^2) + 2*a)/b) + b)/b + 1) + b*sqrt((4*a^2 + b^2)/b^2)*dilog(((2*a*cosh(x) + 2*a*sinh(x) - (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqrt(-(b*sqrt((4*a^2 + b ^2)/b^2) + 2*a)/b) - b)/b + 1) - b*sqrt((4*a^2 + b^2)/b^2)*dilog(-((2*a*co sh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a^2 + b^2)/b^2))*sqr t((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) + b)/b + 1) - b*sqrt((4*a^2 + b^2)/ b^2)*dilog(((2*a*cosh(x) + 2*a*sinh(x) + (b*cosh(x) + b*sinh(x))*sqrt((4*a ^2 + b^2)/b^2))*sqrt((b*sqrt((4*a^2 + b^2)/b^2) - 2*a)/b) - b)/b + 1))/(4* a^2 + b^2)
\[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\int \frac {x}{a + b \sinh {\left (x \right )} \cosh {\left (x \right )}}\, dx \]
\[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\int { \frac {x}{b \cosh \left (x\right ) \sinh \left (x\right ) + a} \,d x } \]
\[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\int { \frac {x}{b \cosh \left (x\right ) \sinh \left (x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x}{a+b \cosh (x) \sinh (x)} \, dx=\int \frac {x}{a+b\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]