Integrand size = 22, antiderivative size = 244 \[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {a x^2}{2 b^2}-\frac {\cosh (c+d x)}{b d^2}+\frac {\sqrt {a^2-b^2} x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2-b^2} x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {\sqrt {a^2-b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {\sqrt {a^2-b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {x \sinh (c+d x)}{b d} \]
-1/2*a*x^2/b^2-cosh(d*x+c)/b/d^2+x*sinh(d*x+c)/b/d+x*ln(1+b*exp(d*x+c)/(a- (a^2-b^2)^(1/2)))*(a^2-b^2)^(1/2)/b^2/d-x*ln(1+b*exp(d*x+c)/(a+(a^2-b^2)^( 1/2)))*(a^2-b^2)^(1/2)/b^2/d+polylog(2,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))* (a^2-b^2)^(1/2)/b^2/d^2-polylog(2,-b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))*(a^2- b^2)^(1/2)/b^2/d^2
Time = 0.78 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {a (c-d x) (c+d x)-2 b \cosh (c+d x)+2 \sqrt {a^2-b^2} \left (d x \left (\log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )-\log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )+2 b d x \sinh (c+d x)}{2 b^2 d^2} \]
(a*(c - d*x)*(c + d*x) - 2*b*Cosh[c + d*x] + 2*Sqrt[a^2 - b^2]*(d*x*(Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])] - Log[1 + (b*E^(c + d*x))/(a + S qrt[a^2 - b^2])]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] - P olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]) + 2*b*d*x*Sinh[c + d* x])/(2*b^2*d^2)
Time = 1.08 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6100, 15, 3042, 3777, 26, 3042, 26, 3118, 3801, 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.
| \(\displaystyle \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6100 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int xdx}{b^2}+\frac {\int x \cosh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\int x \cosh (c+d x)dx}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\int x \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {i \int -i \sinh (c+d x)dx}{d}}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\int \sinh (c+d x)dx}{d}}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\int -i \sin (i c+i d x)dx}{d}}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x \sinh (c+d x)}{d}+\frac {i \int \sin (i c+i d x)dx}{d}}{b}-\frac {a x^2}{2 b^2}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 3801 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \int \frac {e^{c+d x} x}{2 e^{c+d x} a+b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x}{2 \left (a+b e^{c+d x}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x}{2 \left (a+b e^{c+d x}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {\int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^2}{2 b^2}+\frac {\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}}{b}\) |
-1/2*(a*x^2)/b^2 + (2*(a^2 - b^2)*((b*((x*Log[1 + (b*E^(c + d*x))/(a - Sqr t[a^2 - b^2])])/(b*d) + PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]) )]/(b*d^2)))/(2*Sqrt[a^2 - b^2]) - (b*((x*Log[1 + (b*E^(c + d*x))/(a + Sqr t[a^2 - b^2])])/(b*d) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]) )]/(b*d^2)))/(2*Sqrt[a^2 - b^2])))/b^2 + (-(Cosh[c + d*x]/d^2) + (x*Sinh[c + d*x])/d)/b
3.3.30.3.1 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[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_))/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Sin h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x], x] + Simp[(a^2 - b^2)/b^2 Int[(e + f*x)^m*(Sinh[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(861\) vs. \(2(222)=444\).
Time = 1.27 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.53
| method | result | size |
| risch | \(-\frac {a \,x^{2}}{2 b^{2}}+\frac {\left (d x -1\right ) {\mathrm e}^{d x +c}}{2 b \,d^{2}}-\frac {\left (d x +1\right ) {\mathrm e}^{-d x -c}}{2 b \,d^{2}}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x \,a^{2}}{d \,b^{2} \sqrt {a^{2}-b^{2}}}-\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{d \sqrt {a^{2}-b^{2}}}-\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x \,a^{2}}{d \,b^{2} \sqrt {a^{2}-b^{2}}}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{d \sqrt {a^{2}-b^{2}}}+\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c \,a^{2}}{d^{2} b^{2} \sqrt {a^{2}-b^{2}}}-\frac {\ln \left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} \sqrt {a^{2}-b^{2}}}-\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c \,a^{2}}{d^{2} b^{2} \sqrt {a^{2}-b^{2}}}+\frac {\ln \left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} \sqrt {a^{2}-b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) a^{2}}{d^{2} b^{2} \sqrt {a^{2}-b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} \sqrt {a^{2}-b^{2}}}-\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) a^{2}}{d^{2} b^{2} \sqrt {a^{2}-b^{2}}}+\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} b +\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} \sqrt {a^{2}-b^{2}}}-\frac {2 c \arctan \left (\frac {2 \,{\mathrm e}^{d x +c} b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right ) a^{2}}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 c \arctan \left (\frac {2 \,{\mathrm e}^{d x +c} b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d^{2} \sqrt {-a^{2}+b^{2}}}\) | \(862\) |
-1/2*a*x^2/b^2+1/2*(d*x-1)/b/d^2*exp(d*x+c)-1/2*(d*x+1)/b/d^2*exp(-d*x-c)+ 1/d/b^2/(a^2-b^2)^(1/2)*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2) ^(1/2)))*x*a^2-1/d/(a^2-b^2)^(1/2)*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(- a+(a^2-b^2)^(1/2)))*x-1/d/b^2/(a^2-b^2)^(1/2)*ln((exp(d*x+c)*b+(a^2-b^2)^( 1/2)+a)/(a+(a^2-b^2)^(1/2)))*x*a^2+1/d/(a^2-b^2)^(1/2)*ln((exp(d*x+c)*b+(a ^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*x+1/d^2/b^2/(a^2-b^2)^(1/2)*ln((-exp (d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*c*a^2-1/d^2/(a^2-b^2)^( 1/2)*ln((-exp(d*x+c)*b+(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))*c-1/d^2/b^ 2/(a^2-b^2)^(1/2)*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2))) *c*a^2+1/d^2/(a^2-b^2)^(1/2)*ln((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b ^2)^(1/2)))*c+1/d^2/b^2/(a^2-b^2)^(1/2)*dilog((-exp(d*x+c)*b+(a^2-b^2)^(1/ 2)-a)/(-a+(a^2-b^2)^(1/2)))*a^2-1/d^2/(a^2-b^2)^(1/2)*dilog((-exp(d*x+c)*b +(a^2-b^2)^(1/2)-a)/(-a+(a^2-b^2)^(1/2)))-1/d^2/b^2/(a^2-b^2)^(1/2)*dilog( (exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))*a^2+1/d^2/(a^2-b^2)^ (1/2)*dilog((exp(d*x+c)*b+(a^2-b^2)^(1/2)+a)/(a+(a^2-b^2)^(1/2)))-2/d^2/b^ 2*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*exp(d*x+c)*b+2*a)/(-a^2+b^2)^(1/2))*a^2 +2/d^2*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*exp(d*x+c)*b+2*a)/(-a^2+b^2)^(1/2) )
Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (220) = 440\).
Time = 0.29 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.74 \[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {a d^{2} x^{2} \cosh \left (d x + c\right ) + b d x - {\left (b d x - b\right )} \cosh \left (d x + c\right )^{2} - {\left (b d x - b\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b} + 1\right ) - 2 \, {\left (b c \cosh \left (d x + c\right ) + b c \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) + 2 \, {\left (b c \cosh \left (d x + c\right ) + b c \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left ({\left (b d x + b c\right )} \cosh \left (d x + c\right ) + {\left (b d x + b c\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) + 2 \, {\left ({\left (b d x + b c\right )} \cosh \left (d x + c\right ) + {\left (b d x + b c\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} \log \left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{b^{2}}} + b}{b}\right ) + {\left (a d^{2} x^{2} - 2 \, {\left (b d x - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{2 \, {\left (b^{2} d^{2} \cosh \left (d x + c\right ) + b^{2} d^{2} \sinh \left (d x + c\right )\right )}} \]
-1/2*(a*d^2*x^2*cosh(d*x + c) + b*d*x - (b*d*x - b)*cosh(d*x + c)^2 - (b*d *x - b)*sinh(d*x + c)^2 - 2*(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) + 2*(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d *x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) - 2*(b*c*cosh(d*x + c) + b*c*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*lo g(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) + 2*(b*c*cosh(d*x + c) + b*c*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log(2*b *cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 - b^2)/b^2) + 2*a) - 2* ((b*d*x + b*c)*cosh(d*x + c) + (b*d*x + b*c)*sinh(d*x + c))*sqrt((a^2 - b^ 2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh (d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) + 2*((b*d*x + b*c)*cosh(d*x + c) + (b*d*x + b*c)*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b ^2) + b)/b) + (a*d^2*x^2 - 2*(b*d*x - b)*cosh(d*x + c))*sinh(d*x + c) + b) /(b^2*d^2*cosh(d*x + c) + b^2*d^2*sinh(d*x + c))
\[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x \sinh ^{2}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-b>0)', see `assume?` for more details)Is
\[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]