Integrand size = 16, antiderivative size = 107 \[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(c+d x) \cosh ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}-\frac {(b c-a d) \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \]
(d*x+c)*cosh((b*x+a)/(d*x+c))^2/d-(-a*d+b*c)*cosh(2*b/d)*Shi(2*(-a*d+b*c)/ d/(d*x+c))/d^2+(-a*d+b*c)*Chi(2*(-a*d+b*c)/d/(d*x+c))*sinh(2*b/d)/d^2
Leaf count is larger than twice the leaf count of optimal. \(475\) vs. \(2(107)=214\).
Time = 3.17 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.44 \[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 (a+b x)}{c+d x}}+c d e^{\frac {2 (a+b x)}{c+d x}}+2 d^2 x+2 d^2 x \cosh \left (\frac {2 b}{d}\right ) \cosh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 (b c-a d) \text {Chi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {2 b}{d}\right )-\sinh \left (\frac {2 b}{d}\right )\right )+2 (b c-a d) \text {Chi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right ) \left (\cosh \left (\frac {2 b}{d}\right )+\sinh \left (\frac {2 b}{d}\right )\right )+2 d^2 x \sinh \left (\frac {2 b}{d}\right ) \sinh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )}{4 d^2} \]
((c*d)/E^((2*(a + b*x))/(c + d*x)) + c*d*E^((2*(a + b*x))/(c + d*x)) + 2*d ^2*x + 2*d^2*x*Cosh[(2*b)/d]*Cosh[(2*(-(b*c) + a*d))/(d*(c + d*x))] - 2*(b *c - a*d)*CoshIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)]*(Cosh[(2*b)/d] - Sin h[(2*b)/d]) + 2*(b*c - a*d)*CoshIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] *(Cosh[(2*b)/d] + Sinh[(2*b)/d]) + 2*d^2*x*Sinh[(2*b)/d]*Sinh[(2*(-(b*c) + a*d))/(d*(c + d*x))] + 2*b*c*Cosh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d) )/(d*(c + d*x))] - 2*a*d*Cosh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d* (c + d*x))] + 2*b*c*Sinh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] - 2*a*d*Sinh[(2*b)/d]*SinhIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x)) ] - 2*b*c*Cosh[(2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] + 2*a* d*Cosh[(2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] + 2*b*c*Sinh[( 2*b)/d]*SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] - 2*a*d*Sinh[(2*b)/d]* SinhIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)])/(4*d^2)
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6142, 3042, 3794, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 6142 |
| \(\displaystyle -\frac {\int (c+d x)^2 \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )^2d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {-\left ((c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )-\frac {2 i (b c-a d) \int -\frac {1}{2} i (c+d x) \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {(b c-a d) \int (c+d x) \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}-\left ((c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\left ((c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )\right )-\frac {(b c-a d) \int -i (c+d x) \sin \left (\frac {2 i b}{d}-\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \int (c+d x) \sin \left (\frac {2 i b}{d}-\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \cosh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}+\cosh \left (\frac {2 b}{d}\right ) \int -i (c+d x) \sinh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \cosh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \int (c+d x) \sinh \left (\frac {2 (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \int -i (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-\cosh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}\right )d\frac {1}{c+d x}\right )}{d}}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \int (c+d x) \sin \left (\frac {2 i (b c-a d)}{d (c+d x)}+\frac {\pi }{2}\right )d\frac {1}{c+d x}-i \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )\right )}{d}}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle -\frac {-(c+d x) \cosh ^2\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )+\frac {i (b c-a d) \left (i \sinh \left (\frac {2 b}{d}\right ) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )-i \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )\right )}{d}}{d}\) |
-((-((c + d*x)*Cosh[b/d - (b*c - a*d)/(d*(c + d*x))]^2) + (I*(b*c - a*d)*( I*CoshIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sinh[(2*b)/d] - I*Cosh[(2*b) /d]*SinhIntegral[(2*(b*c - a*d))/(d*(c + d*x))]))/d)/d)
3.3.63.3.1 Defintions of rubi rules used
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[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[Cosh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol ] :> Simp[-d^(-1) Subst[Int[Cosh[b*(e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x] , x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(107)=214\).
Time = 0.85 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.35
| method | result | size |
| risch | \(\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} a}{\frac {4 d a}{d x +c}-\frac {4 c b}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} c b}{4 d \left (\frac {d a}{d x +c}-\frac {c b}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {Ei}_{1}\left (\frac {2 d a -2 c b}{\left (d x +c \right ) d}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {Ei}_{1}\left (\frac {2 d a -2 c b}{\left (d x +c \right ) d}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x c b}{4 \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c a}{4 d a -4 c b}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c^{2} b}{4 d \left (d a -c b \right )}+\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (d a -c b \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\) | \(358\) |
1/2*x+1/4*exp(-2*(b*x+a)/(d*x+c))/(d/(d*x+c)*a-1/(d*x+c)*c*b)*a-1/4/d*exp( -2*(b*x+a)/(d*x+c))/(d/(d*x+c)*a-1/(d*x+c)*c*b)*c*b-1/2/d*exp(-2/d*b)*Ei(1 ,2*(a*d-b*c)/d/(d*x+c))*a+1/2/d^2*exp(-2/d*b)*Ei(1,2*(a*d-b*c)/d/(d*x+c))* b*c+1/4*d*exp(2*(b*x+a)/(d*x+c))/(a*d-b*c)*x*a-1/4*exp(2*(b*x+a)/(d*x+c))/ (a*d-b*c)*x*c*b+1/4*exp(2*(b*x+a)/(d*x+c))/(a*d-b*c)*c*a-1/4/d*exp(2*(b*x+ a)/(d*x+c))/(a*d-b*c)*c^2*b+1/2/d*exp(2/d*b)*Ei(1,-2*(a*d-b*c)/d/(d*x+c))* a-1/2/d^2*exp(2/d*b)*Ei(1,-2*(a*d-b*c)/d/(d*x+c))*b*c
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (107) = 214\).
Time = 0.30 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.42 \[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {d^{2} x + {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (d^{2} x - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) + {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2}\right )}} \]
1/2*(d^2*x + (d^2*x + c*d)*cosh((b*x + a)/(d*x + c))^2 + (d^2*x - (b*c - a *d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh(2*b/d) + c*d)*sinh((b*x + a)/(d* x + c))^2 + ((b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/( d*x + c))^2 - (b*c - a*d)*Ei(2*(b*c - a*d)/(d^2*x + c*d)))*cosh(2*b/d) + ( (b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2 - (b*c - a*d)*Ei(-2*(b*c - a*d)/(d^2*x + c*d))*sinh((b*x + a)/(d*x + c))^2 + (b*c - a*d)*Ei(2*(b*c - a*d)/(d^2*x + c*d)))*sinh(2*b/d))/(d^2*cosh((b*x + a)/(d*x + c))^2 - d^2*sinh((b*x + a)/(d*x + c))^2)
\[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int \cosh ^{2}{\left (\frac {a + b x}{c + d x} \right )}\, dx \]
\[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \,d x } \]
1/2*x + 1/4*integrate(e^(2*b*c/(d^2*x + c*d) - 2*a/(d*x + c) - 2*b/d), x) + 1/4*integrate(e^(-2*b*c/(d^2*x + c*d) + 2*a/(d*x + c) + 2*b/d), x)
Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (107) = 214\).
Time = 6.84 (sec) , antiderivative size = 749, normalized size of antiderivative = 7.00 \[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (2 \, b^{3} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - 4 \, a b^{2} c d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} + 2 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} + \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, b^{3} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + 4 \, a b^{2} c d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, a^{2} b d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} - \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + b^{2} c^{2} d e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + 2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{4 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]
1/4*(2*b^3*c^2*Ei(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) - 4*a*b^2*c* d*Ei(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) - 2*(b*x + a)*b^2*c^2*d*E i(-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) + 2*a^2*b*d^2*Ei(- 2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d) + 4*(b*x + a)*a*b*c*d^2*Ei(-2*( b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) - 2*(b*x + a)*a^2*d^3*Ei (-2*(b - (b*x + a)*d/(d*x + c))/d)*e^(2*b/d)/(d*x + c) - 2*b^3*c^2*Ei(2*(b - (b*x + a)*d/(d*x + c))/d)*e^(-2*b/d) + 4*a*b^2*c*d*Ei(2*(b - (b*x + a)* d/(d*x + c))/d)*e^(-2*b/d) + 2*(b*x + a)*b^2*c^2*d*Ei(2*(b - (b*x + a)*d/( d*x + c))/d)*e^(-2*b/d)/(d*x + c) - 2*a^2*b*d^2*Ei(2*(b - (b*x + a)*d/(d*x + c))/d)*e^(-2*b/d) - 4*(b*x + a)*a*b*c*d^2*Ei(2*(b - (b*x + a)*d/(d*x + c))/d)*e^(-2*b/d)/(d*x + c) + 2*(b*x + a)*a^2*d^3*Ei(2*(b - (b*x + a)*d/(d *x + c))/d)*e^(-2*b/d)/(d*x + c) + b^2*c^2*d*e^(2*(b*x + a)/(d*x + c)) - 2 *a*b*c*d^2*e^(2*(b*x + a)/(d*x + c)) + a^2*d^3*e^(2*(b*x + a)/(d*x + c)) + b^2*c^2*d*e^(-2*(b*x + a)/(d*x + c)) - 2*a*b*c*d^2*e^(-2*(b*x + a)/(d*x + c)) + a^2*d^3*e^(-2*(b*x + a)/(d*x + c)) + 2*b^2*c^2*d - 4*a*b*c*d^2 + 2* a^2*d^3)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d *x + c))
Timed out. \[ \int \cosh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\mathrm {cosh}\left (\frac {a+b\,x}{c+d\,x}\right )}^2 \,d x \]