Integrand size = 16, antiderivative size = 194 \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {3 (b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}+\frac {3 (b c-a d) \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2} \]
-3/4*(-a*d+b*c)*Chi((-a*d+b*c)/d/(d*x+c))*cosh(b/d)/d^2+3/4*(-a*d+b*c)*Chi (3*(-a*d+b*c)/d/(d*x+c))*cosh(3*b/d)/d^2+3/4*(-a*d+b*c)*Shi((-a*d+b*c)/d/( d*x+c))*sinh(b/d)/d^2-3/4*(-a*d+b*c)*Shi(3*(-a*d+b*c)/d/(d*x+c))*sinh(3*b/ d)/d^2+(d*x+c)*sinh((b*x+a)/(d*x+c))^3/d
Leaf count is larger than twice the leaf count of optimal. \(651\) vs. \(2(194)=388\).
Time = 5.82 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.36 \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {3 (a+b x)}{c+d x}}+3 c d e^{-\frac {a+b x}{c+d x}}-3 c d e^{\frac {a+b x}{c+d x}}+c d e^{\frac {3 (a+b x)}{c+d x}}-6 d^2 x \cosh \left (\frac {-b c+a d}{d (c+d x)}\right ) \sinh \left (\frac {b}{d}\right )+2 d^2 x \cosh \left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \sinh \left (\frac {3 b}{d}\right )-6 d^2 x \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {-b c+a d}{d (c+d x)}\right )+2 d^2 x \cosh \left (\frac {3 b}{d}\right ) \sinh \left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+3 (b c-a d) \left (\cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )-\cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right )+\text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \sinh \left (\frac {b}{d}\right )-\text {Chi}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cosh \left (\frac {b}{d}\right )+\sinh \left (\frac {b}{d}\right )\right )-\text {Chi}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right ) \sinh \left (\frac {3 b}{d}\right )+\text {Chi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \left (\cosh \left (\frac {3 b}{d}\right )+\sinh \left (\frac {3 b}{d}\right )\right )-\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )-\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\cosh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+\sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+\cosh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )-\sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )-\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )+\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{8 d^2} \]
(-((c*d)/E^((3*(a + b*x))/(c + d*x))) + (3*c*d)/E^((a + b*x)/(c + d*x)) - 3*c*d*E^((a + b*x)/(c + d*x)) + c*d*E^((3*(a + b*x))/(c + d*x)) - 6*d^2*x* Cosh[(-(b*c) + a*d)/(d*(c + d*x))]*Sinh[b/d] + 2*d^2*x*Cosh[(3*(-(b*c) + a *d))/(d*(c + d*x))]*Sinh[(3*b)/d] - 6*d^2*x*Cosh[b/d]*Sinh[(-(b*c) + a*d)/ (d*(c + d*x))] + 2*d^2*x*Cosh[(3*b)/d]*Sinh[(3*(-(b*c) + a*d))/(d*(c + d*x ))] + 3*(b*c - a*d)*(Cosh[(3*b)/d]*CoshIntegral[(3*b*c - 3*a*d)/(c*d + d^2 *x)] - Cosh[b/d]*CoshIntegral[(b*c - a*d)/(c*d + d^2*x)] + CoshIntegral[(b *c - a*d)/(c*d + d^2*x)]*Sinh[b/d] - CoshIntegral[(-(b*c) + a*d)/(d*(c + d *x))]*(Cosh[b/d] + Sinh[b/d]) - CoshIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x) ]*Sinh[(3*b)/d] + CoshIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))]*(Cosh[(3* b)/d] + Sinh[(3*b)/d]) - Cosh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x ))] - Sinh[b/d]*SinhIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + Cosh[(3*b)/d] *SinhIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] + Sinh[(3*b)/d]*SinhIntegr al[(3*(-(b*c) + a*d))/(d*(c + d*x))] + Cosh[(3*b)/d]*SinhIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)] - Sinh[(3*b)/d]*SinhIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)] - Cosh[b/d]*SinhIntegral[(b*c - a*d)/(c*d + d^2*x)] + Sinh[b/d]*S inhIntegral[(b*c - a*d)/(c*d + d^2*x)]))/(8*d^2)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6141, 3042, 26, 3794, 2009}
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.
| \(\displaystyle \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 6141 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sinh ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int i (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i (b c-a d)}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (c+d x)^2 \sin \left (\frac {i b}{d}-\frac {i (b c-a d)}{d (c+d x)}\right )^3d\frac {1}{c+d x}}{d}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {i \left (i (c+d x) \sinh ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )-\frac {3 i (b c-a d) \int \left (\frac {1}{4} (c+d x) \cosh \left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )-\frac {1}{4} (c+d x) \cosh \left (\frac {3 b}{d}-\frac {3 (b c-a d)}{d (c+d x)}\right )\right )d\frac {1}{c+d x}}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \left (i (c+d x) \sinh ^3\left (\frac {b}{d}-\frac {b c-a d}{d (c+d x)}\right )-\frac {3 i (b c-a d) \left (\frac {1}{4} \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )-\frac {1}{4} \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )-\frac {1}{4} \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )+\frac {1}{4} \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )\right )}{d}\right )}{d}\) |
((-I)*(I*(c + d*x)*Sinh[b/d - (b*c - a*d)/(d*(c + d*x))]^3 - ((3*I)*(b*c - a*d)*((Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/4 - (Cosh[(3*b) /d]*CoshIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/4 - (Sinh[b/d]*SinhIntegr al[(b*c - a*d)/(d*(c + d*x))])/4 + (Sinh[(3*b)/d]*SinhIntegral[(3*(b*c - a *d))/(d*(c + d*x))])/4))/d))/d
3.3.97.3.1 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[((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[Sinh[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol ] :> Simp[-d^(-1) Subst[Int[Sinh[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. \(699\) vs. \(2(186)=372\).
Time = 1.89 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.61
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{-\frac {3 \left (b x +a \right )}{d x +c}} a}{8 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {3 \left (b x +a \right )}{d x +c}} b c}{8 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \operatorname {Ei}_{1}\left (\frac {3 a d -3 b c}{\left (d x +c \right ) d}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {3 b}{d}} \operatorname {Ei}_{1}\left (\frac {3 a d -3 b c}{\left (d x +c \right ) d}\right ) b c}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{8 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b x +a}{d x +c}} b c}{8 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {3 \,{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {d \,{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} x a}{8 a d -8 b c}-\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} x b c}{8 \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} c a}{8 a d -8 b c}-\frac {{\mathrm e}^{\frac {3 b x +3 a}{d x +c}} c^{2} b}{8 d \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{\frac {3 b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{8 \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} x b c}{8 \left (a d -b c \right )}-\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{8 \left (a d -b c \right )}+\frac {3 \,{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{8 d \left (a d -b c \right )}-\frac {3 \,{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}\) | \(700\) |
-1/8*exp(-3*(b*x+a)/(d*x+c))/(d/(d*x+c)*a-b*c/(d*x+c))*a+1/8/d*exp(-3*(b*x +a)/(d*x+c))/(d/(d*x+c)*a-b*c/(d*x+c))*b*c+3/8/d*exp(-3*b/d)*Ei(1,3*(a*d-b *c)/d/(d*x+c))*a-3/8/d^2*exp(-3*b/d)*Ei(1,3*(a*d-b*c)/d/(d*x+c))*b*c+3/8*e xp(-(b*x+a)/(d*x+c))/(d/(d*x+c)*a-b*c/(d*x+c))*a-3/8/d*exp(-(b*x+a)/(d*x+c ))/(d/(d*x+c)*a-b*c/(d*x+c))*b*c-3/8/d*exp(-b/d)*Ei(1,(a*d-b*c)/d/(d*x+c)) *a+3/8/d^2*exp(-b/d)*Ei(1,(a*d-b*c)/d/(d*x+c))*b*c+1/8*d*exp(3*(b*x+a)/(d* x+c))/(a*d-b*c)*x*a-1/8*exp(3*(b*x+a)/(d*x+c))/(a*d-b*c)*x*b*c+1/8*exp(3*( b*x+a)/(d*x+c))/(a*d-b*c)*c*a-1/8/d*exp(3*(b*x+a)/(d*x+c))/(a*d-b*c)*c^2*b +3/8/d*exp(3*b/d)*Ei(1,-3*(a*d-b*c)/d/(d*x+c))*a-3/8/d^2*exp(3*b/d)*Ei(1,- 3*(a*d-b*c)/d/(d*x+c))*b*c-3/8*d*exp((b*x+a)/(d*x+c))/(a*d-b*c)*x*a+3/8*ex p((b*x+a)/(d*x+c))/(a*d-b*c)*x*b*c-3/8*exp((b*x+a)/(d*x+c))/(a*d-b*c)*c*a+ 3/8/d*exp((b*x+a)/(d*x+c))/(a*d-b*c)*c^2*b-3/8/d*exp(b/d)*Ei(1,-(a*d-b*c)/ d/(d*x+c))*a+3/8/d^2*exp(b/d)*Ei(1,-(a*d-b*c)/d/(d*x+c))*b*c
Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (186) = 372\).
Time = 0.28 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.70 \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {6 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \cosh \left (\frac {3 \, b}{d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} - 3 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {3 \, b}{d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{3} - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{4} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {3 \, b}{d}\right ) + 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) + 6 \, {\left (d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{4} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{8 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{4} - 2 \, d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{4}\right )}} \]
-1/8*(6*(b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2*cosh(3*b/d)*sinh((b*x + a)/(d*x + c))^2 - 3*(b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh(3*b/d)*sinh((b*x + a)/(d*x + c))^4 - 2*(d^2*x + c*d)*sinh((b*x + a)/(d*x + c))^3 - 3*((b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2* x + c*d))*cosh((b*x + a)/(d*x + c))^4 + (b*c - a*d)*Ei(3*(b*c - a*d)/(d^2* x + c*d)))*cosh(3*b/d) + 3*((b*c - a*d)*Ei((b*c - a*d)/(d^2*x + c*d)) + (b *c - a*d)*Ei(-(b*c - a*d)/(d^2*x + c*d)))*cosh(b/d) + 6*(d^2*x - (d^2*x + c*d)*cosh((b*x + a)/(d*x + c))^2 + c*d)*sinh((b*x + a)/(d*x + c)) - 3*((b* c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^4 - 2* (b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))*cosh((b*x + a)/(d*x + c))^2*s inh((b*x + a)/(d*x + c))^2 + (b*c - a*d)*Ei(-3*(b*c - a*d)/(d^2*x + c*d))* sinh((b*x + a)/(d*x + c))^4 - (b*c - a*d)*Ei(3*(b*c - a*d)/(d^2*x + c*d))) *sinh(3*b/d) - 3*((b*c - a*d)*Ei((b*c - a*d)/(d^2*x + c*d)) - (b*c - a*d)* Ei(-(b*c - a*d)/(d^2*x + c*d)))*sinh(b/d))/(d^2*cosh((b*x + a)/(d*x + c))^ 4 - 2*d^2*cosh((b*x + a)/(d*x + c))^2*sinh((b*x + a)/(d*x + c))^2 + d^2*si nh((b*x + a)/(d*x + c))^4)
Timed out. \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Timed out} \]
\[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x + a}{d x + c}\right )^{3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1383 vs. \(2 (186) = 372\).
Time = 9.58 (sec) , antiderivative size = 1383, normalized size of antiderivative = 7.13 \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Too large to display} \]
1/8*(3*b^3*c^2*Ei(-3*(b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d) - 6*a*b^2*c* d*Ei(-3*(b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d) - 3*(b*x + a)*b^2*c^2*d*E i(-3*(b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d)/(d*x + c) + 3*a^2*b*d^2*Ei(- 3*(b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d) + 6*(b*x + a)*a*b*c*d^2*Ei(-3*( b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d)/(d*x + c) - 3*(b*x + a)*a^2*d^3*Ei (-3*(b - (b*x + a)*d/(d*x + c))/d)*e^(3*b/d)/(d*x + c) - 3*b^3*c^2*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d) + 6*a*b^2*c*d*Ei(-(b - (b*x + a)*d/(d* x + c))/d)*e^(b/d) + 3*(b*x + a)*b^2*c^2*d*Ei(-(b - (b*x + a)*d/(d*x + c)) /d)*e^(b/d)/(d*x + c) - 3*a^2*b*d^2*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^( b/d) - 6*(b*x + a)*a*b*c*d^2*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d)/(d *x + c) + 3*(b*x + a)*a^2*d^3*Ei(-(b - (b*x + a)*d/(d*x + c))/d)*e^(b/d)/( d*x + c) - 3*b^3*c^2*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d) + 6*a*b^2* c*d*Ei((b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d) + 3*(b*x + a)*b^2*c^2*d*Ei( (b - (b*x + a)*d/(d*x + c))/d)*e^(-b/d)/(d*x + c) - 3*a^2*b*d^2*Ei((b - (b *x + a)*d/(d*x + c))/d)*e^(-b/d) - 6*(b*x + a)*a*b*c*d^2*Ei((b - (b*x + a) *d/(d*x + c))/d)*e^(-b/d)/(d*x + c) + 3*(b*x + a)*a^2*d^3*Ei((b - (b*x + a )*d/(d*x + c))/d)*e^(-b/d)/(d*x + c) + 3*b^3*c^2*Ei(3*(b - (b*x + a)*d/(d* x + c))/d)*e^(-3*b/d) - 6*a*b^2*c*d*Ei(3*(b - (b*x + a)*d/(d*x + c))/d)*e^ (-3*b/d) - 3*(b*x + a)*b^2*c^2*d*Ei(3*(b - (b*x + a)*d/(d*x + c))/d)*e^(-3 *b/d)/(d*x + c) + 3*a^2*b*d^2*Ei(3*(b - (b*x + a)*d/(d*x + c))/d)*e^(-3...
Timed out. \[ \int \sinh ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\mathrm {sinh}\left (\frac {a+b\,x}{c+d\,x}\right )}^3 \,d x \]