Integrand size = 23, antiderivative size = 213 \[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b d}-\frac {3 e^4 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b d}-\frac {e^4 \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 b d}+\frac {3 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 b d} \]
1/8*e^4*cosh(a/b)*Shi((a+b*arccosh(d*x+c))/b)/b/d+3/16*e^4*cosh(3*a/b)*Shi (3*(a+b*arccosh(d*x+c))/b)/b/d+1/16*e^4*cosh(5*a/b)*Shi(5*(a+b*arccosh(d*x +c))/b)/b/d-1/8*e^4*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b/d-3/16*e^4*Chi (3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b/d-1/16*e^4*Chi(5*(a+b*arccosh(d*x +c))/b)*sinh(5*a/b)/b/d
Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.71 \[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=\frac {e^4 \left (-2 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{16 b d} \]
(e^4*(-2*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 3*CoshIntegral[3 *(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcCosh[c + d*x])]*Sinh[(5*a)/b] + 2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])] + Cosh[(5*a)/b ]*SinhIntegral[5*(a/b + ArcCosh[c + d*x])]))/(16*b*d)
Time = 0.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.79, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6411, 27, 6302, 25, 5971, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{a+b \text {arccosh}(c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{a+b \text {arccosh}(c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle \frac {e^4 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e^4 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {e^4 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{16 (a+b \text {arccosh}(c+d x))}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{b d}\) |
(e^4*(-1/8*(CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b]) - (3*CoshI ntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/16 - (CoshIntegral[ (5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]*SinhIntegra l[(a + b*ArcCosh[c + d*x])/b])/8 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b *ArcCosh[c + d*x]))/b])/16 + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh [c + d*x]))/b])/16))/(b*d)
3.2.30.3.1 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[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 2.56 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b}+\frac {3 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b}+\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b}-\frac {3 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b}-\frac {e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b}}{d}\) | \(194\) |
| default | \(\frac {\frac {e^{4} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (d x +c \right )+\frac {5 a}{b}\right )}{32 b}+\frac {3 e^{4} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{32 b}+\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b}-\frac {3 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{32 b}-\frac {e^{4} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (d x +c \right )-\frac {5 a}{b}\right )}{32 b}}{d}\) | \(194\) |
1/d*(1/32*e^4/b*exp(5*a/b)*Ei(1,5*arccosh(d*x+c)+5*a/b)+3/32*e^4/b*exp(3*a /b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/16*e^4/b*exp(a/b)*Ei(1,arccosh(d*x+c)+a /b)-1/16*e^4/b*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-3/32*e^4/b*exp(-3*a/b)* Ei(1,-3*arccosh(d*x+c)-3*a/b)-1/32*e^4/b*exp(-5*a/b)*Ei(1,-5*arccosh(d*x+c )-5*a/b))
\[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
integral((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4* x + c^4*e^4)/(b*arccosh(d*x + c) + a), x)
\[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=e^{4} \left (\int \frac {c^{4}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(a + b*acosh(c + d*x)), x) + Integral(d**4*x**4/(a + b *acosh(c + d*x)), x) + Integral(4*c*d**3*x**3/(a + b*acosh(c + d*x)), x) + Integral(6*c**2*d**2*x**2/(a + b*acosh(c + d*x)), x) + Integral(4*c**3*d* x/(a + b*acosh(c + d*x)), x))
\[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{a+b \text {arccosh}(c+d x)} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]