Integrand size = 23, antiderivative size = 191 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b^2 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b^2 d}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b^2 d} \]
1/4*e^2*Chi((a+b*arccosh(d*x+c))/b)*cosh(a/b)/b^2/d+3/4*e^2*Chi(3*(a+b*arc cosh(d*x+c))/b)*cosh(3*a/b)/b^2/d-1/4*e^2*Shi((a+b*arccosh(d*x+c))/b)*sinh (a/b)/b^2/d-3/4*e^2*Shi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^2/d-e^2*(d *x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))
Time = 1.65 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {e^2 \left (-\frac {4 b (c+d x)^2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \text {arccosh}(c+d x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{4 b^2 d} \]
(e^2*((-4*b*(c + d*x)^2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))/ (a + b*ArcCosh[c + d*x]) + Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c + d*x])] - Sinh[a/b]*Sin hIntegral[a/b + ArcCosh[c + d*x]] - 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])]))/(4*b^2*d)
Time = 0.46 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6411, 27, 6300, 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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e^2 \left (-\frac {\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b (a+b \text {arccosh}(c+d x))}\right )}{d}\) |
(e^2*(-((Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*(a + b*ArcCo sh[c + d*x]))) - (-1/4*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b] ) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/4 + (Si nh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/4 + (3*Sinh[(3*a)/b]*Sin hIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/4)/b^2))/d
3.2.38.3.1 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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(179)=358\).
Time = 0.61 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.96
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\) | \(374\) |
| default | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\) | \(374\) |
1/d*(1/8*(-4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+(d*x+c-1)^(1/2)*(d* x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*e^2/b/(a+b*arccosh(d*x+c))-3/8*e^2/b^2 *exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/8*(-(d*x+c-1)^(1/2)*(d*x+c+1)^( 1/2)+d*x+c)*e^2/b/(a+b*arccosh(d*x+c))-1/8*e^2/b^2*exp(a/b)*Ei(1,arccosh(d *x+c)+a/b)-1/8/b*e^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh( d*x+c))-1/8/b^2*e^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/8/b*e^2*(4*(d*x+ c)^3-3*d*x-3*c+4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2) *(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-3/8/b^2*e^2*exp(-3*a/b)*Ei(1,-3*arc cosh(d*x+c)-3*a/b))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^2*arccosh(d*x + c)^2 + 2 *a*b*arccosh(d*x + c) + a^2), x)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{2} + 2 a b \operatorname {acosh}{\left (c + d x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**2/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2) , x) + Integral(d**2*x**2/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d* x)**2), x) + Integral(2*c*d*x/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2), x))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
-(d^5*e^2*x^5 + 5*c*d^4*e^2*x^4 + c^5*e^2 - c^3*e^2 + (10*c^2*d^3*e^2 - d^ 3*e^2)*x^3 + (10*c^3*d^2*e^2 - 3*c*d^2*e^2)*x^2 + (d^4*e^2*x^4 + 4*c*d^3*e ^2*x^3 + c^4*e^2 - c^2*e^2 + (6*c^2*d^2*e^2 - d^2*e^2)*x^2 + 2*(2*c^3*d*e^ 2 - c*d*e^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (5*c^4*d*e^2 - 3*c^2 *d*e^2)*x)/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d - d)*a*b + (a*b*d^2*x + a *b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + integrate((3*d ^6*e^2*x^6 + 18*c*d^5*e^2*x^5 + 3*c^6*e^2 - 6*c^4*e^2 + 3*(15*c^2*d^4*e^2 - 2*d^4*e^2)*x^4 + 3*c^2*e^2 + 12*(5*c^3*d^3*e^2 - 2*c*d^3*e^2)*x^3 + (3*d ^4*e^2*x^4 + 12*c*d^3*e^2*x^3 + 3*c^4*e^2 - c^2*e^2 + (18*c^2*d^2*e^2 - d^ 2*e^2)*x^2 + 2*(6*c^3*d*e^2 - c*d*e^2)*x)*(d*x + c + 1)*(d*x + c - 1) + 3* (15*c^4*d^2*e^2 - 12*c^2*d^2*e^2 + d^2*e^2)*x^2 + (6*d^5*e^2*x^5 + 30*c*d^ 4*e^2*x^4 + 6*c^5*e^2 - 7*c^3*e^2 + (60*c^2*d^3*e^2 - 7*d^3*e^2)*x^3 + 2*c *e^2 + 3*(20*c^3*d^2*e^2 - 7*c*d^2*e^2)*x^2 + (30*c^4*d*e^2 - 21*c^2*d*e^2 + 2*d*e^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 6*(3*c^5*d*e^2 - 4*c^ 3*d*e^2 + c*d*e^2)*x)/(a*b*d^4*x^4 + 4*a*b*c*d^3*x^3 + 2*(3*c^2*d^2 - d^2) *a*b*x^2 + 4*(c^3*d - c*d)*a*b*x + (a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*( d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*a*b + 2*(a*b*d^3*x^3 + 3*a* b*c*d^2*x^2 + (3*c^2*d - d)*a*b*x + (c^3 - c)*a*b)*sqrt(d*x + c + 1)*sq...
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]