\(\int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx\) [132]

Optimal result

Integrand size = 23, antiderivative size = 141 \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {e^2 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b d}-\frac {e^2 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d} \]

[Out]

1/4*e^2*cosh(a/b)*Shi((a+b*arccosh(d*x+c))/b)/b/d+1/4*e^2*cosh(3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b/d-1/4*e^
2*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b/d-1/4*e^2*Chi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5996, 12, 5887, 5556, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b d}-\frac {e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d} \]

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x]),x]

[Out]

-1/4*(e^2*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(b*d) - (e^2*CoshIntegral[(3*(a + b*ArcCosh[c +
d*x]))/b]*Sinh[(3*a)/b])/(4*b*d) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b*d) + (e^2*Cos
h[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 5556

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]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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]

Rule 5996

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^2 x^2}{a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{a+b \text {arccosh}(x)} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b d} \\ & = -\frac {e^2 \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arccosh}(c+d x)\right )}{b d} \\ & = -\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d}-\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d} \\ & = \frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d}+\frac {\left (e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d}-\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d}-\frac {\left (e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c+d x)\right )}{4 b d} \\ & = -\frac {e^2 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b d}-\frac {e^2 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 b d}+\frac {e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.72 \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\frac {e^2 \left (-\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{4 b d} \]

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x]),x]

[Out]

(e^2*(-(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b]) - CoshIntegral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/
b] + Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])]))
/(4*b*d)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b}}{d}\)	\(130\)
default	\(\frac {\frac {e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b}-\frac {e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b}}{d}\)	\(130\)

[In]

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/8*e^2/b*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/8*e^2/b*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/8*e^2/b*e
xp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/8*e^2/b*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))

Fricas [F]

\[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c)),x, algorithm="fricas")

[Out]

integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b*arccosh(d*x + c) + a), x)

Sympy [F]

\[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=e^{2} \left (\int \frac {c^{2}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**2/(a+b*acosh(d*x+c)),x)

[Out]

e**2*(Integral(c**2/(a + b*acosh(c + d*x)), x) + Integral(d**2*x**2/(a + b*acosh(c + d*x)), x) + Integral(2*c*
d*x/(a + b*acosh(c + d*x)), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c)),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a), x)

Giac [F]

\[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^2}{a+b \text {arccosh}(c+d x)} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]

[In]

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x)),x)

[Out]

int((c*e + d*e*x)^2/(a + b*acosh(c + d*x)), x)