\(\int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^2} \, dx\) [164]

Optimal result

Integrand size = 23, antiderivative size = 184 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 d}-\frac {3 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 b^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5859, 12, 5778, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 b^2 d}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 b^2 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 b^2 d}-\frac {e^2 (c+d x)^2 \sqrt {(c+d x)^2+1}}{b d (a+b \text {arcsinh}(c+d x))} \]

[In]

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

[Out]

-((e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) + (e^2*CoshIntegral[(a + b*ArcSinh[c
+ d*x])/b]*Sinh[a/b])/(4*b^2*d) - (3*e^2*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/(4*b^2*d
) - (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*
(a + b*ArcSinh[c + d*x]))/b])/(4*b^2*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 5778

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5859

Int[((a_.) + ArcSinh[(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*ArcSinh[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 {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Subst}\left (\int \left (-\frac {3 \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 {arcsinh}(c+d x)\right )}{b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 b^2 d}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{4 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\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 {arcsinh}(c+d x)\right )}{4 b^2 d}+\frac {\left (3 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 {arcsinh}(c+d x)\right )}{4 b^2 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 {arcsinh}(c+d x)\right )}{4 b^2 d}-\frac {\left (3 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 {arcsinh}(c+d x)\right )}{4 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 d}-\frac {3 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 b^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {e^2 \left (-\frac {4 b (c+d x)^2 \sqrt {1+(c+d x)^2}}{a+b \text {arcsinh}(c+d x)}+\text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-3 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )}{4 b^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.86

method	result	size
derivativedivides	\(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\)	\(342\)
default	\(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2}}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\)	\(342\)

[In]

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

[Out]

1/d*(1/8*(-4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x+c)^2)^(1/2)+3*d*x+3*c)*e^2/b/(a+b*arcsinh(d*x+c
))+3/8*e^2/b^2*exp(3*a/b)*Ei(1,3*arcsinh(d*x+c)+3*a/b)-1/8*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^2/b/(a+b*arcsinh(d*x
+c))-1/8*e^2/b^2*exp(a/b)*Ei(1,arcsinh(d*x+c)+a/b)+1/8/b*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+
1/8/b^2*e^2*exp(-a/b)*Ei(1,-arcsinh(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*(1+(d*x+c)^2)^(1/
2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-3/8/b^2*e^2*exp(-3*a/b)*Ei(1,-3*arcsinh(d*x+c)-3*a/b))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-(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 + (5*c^4*d*e^2 + 3*c^2*d*e^2)*x + (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^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b*d^3*x^2 + 2*a*b*c*d^
2*x + (c^2*d + d)*a*b + (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^2*x^2 +
2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + (a*b*d^2*x + a*b*c*d)*sqrt(d^2*x^2 + 2*
c*d*x + c^2 + 1)) + 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*(15*c^4*d^2*e^2 + 12*c^2*d^2*e^2 + d^2*e
^2)*x^2 + (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^2*x^2 + 2*c*d*x + c^2 + 1) + 6*(3*c^5*d*e^2 + 4*c^3*d*e^2 + c*d*e^2)*x + (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^2*x^2 + 2*c*d*x + c^2 + 1))/(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 + (c^4 + 2*c^2 + 1)*a*b + (a*b*d^2*x^
2 + 2*a*b*c*d*x + a*b*c^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 2*(3*c^2*d^2 + d^2
)*b^2*x^2 + 4*(c^3*d + c*d)*b^2*x + (c^4 + 2*c^2 + 1)*b^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*(d^2*x^2 + 2
*c*d*x + c^2 + 1) + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d + d)*b^2*x + (c^3 + c)*b^2)*sqrt(d^2*x^2 + 2*c
*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 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^2*x^2 + 2*c*d*x + c^2 + 1)), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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