\(\int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx\) [166]

Optimal result

Integrand size = 12, antiderivative size = 91 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {\sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5773, 5819, 3384, 3379, 3382} \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sqrt {(c+d x)^2+1}}{b d (a+b \text {arcsinh}(c+d x))} \]

[In]

Int[(a + b*ArcSinh[c + d*x])^(-2),x]

[Out]

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

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 5773

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1
)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rule 5858

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {\sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {\sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {\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 )}{b^2 d} \\ & = -\frac {\sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^2 d} \\ & = -\frac {\sqrt {1+(c+d x)^2}}{b d (a+b \text {arcsinh}(c+d x))}-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\frac {-\frac {b \sqrt {1+(c+d x)^2}}{a+b \text {arcsinh}(c+d x)}-\text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{b^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {\frac {-\sqrt {1+\left (d x +c \right )^{2}}+d x +c}{2 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {1+\left (d x +c \right )^{2}}}{2 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\)	\(128\)
default	\(\frac {\frac {-\sqrt {1+\left (d x +c \right )^{2}}+d x +c}{2 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {1+\left (d x +c \right )^{2}}}{2 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\)	\(128\)

[In]

int(1/(a+b*arcsinh(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2), x)

Sympy [F]

\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(a+b*asinh(d*x+c))**2,x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")

[Out]

-(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c)/(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((d^4*x^4 + 4*c*d^3*x^3 + c^4 + 2*(3*c^2*d^2 + d^2)*x^2 + (d^2*x^2 + 2*c*d*
x + c^2 + 1)*(d^2*x^2 + 2*c*d*x + c^2 - 1) + 2*c^2 + 4*(c^3*d + c*d)*x + (2*d^3*x^3 + 6*c*d^2*x^2 + 2*c^3 + (6
*c^2*d + d)*x + c)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 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]

\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int(1/(a + b*asinh(c + d*x))^2,x)

[Out]

int(1/(a + b*asinh(c + d*x))^2, x)