\(\int \frac {1}{\text {arcsinh}(a+b x)^2} \, dx\) [87]

Optimal result

Integrand size = 8, antiderivative size = 38 \[ \int \frac {1}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {\sqrt {1+(a+b x)^2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Shi}(\text {arcsinh}(a+b x))}{b} \]

[Out]

Shi(arcsinh(b*x+a))/b-(1+(b*x+a)^2)^(1/2)/b/arcsinh(b*x+a)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5773, 5819, 3379} \[ \int \frac {1}{\text {arcsinh}(a+b x)^2} \, dx=\frac {\text {Shi}(\text {arcsinh}(a+b x))}{b}-\frac {\sqrt {(a+b x)^2+1}}{b \text {arcsinh}(a+b x)} \]

[In]

Int[ArcSinh[a + b*x]^(-2),x]

[Out]

-(Sqrt[1 + (a + b*x)^2]/(b*ArcSinh[a + b*x])) + SinhIntegral[ArcSinh[a + b*x]]/b

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 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}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\sqrt {1+(a+b x)^2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\sqrt {1+(a+b x)^2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {\sqrt {1+(a+b x)^2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Shi}(\text {arcsinh}(a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\text {arcsinh}(a+b x)^2} \, dx=\frac {-\frac {\sqrt {1+(a+b x)^2}}{\text {arcsinh}(a+b x)}+\text {Shi}(\text {arcsinh}(a+b x))}{b} \]

[In]

Integrate[ArcSinh[a + b*x]^(-2),x]

[Out]

(-(Sqrt[1 + (a + b*x)^2]/ArcSinh[a + b*x]) + SinhIntegral[ArcSinh[a + b*x]])/b

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {1+\left (b x +a \right )^{2}}}{\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b}\)	\(34\)
default	\(\frac {-\frac {\sqrt {1+\left (b x +a \right )^{2}}}{\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b}\)	\(34\)

[In]

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

[Out]

1/b*(-1/arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)+Shi(arcsinh(b*x+a)))

Fricas [F]

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

[In]

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

[Out]

integral(arcsinh(b*x + a)^(-2), x)

Sympy [F]

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

[In]

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

[Out]

Integral(asinh(a + b*x)**(-2), x)

Maxima [F]

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

[In]

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

[Out]

-(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a)/((b^3*x^2 + 2*a*b^2
*x + a^2*b + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x + a*b) + b)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1))) + integrate((b^4*x^4 + 4*a*b^3*x^3 + a^4 + 2*(3*a^2*b^2 + b^2)*x^2 + (b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*
x^2 + 2*a*b*x + a^2 - 1) + 2*a^2 + 4*(a^3*b + a*b)*x + (2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2*b + b)*x + a)
*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1)/((b^4*x^4 + 4*a*b^3*x^3 + a^4 + 2*(3*a^2*b^2 + b^2)*x^2 + (b^2*x^2 + 2
*a*b*x + a^2 + 1)*(b^2*x^2 + 2*a*b*x + a^2) + 2*a^2 + 4*(a^3*b + a*b)*x + 2*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*
a^2*b + b)*x + a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))), x)

Giac [F]

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

[In]

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

[Out]

integrate(arcsinh(b*x + a)^(-2), x)

Mupad [F(-1)]

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

[In]

int(1/asinh(a + b*x)^2,x)

[Out]

int(1/asinh(a + b*x)^2, x)