3.1.57 \(\int \frac {1}{\sinh ^{-1}(a x)^2} \, dx\) [57]

Optimal. Leaf size=34 \[ -\frac {\sqrt {1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5773, 5819, 3379} \begin {gather*} \frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a}-\frac {\sqrt {a^2 x^2+1}}{a \sinh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^(-2),x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 31, normalized size = 0.91 \begin {gather*} \frac {-\frac {\sqrt {1+a^2 x^2}}{\sinh ^{-1}(a x)}+\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^(-2),x]

[Out]

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

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Maple [A]
time = 1.20, size = 30, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{\arcsinh \left (a x \right )}+\hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )}{a}\)	\(30\)
default	\(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{\arcsinh \left (a x \right )}+\hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )}{a}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsinh(a*x)^(-2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a*x)**(-2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^(-2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/asinh(a*x)^2,x)

[Out]

int(1/asinh(a*x)^2, x)

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