3.1.46 \(\int \frac {x^2}{\sinh ^{-1}(a x)} \, dx\) [46]

Optimal. Leaf size=27 \[ -\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3} \]

[Out]

-1/4*Chi(arcsinh(a*x))/a^3+1/4*Chi(3*arcsinh(a*x))/a^3

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Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556, 3382} \begin {gather*} \frac {\text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}-\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSinh[a*x],x]

[Out]

-1/4*CoshIntegral[ArcSinh[a*x]]/a^3 + CoshIntegral[3*ArcSinh[a*x]]/(4*a^3)

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 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 5780

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

Rubi steps

\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} \frac {-\text {Chi}\left (\sinh ^{-1}(a x)\right )+\text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSinh[a*x],x]

[Out]

(-CoshIntegral[ArcSinh[a*x]] + CoshIntegral[3*ArcSinh[a*x]])/(4*a^3)

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Maple [A]
time = 1.35, size = 22, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {-\frac {\hyperbolicCosineIntegral \left (\arcsinh \left (a x \right )\right )}{4}+\frac {\hyperbolicCosineIntegral \left (3 \arcsinh \left (a x \right )\right )}{4}}{a^{3}}\)	\(22\)
default	\(\frac {-\frac {\hyperbolicCosineIntegral \left (\arcsinh \left (a x \right )\right )}{4}+\frac {\hyperbolicCosineIntegral \left (3 \arcsinh \left (a x \right )\right )}{4}}{a^{3}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/4*Chi(arcsinh(a*x))+1/4*Chi(3*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(x^2/arcsinh(a*x),x, algorithm="maxima")

[Out]

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

[Out]

integral(x^2/arcsinh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/asinh(a*x), 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(x^2/arcsinh(a*x),x, algorithm="giac")

[Out]

integrate(x^2/arcsinh(a*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2}{\mathrm {asinh}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/asinh(a*x), x)

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