3.1.19 \(\int \frac {\sinh ^{-1}(a x)^2}{x^3} \, dx\) [19]

Optimal. Leaf size=43 \[ -\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5776, 5800, 29} \begin {gather*} -\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sinh ^{-1}(a x)^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^2/x^3,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x^3} \, dx &=-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a \int \frac {\sinh ^{-1}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^2/x^3,x]

[Out]

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

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Maple [A]
time = 3.17, size = 72, normalized size = 1.67

method	result	size
derivativedivides	\(a^{2} \left (-2 \arcsinh \left (a x \right )-\frac {\arcsinh \left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\)	\(72\)
default	\(a^{2} \left (-2 \arcsinh \left (a x \right )-\frac {\arcsinh \left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 39, normalized size = 0.91 \begin {gather*} a^{2} \log \left (x\right ) - \frac {\sqrt {a^{2} x^{2} + 1} a \operatorname {arsinh}\left (a x\right )}{x} - \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 67, normalized size = 1.56 \begin {gather*} \frac {2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(2*a^2*x^2*log(x) - 2*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1)) - log(a*x + sqrt(a^2*x^2 + 1))^2)
/x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**2/x**3,x)

[Out]

Integral(asinh(a*x)**2/x**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (39) = 78\).
time = 0.46, size = 98, normalized size = 2.28 \begin {gather*} -{\left (a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) - a \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="giac")

[Out]

-(a*log(-x*abs(a) + sqrt(a^2*x^2 + 1)) - a*log(abs(x)) - 2*abs(a)*log(a*x + sqrt(a^2*x^2 + 1))/((x*abs(a) - sq
rt(a^2*x^2 + 1))^2 - 1))*a - 1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/x^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^2/x^3,x)

[Out]

int(asinh(a*x)^2/x^3, x)

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