3.4.85 \(\int \frac {1}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx\) [385]

Optimal. Leaf size=9 \[ \frac {\log \left (\sinh ^{-1}(a x)\right )}{a} \]

[Out]

ln(arcsinh(a*x))/a

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Rubi [A]
time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5782} \begin {gather*} \frac {\log \left (\sinh ^{-1}(a x)\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]

[Out]

Log[ArcSinh[a*x]]/a

Rule 5782

Int[1/(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(1/(b*c))*Simp[Sqrt[1
+ c^2*x^2]/Sqrt[d + e*x^2]]*Log[a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sinh ^{-1}(a x)\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]

[Out]

Log[ArcSinh[a*x]]/a

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Maple [A]
time = 0.28, size = 10, normalized size = 1.11

method	result	size
derivativedivides	\(\frac {\ln \left (\arcsinh \left (a x \right )\right )}{a}\)	\(10\)
default	\(\frac {\ln \left (\arcsinh \left (a x \right )\right )}{a}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(arcsinh(a*x))/a

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Maxima [A]
time = 0.27, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\operatorname {arsinh}\left (a x\right )\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arcsinh(a*x))/a

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
time = 0.34, size = 21, normalized size = 2.33 \begin {gather*} \frac {\log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(a*x + sqrt(a^2*x^2 + 1)))/a

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Sympy [A]
time = 0.24, size = 7, normalized size = 0.78 \begin {gather*} \frac {\log {\left (\operatorname {asinh}{\left (a x \right )} \right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(asinh(a*x))/a

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

[Out]

integrate(1/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)), x)

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Mupad [B]
time = 0.10, size = 9, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {asinh}\left (a\,x\right )\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(asinh(a*x))/a

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