3.116 \(\int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx\)

Optimal. Leaf size=13 \[ \frac {\sinh ^{-1}(a x)^2}{2 a} \]

[Out]

1/2*arcsinh(a*x)^2/a

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5675} \[ \frac {\sinh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/Sqrt[1 + a^2*x^2],x]

[Out]

ArcSinh[a*x]^2/(2*a)

Rule 5675

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\sinh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[a*x]^2/(2*a)

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fricas [B]  time = 0.66, size = 23, normalized size = 1.77 \[ \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \frac {\arcsinh \left (a x \right )^{2}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsinh(a*x)^2/a

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maxima [A]  time = 0.35, size = 11, normalized size = 0.85 \[ \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsinh(a*x)^2/a

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mupad [B]  time = 0.11, size = 11, normalized size = 0.85 \[ \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)/(a^2*x^2 + 1)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.34, size = 10, normalized size = 0.77 \[ \begin {cases} \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{2 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((asinh(a*x)**2/(2*a), Ne(a, 0)), (0, True))

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