3.4.46 \(\int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\) [346]

Optimal. Leaf size=13 \[ \frac {\sinh ^{-1}(a x)^4}{4 a} \]

[Out]

1/4*arcsinh(a*x)^4/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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5783} \begin {gather*} \frac {\sinh ^{-1}(a x)^4}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/Sqrt[1 + a^2*x^2],x]

[Out]

ArcSinh[a*x]^4/(4*a)

Rule 5783

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

Rubi steps

\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sinh ^{-1}(a x)^4}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sinh ^{-1}(a x)^4}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^3/Sqrt[1 + a^2*x^2],x]

[Out]

ArcSinh[a*x]^4/(4*a)

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Maple [A]
time = 0.29, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\arcsinh \left (a x \right )^{4}}{4 a}\)	\(12\)
default	\(\frac {\arcsinh \left (a x \right )^{4}}{4 a}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*arcsinh(a*x)^4/a

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.85 \begin {gather*} \frac {\operatorname {arsinh}\left (a x\right )^{4}}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsinh(a*x)^4/a

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
time = 0.37, size = 23, normalized size = 1.77 \begin {gather*} \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4}}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/4*log(a*x + sqrt(a^2*x^2 + 1))^4/a

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Sympy [A]
time = 0.25, size = 10, normalized size = 0.77 \begin {gather*} \begin {cases} \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{4 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((asinh(a*x)**4/(4*a), Ne(a, 0)), (0, True))

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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(arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/sqrt(a^2*x^2 + 1), x)

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Mupad [B]
time = 0.10, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{4\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^3/(a^2*x^2 + 1)^(1/2),x)

[Out]

asinh(a*x)^4/(4*a)

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