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

Optimal. Leaf size=17 \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

[Out]

arcsinh(a*x)^(1+n)/a/(1+n)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 17, 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 = {5675} \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 83, normalized size = 4.88 \[ \frac {\cosh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cosh(n*log(log(a*x + sqrt(a^2*x^2 + 1))))*log(a*x + sqrt(a^2*x^2 + 1)) + log(a*x + sqrt(a^2*x^2 + 1))*sinh(n*
log(log(a*x + sqrt(a^2*x^2 + 1)))))/(a*n + a)

________________________________________________________________________________________

giac [A]  time = 0.88, size = 29, normalized size = 1.71 \[ \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{n + 1}}{a {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 18, normalized size = 1.06 \[ \frac {\arcsinh \left (a x \right )^{1+n}}{a \left (1+n \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(a*x)^(1+n)/a/(1+n)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 17, normalized size = 1.00 \[ \frac {\operatorname {arsinh}\left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(a*x)^(n + 1)/(a*(n + 1))

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 33, normalized size = 1.94 \[ \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asinh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asinh}\left (a\,x\right )}^{n+1}}{a\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(n == -1, log(asinh(a*x))/a, n ~= -1, asinh(a*x)^(n + 1)/(a*(n + 1)))

________________________________________________________________________________________

sympy [A]  time = 0.87, size = 34, normalized size = 2.00 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge n = -1 \\0^{n} x & \text {for}\: a = 0 \\\frac {\log {\left (\operatorname {asinh}{\left (a x \right )} \right )}}{a} & \text {for}\: n = -1 \\\frac {\operatorname {asinh}{\left (a x \right )} \operatorname {asinh}^{n}{\left (a x \right )}}{a n + a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(n, -1)), (0**n*x, Eq(a, 0)), (log(asinh(a*x))/a, Eq(n, -1)), (asinh(a*x)*asinh
(a*x)**n/(a*n + a), True))

________________________________________________________________________________________