\(\int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx\) [531]

Optimal result

Integrand size = 20, antiderivative size = 17 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5783} \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{n+1}}{a (n+1)} \]

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

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*} \text {integral}& = \frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^{1+n}}{a (1+n)} \]

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\operatorname {arcsinh}\left (a x \right )^{1+n}}{a \left (1+n \right )}\)	\(18\)
default	\(\frac {\operatorname {arcsinh}\left (a x \right )^{1+n}}{a \left (1+n \right )}\)	\(18\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\cosh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).

Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge n = -1 \\0^{n} x & \text {for}\: a = 0 \\\frac {\log {\left (\operatorname {asinh}{\left (a x \right )} \right )}}{a} & \text {for}\: n = -1 \\\frac {\operatorname {asinh}{\left (a x \right )} \operatorname {asinh}^{n}{\left (a x \right )}}{a n + a} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\operatorname {arsinh}\left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{n + 1}}{a {\left (n + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 2.93 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {\text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asinh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asinh}\left (a\,x\right )}^{n+1}}{a\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

[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)))