\(\int x^{-1+n} \text {arcsinh}(a+b x^n) \, dx\) [367]

Optimal result

Integrand size = 14, antiderivative size = 46 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=-\frac {\sqrt {1+\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \text {arcsinh}\left (a+b x^n\right )}{b n} \]

[Out]

(a+b*x^n)*arcsinh(a+b*x^n)/b/n-(1+(a+b*x^n)^2)^(1/2)/b/n

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6847, 5858, 5772, 267} \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {arcsinh}\left (a+b x^n\right )}{b n}-\frac {\sqrt {\left (a+b x^n\right )^2+1}}{b n} \]

[In]

Int[x^(-1 + n)*ArcSinh[a + b*x^n],x]

[Out]

-(Sqrt[1 + (a + b*x^n)^2]/(b*n)) + ((a + b*x^n)*ArcSinh[a + b*x^n])/(b*n)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5772

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

Rule 5858

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {arcsinh}(a+b x) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \text {arcsinh}(x) \, dx,x,a+b x^n\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \text {arcsinh}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,a+b x^n\right )}{b n} \\ & = -\frac {\sqrt {1+\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \text {arcsinh}\left (a+b x^n\right )}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\frac {-\sqrt {1+\left (a+b x^n\right )^2}+\left (a+b x^n\right ) \text {arcsinh}\left (a+b x^n\right )}{b n} \]

[In]

Integrate[x^(-1 + n)*ArcSinh[a + b*x^n],x]

[Out]

(-Sqrt[1 + (a + b*x^n)^2] + (a + b*x^n)*ArcSinh[a + b*x^n])/(b*n)

Maple [F]

\[\int x^{n -1} \operatorname {arcsinh}\left (a +b \,x^{n}\right )d x\]

[In]

int(x^(n-1)*arcsinh(a+b*x^n),x)

[Out]

int(x^(n-1)*arcsinh(a+b*x^n),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (44) = 88\).

Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 3.30 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\frac {{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right ) - \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}}{b n} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).

Time = 9.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\begin {cases} \log {\left (x \right )} \operatorname {asinh}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x x^{n - 1} \operatorname {asinh}{\left (a \right )}}{n} & \text {for}\: b = 0 \\\log {\left (x \right )} \operatorname {asinh}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {a \operatorname {asinh}{\left (a + b x^{n} \right )}}{b n} + \frac {x^{n} \operatorname {asinh}{\left (a + b x^{n} \right )}}{n} - \frac {\sqrt {a^{2} + 2 a b x^{n} + b^{2} x^{2 n} + 1}}{b n} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+n)*asinh(a+b*x**n),x)

[Out]

Piecewise((log(x)*asinh(a), Eq(b, 0) & Eq(n, 0)), (x*x**(n - 1)*asinh(a)/n, Eq(b, 0)), (log(x)*asinh(a + b), E
q(n, 0)), (a*asinh(a + b*x**n)/(b*n) + x**n*asinh(a + b*x**n)/n - sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n) + 1)/
(b*n), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\frac {{\left (b x^{n} + a\right )} \operatorname {arsinh}\left (b x^{n} + a\right ) - \sqrt {{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \]

[In]

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

[Out]

((b*x^n + a)*arcsinh(b*x^n + a) - sqrt((b*x^n + a)^2 + 1))/(b*n)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (44) = 88\).

Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.46 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=-\frac {b {\left (\frac {a \log \left (-a b - {\left (x^{n} {\left | b \right |} - \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 1}\right )} {\left | b \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 1}}{b^{2}}\right )} - x^{n} \log \left (b x^{n} + a + \sqrt {{\left (b x^{n} + a\right )}^{2} + 1}\right )}{n} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.15 \[ \int x^{-1+n} \text {arcsinh}\left (a+b x^n\right ) \, dx=\frac {x^n\,\mathrm {asinh}\left (a+b\,x^n\right )}{n}-\frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1}}{b\,n}+\frac {a\,\ln \left (\frac {a\,b+b^2\,x^n}{\sqrt {b^2}}+\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1}\right )}{n\,\sqrt {b^2}} \]

[In]

int(x^(n - 1)*asinh(a + b*x^n),x)

[Out]

(x^n*asinh(a + b*x^n))/n - (a^2 + b^2*x^(2*n) + 2*a*b*x^n + 1)^(1/2)/(b*n) + (a*log((a*b + b^2*x^n)/(b^2)^(1/2
) + (a^2 + b^2*x^(2*n) + 2*a*b*x^n + 1)^(1/2)))/(n*(b^2)^(1/2))