3.367 \(\int x^{-1+n} \sinh ^{-1}(a+b x^n) \, dx\)

Optimal. Leaf size=46 \[ \frac{\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{\left (a+b x^n\right )^2+1}}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0504628, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6715, 5863, 5653, 261} \[ \frac{\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{\left (a+b x^n\right )^2+1}}{b n} \]

Antiderivative was successfully verified.

[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 6715

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]

Rule 5863

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 5653

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 261

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]

Rubi steps

\begin{align*} \int x^{-1+n} \sinh ^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )}{b n}-\frac{\operatorname{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 ) \sinh ^{-1}\left (a+b x^n\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0370183, size = 41, normalized size = 0.89 \[ \frac{\left (a+b x^n\right ) \sinh ^{-1}\left (a+b x^n\right )-\sqrt{\left (a+b x^n\right )^2+1}}{b n} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\it Arcsinh} \left ( a+b{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13509, size = 53, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.78078, size = 443, normalized size = 9.63 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.35281, size = 153, normalized size = 3.33 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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