### 3.71 $$\int x^{-1+n} \text{csch}^{-1}(a+b x^n) \, dx$$

Optimal. Leaf size=46 $\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}\right )}{b n}+\frac{\left (a+b x^n\right ) \text{csch}^{-1}\left (a+b x^n\right )}{b n}$

[Out]

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

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Rubi [A]  time = 0.0702649, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {6715, 6314, 372, 266, 63, 207} $\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}\right )}{b n}+\frac{\left (a+b x^n\right ) \text{csch}^{-1}\left (a+b x^n\right )}{b n}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^(-1 + n)*ArcCsch[a + b*x^n],x]

[Out]

((a + b*x^n)*ArcCsch[a + b*x^n])/(b*n) + ArcTanh[Sqrt[1 + (a + b*x^n)^(-2)]]/(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 6314

Int[ArcCsch[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[((c + d*x)*ArcCsch[c + d*x])/d, x] + Int[1/((c + d*x)*Sqrt[1
+ 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.173554, size = 74, normalized size = 1.61 $\frac{\frac{\sqrt{\left (a+b x^n\right )^2+1} \sinh ^{-1}\left (a+b x^n\right )}{\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}}+\left (a+b x^n\right )^2 \text{csch}^{-1}\left (a+b x^n\right )}{b n \left (a+b x^n\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F]  time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arccsch} \left (a+b{x}^{n}\right )\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03147, size = 81, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )} \operatorname{arcsch}\left (b x^{n} + a\right ) + \log \left (\sqrt{\frac{1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{{\left (b x^{n} + a\right )}^{2}} + 1} - 1\right )}{2 \, b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.01241, size = 969, normalized size = 21.07 \begin{align*} \frac{a \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 )}} + 1\right ) - a \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 )}} - 1\right ) +{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac{\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 )}} + 1}{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 )}{b n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)))) + 1) - 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)
))) - 1) + (b*cosh(n*log(x)) + b*sinh(n*log(x)))*log((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)))) + 1)/(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)) - lo
g(-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)*si
nh(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)*acsch(a+b*x**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{n - 1} \operatorname{arcsch}\left (b x^{n} + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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