\(\int x^{-1+n} \text {csch}^{-1}(a+b x^n) \, dx\) [28]

Optimal result

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

(a+b*x^n)*arccsch(a+b*x^n)/b/n+arctanh((1+1/(a+b*x^n)^2)^(1/2))/b/n
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(46)=92\).

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.02 \[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right )^2 \text {csch}^{-1}\left (a+b x^n\right )-\frac {\sqrt {1+\left (a+b x^n\right )^2} \log \left (-a-b x^n+\sqrt {1+\left (a+b x^n\right )^2}\right )}{\sqrt {1+\frac {1}{\left (a+b x^n\right )^2}}}}{b n \left (a+b x^n\right )} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {7266, 6868, 895, 798, 73, 220}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^{n-1} \text {csch}^{-1}\left (a+b x^n\right ) \, dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {\int \text {csch}^{-1}\left (b x^n+a\right )dx^n}{n}\)

\(\Big \downarrow \) 6868

\(\displaystyle \frac {\int \frac {1}{\left (b x^n+a\right ) \sqrt {1+\frac {1}{\left (b x^n+a\right )^2}}}dx^n+\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b}}{n}\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\frac {\int \frac {x^{-n}}{\sqrt {x^{-2 n}+1}}d\left (b x^n+a\right )}{b}+\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b}}{n}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b}-\frac {\int \frac {x^{-n}}{\sqrt {x^{-2 n}+1}}dx^{-2 n}}{2 b}}{n}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b}-\frac {\int \frac {1}{x^{2 n}-1}d\sqrt {x^{-2 n}+1}}{b}}{n}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b}+\frac {\text {arctanh}\left (\sqrt {x^{-2 n}+1}\right )}{b}}{n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 220

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

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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 895

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff 
icient[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 6868

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 7266

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

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (44) = 88\).

Time = 0.12 (sec) , antiderivative size = 334, normalized size of antiderivative = 7.26 \[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\frac {a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1\right ) - a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} - 1\right ) + {\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac {\sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a}\right ) - \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right )}{b n} \] Input:

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

Output:

(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)) - s 
inh(n*log(x)))) + 1) - a*log(-b*cosh(n*log(x)) - b*sinh(n*log(x)) - a + sq 
rt((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)) - 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*l 
og(x)))/(cosh(n*log(x)) - sinh(n*log(x))))))/(b*n)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcsch}\left (b x^{n} + a\right ) + \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} - 1\right )}{2 \, b n} \] Input:

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

Output:

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)
 

Giac [F]

\[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\int { x^{n - 1} \operatorname {arcsch}\left (b x^{n} + a\right ) \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.51 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\frac {\mathrm {atanh}\left (\sqrt {\frac {1}{{\left (a+b\,x^n\right )}^2}+1}\right )+\mathrm {asinh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \] Input:

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

Output:

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

Reduce [F]

\[ \int x^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx=\int \frac {x^{n} \mathit {acsch} \left (x^{n} b +a \right )}{x}d x \] Input:

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

Output:

int((x**n*acsch(x**n*b + a))/x,x)