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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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])

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 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 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 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]

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 = 0.18, 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))

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 334, normalized size = 7.26 \[ \frac {a \log \left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} + 1\right ) - a \log \left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} - 1\right ) + {\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right )} \log \left (\frac {\sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}} + 1}{b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a}\right ) - \log \left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \relax (x)\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \relax (x)\right )}{\cosh \left (n \log \relax (x)\right ) - \sinh \left (n \log \relax (x)\right )}}\right )}{b n} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{n - 1} \operatorname {arcsch}\left (b x^{n} + a\right )\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int x^{-1+n} \mathrm {arccsch}\left (a +b \,x^{n}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 60, normalized size = 1.30 \[ \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcsch}\left (b x^{n} + a\right ) + \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} - 1\right )}{2 \, b n} \]

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)

________________________________________________________________________________________

mupad [B]  time = 2.21, size = 40, normalized size = 0.87 \[ \frac {\mathrm {atanh}\left (\sqrt {\frac {1}{{\left (a+b\,x^n\right )}^2}+1}\right )+\mathrm {asinh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*asinh(1/(a + b*x^n)),x)

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________