3.100 \(\int x^{-1+n} \text{sech}^{-1}(a+b x^n) \, dx\)

Optimal. Leaf size=58 \[ \frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{-a-b x^n+1}{a+b x^n+1}}\right )}{b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.113096, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6715, 6313, 1961, 12, 203} \[ \frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{-a-b x^n+1}{a+b x^n+1}}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcSech[a + b*x^n],x]

[Out]

((a + b*x^n)*ArcSech[a + b*x^n])/(b*n) - (2*ArcTan[Sqrt[(1 - a - b*x^n)/(1 + 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 6313

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

Rule 1961

Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n -
1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^(1/n + 1), x], x], x, ((e*(a + b*x
^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/
n] && IntegerQ[r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

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

Rubi steps

\begin{align*} \int x^{-1+n} \text{sech}^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \text{sech}^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{\frac{1-a-b x}{1+a+b x}}}{1-a-b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a-b x^n}{1+a+b x^n}}\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a-b x^n}{1+a+b x^n}}\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-a-b x^n}{1+a+b x^n}}\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.247847, size = 84, normalized size = 1.45 \[ \frac{\frac{\sqrt{1-\left (a+b x^n\right )^2} \sin ^{-1}\left (a+b x^n\right )}{\sqrt{-\frac{a+b x^n-1}{a+b x^n+1}} \left (a+b x^n+1\right )}+\left (a+b x^n\right ) \text{sech}^{-1}\left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arcsech} \left (a+b{x}^{n}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.99902, size = 54, normalized size = 0.93 \begin{align*} \frac{{\left (b x^{n} + a\right )} \operatorname{arsech}\left (b x^{n} + a\right ) - \arctan \left (\sqrt{\frac{1}{{\left (b x^{n} + a\right )}^{2}} - 1}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.60826, size = 1138, normalized size = 19.62 \begin{align*} \frac{2 \,{\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 ) + a \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}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - a \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}{\cosh \left (n \log \left (x\right )\right ) + \sinh \left (n \log \left (x\right )\right )}\right ) - 2 \, \arctan \left (\frac{{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\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^{2} \cosh \left (n \log \left (x\right )\right )^{2} + b^{2} \sinh \left (n \log \left (x\right )\right )^{2} + 2 \, a b \cosh \left (n \log \left (x\right )\right ) + a^{2} + 2 \,{\left (b^{2} \cosh \left (n \log \left (x\right )\right ) + a b\right )} \sinh \left (n \log \left (x\right )\right ) - 1}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)*asech(a+b*x**n),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{n - 1} \operatorname{arsech}\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)*arcsech(a+b*x^n),x, algorithm="giac")

[Out]

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