Integrand size = 14, antiderivative size = 58 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b n}-\frac {2 \arctan \left (\sqrt {\frac {1-a-b x^n}{1+a+b x^n}}\right )}{b n} \]
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.83 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )+\frac {2 \sqrt {-\frac {-1+a+b x^n}{1+a+b x^n}} \sqrt {1-\left (a+b x^n\right )^2} \arctan \left (\frac {\sqrt {1-\left (a+b x^n\right )^2}}{-1+a+b x^n}\right )}{-1+a+b x^n}}{b n} \]
((a + b*x^n)*ArcSech[a + b*x^n] + (2*Sqrt[-((-1 + a + b*x^n)/(1 + a + b*x^ n))]*Sqrt[1 - (a + b*x^n)^2]*ArcTan[Sqrt[1 - (a + b*x^n)^2]/(-1 + a + b*x^ n)])/(-1 + a + b*x^n))/(b*n)
Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {7266, 6867, 2055, 27, 216}
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.
\(\displaystyle \int x^{n-1} \text {sech}^{-1}\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {\int \text {sech}^{-1}\left (b x^n+a\right )dx^n}{n}\) |
\(\Big \downarrow \) 6867 |
\(\displaystyle \frac {\int \frac {\sqrt {\frac {-b x^n-a+1}{b x^n+a+1}}}{-b x^n-a+1}dx^n+\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b}}{n}\) |
\(\Big \downarrow \) 2055 |
\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b}-4 b \int \frac {1}{2 b^2 \left (x^{2 n}+1\right )}d\sqrt {\frac {-b x^n-a+1}{b x^n+a+1}}}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b}-\frac {2 \int \frac {1}{x^{2 n}+1}d\sqrt {\frac {-b x^n-a+1}{b x^n+a+1}}}{b}}{n}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {\left (a+b x^n\right ) \text {sech}^{-1}\left (a+b x^n\right )}{b}-\frac {2 \arctan \left (\sqrt {\frac {-a-b x^n+1}{a+b x^n+1}}\right )}{b}}{n}\) |
3.1.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/ (b*e - d*x^q)^(1/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1 /n))^r, 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] && Int egerQ[r]
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] /; Free Q[{c, d}, x]
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]
\[\int x^{-1+n} \operatorname {arcsech}\left (a +b \,x^{n}\right )d x\]
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (54) = 108\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 6.64 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\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} \]
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*arctan((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^2*cosh(n*log(x))^2 + b^2*sin h(n*log(x))^2 + 2*a*b*cosh(n*log(x)) + a^2 + 2*(b^2*cosh(n*log(x)) + a*b)* sinh(n*log(x)) - 1)))/(b*n)
Timed out. \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {{\left (b x^{n} + a\right )} \operatorname {arsech}\left (b x^{n} + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} - 1}\right )}{b n} \]
\[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\int { x^{n - 1} \operatorname {arsech}\left (b x^{n} + a\right ) \,d x } \]
Time = 5.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int x^{-1+n} \text {sech}^{-1}\left (a+b x^n\right ) \, dx=\frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{a+b\,x^n}-1}\,\sqrt {\frac {1}{a+b\,x^n}+1}}\right )+\mathrm {acosh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]