3.63 \(\int \frac{e^{\text{sech}^{-1}(a x^p)}}{x} \, dx\)

Optimal. Leaf size=87 \[ -\frac{x^{-p} \sqrt{1-a x^p}}{a p \sqrt{\frac{1}{a x^p+1}}}-\frac{x^{-p}}{a p}-\frac{\sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \sin ^{-1}\left (a x^p\right )}{p} \]

[Out]

-(1/(a*p*x^p)) - Sqrt[1 - a*x^p]/(a*p*x^p*Sqrt[(1 + a*x^p)^(-1)]) - (Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Ar
cSin[a*x^p])/p

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Rubi [A]  time = 0.0689316, antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6334, 259, 345, 242, 277, 216} \[ -\frac{x^{-p} \sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \sqrt{1-a^2 x^{2 p}}}{a p}-\frac{x^{-p}}{a p}-\frac{\sqrt{\frac{1}{a x^p+1}} \sqrt{a x^p+1} \csc ^{-1}\left (\frac{x^{-p}}{a}\right )}{p} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x^p]/x,x]

[Out]

-(1/(a*p*x^p)) - (Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Sqrt[1 - a^2*x^(2*p)])/(a*p*x^p) - (Sqrt[(1 + a*x^p)^
(-1)]*Sqrt[1 + a*x^p]*ArcCsc[1/(a*x^p)])/p

Rule 6334

Int[E^ArcSech[(a_.)*(x_)^(p_.)]/(x_), x_Symbol] :> -Simp[(a*p*x^p)^(-1), x] + Dist[(Sqrt[1 + a*x^p]*Sqrt[1/(1
+ a*x^p)])/a, Int[(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p])/x^(p + 1), x], x] /; FreeQ[{a, p}, x]

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 345

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

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

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

Rubi steps

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

Mathematica [C]  time = 0.137093, size = 96, normalized size = 1.1 \[ -\frac{i \left (-i \left (a+x^{-p}\right ) \sqrt{\frac{1-a x^p}{a x^p+1}}+a \log \left (2 \sqrt{\frac{1-a x^p}{a x^p+1}} \left (a x^p+1\right )-2 i a x^p\right )-i x^{-p}\right )}{a p} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^p]/x,x]

[Out]

((-I)*((-I)/x^p - I*(a + x^(-p))*Sqrt[(1 - a*x^p)/(1 + a*x^p)] + a*Log[(-2*I)*a*x^p + 2*Sqrt[(1 - a*x^p)/(1 +
a*x^p)]*(1 + a*x^p)]))/(a*p)

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Maple [C]  time = 0.326, size = 145, normalized size = 1.7 \begin{align*} -{\frac{{\it csgn} \left ( a \right ){x}^{p}a}{p}\sqrt{-{\frac{a{x}^{p}-1}{a{x}^{p}}}}\sqrt{{\frac{1+a{x}^{p}}{a{x}^{p}}}}\arctan \left ({{\it csgn} \left ( a \right ){x}^{p}a{\frac{1}{\sqrt{- \left ({x}^{p} \right ) ^{2}{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{- \left ({x}^{p} \right ) ^{2}{a}^{2}+1}}}}-{\frac{ \left ({\it csgn} \left ( a \right ) \right ) ^{2}}{p}\sqrt{-{\frac{a{x}^{p}-1}{a{x}^{p}}}}\sqrt{{\frac{1+a{x}^{p}}{a{x}^{p}}}}}-{\frac{1}{ap{x}^{p}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))/x,x)

[Out]

-1/p*(-(a*x^p-1)/a/(x^p))^(1/2)*((1+a*x^p)/a/(x^p))^(1/2)*csgn(a)/(-(x^p)^2*a^2+1)^(1/2)*arctan(csgn(a)*a*x^p/
(-(x^p)^2*a^2+1)^(1/2))*x^p*a-1/p*(-(a*x^p-1)/a/(x^p))^(1/2)*((1+a*x^p)/a/(x^p))^(1/2)*csgn(a)^2-1/a/p/(x^p)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.45426, size = 197, normalized size = 2.26 \begin{align*} -\frac{a x^{p} \sqrt{\frac{a x^{p} + 1}{a x^{p}}} \sqrt{-\frac{a x^{p} - 1}{a x^{p}}} - a x^{p} \arctan \left (\sqrt{\frac{a x^{p} + 1}{a x^{p}}} \sqrt{-\frac{a x^{p} - 1}{a x^{p}}}\right ) + 1}{a p x^{p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))/x,x, algorithm="fricas")

[Out]

-(a*x^p*sqrt((a*x^p + 1)/(a*x^p))*sqrt(-(a*x^p - 1)/(a*x^p)) - a*x^p*arctan(sqrt((a*x^p + 1)/(a*x^p))*sqrt(-(a
*x^p - 1)/(a*x^p))) + 1)/(a*p*x^p)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{- p}}{x}\, dx + \int \frac{a \sqrt{-1 + \frac{x^{- p}}{a}} \sqrt{1 + \frac{x^{- p}}{a}}}{x}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/(x**p)+(1/a/(x**p)-1)**(1/2)*(1/a/(x**p)+1)**(1/2))/x,x)

[Out]

(Integral(x**(-p)/x, x) + Integral(a*sqrt(-1 + x**(-p)/a)*sqrt(1 + x**(-p)/a)/x, x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{1}{a x^{p}} + 1} \sqrt{\frac{1}{a x^{p}} - 1} + \frac{1}{a x^{p}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))/x,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x^p) + 1)*sqrt(1/(a*x^p) - 1) + 1/(a*x^p))/x, x)