3.51 \(\int e^{\text{sech}^{-1}(a x^2)} x \, dx\)

Optimal. Leaf size=68 \[ -\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )}{2 a}+\frac{1}{2} x^2 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{\log (x)}{a} \]

[Out]

(E^ArcSech[a*x^2]*x^2)/2 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]])/(2*a) + Log[x]/
a

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Rubi [A]  time = 0.0425703, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6335, 29, 259, 266, 63, 208} \[ -\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )}{2 a}+\frac{1}{2} x^2 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x^2]*x,x]

[Out]

(E^ArcSech[a*x^2]*x^2)/2 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]])/(2*a) + Log[x]/
a

Rule 6335

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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 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 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 208

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

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x \, dx &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^2\right )} x^2+\frac{\int \frac{1}{x} \, dx}{a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x \sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^2\right )} x^2+\frac{\log (x)}{a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x \sqrt{1-a^2 x^4}} \, dx}{a}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^2\right )} x^2+\frac{\log (x)}{a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^2\right )} x^2+\frac{\log (x)}{a}-\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^4}\right )}{2 a^3}\\ &=\frac{1}{2} e^{\text{sech}^{-1}\left (a x^2\right )} x^2-\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )}{2 a}+\frac{\log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0633233, size = 100, normalized size = 1.47 \[ \frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )+2 \log \left (a x^2\right )-\log \left (a x^2 \sqrt{\frac{1-a x^2}{a x^2+1}}+\sqrt{\frac{1-a x^2}{a x^2+1}}+1\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]*x,x]

[Out]

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

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Maple [C]  time = 0.312, size = 127, normalized size = 1.9 \begin{align*}{\frac{{x}^{2}{\it csgn} \left ({a}^{-1} \right ) }{2\,a}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({\it csgn} \left ({a}^{-1} \right ) a\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}-\ln \left ( 2\,{\frac{1}{{a}^{2}{x}^{2}} \left ({\it csgn} \left ({a}^{-1} \right ) a\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}+1 \right ) } \right ) \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}}+{\frac{\ln \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(csgn(1/a)*a*(-(a^2*x^4-1)/a^2)^(1/2)-ln(2*(csgn(1/a)
*a*(-(a^2*x^4-1)/a^2)^(1/2)+1)/a^2/x^2))*csgn(1/a)/a/(-(a^2*x^4-1)/a^2)^(1/2)+ln(x)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{1}{2} \, \sqrt{-a^{2} x^{4} + 1} - \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-a^{2} x^{4} + 1}}{x^{2}} + \frac{2}{x^{2}}\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/x, x)/a + log(x)/a

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Fricas [B]  time = 2.09933, size = 298, normalized size = 4.38 \begin{align*} \frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - \log \left (a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 1\right ) + \log \left (a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 1\right ) + 4 \, \log \left (x\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - log(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-
(a*x^2 - 1)/(a*x^2)) + 1) + log(a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) - 1) + 4*log(x))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x}\, dx + \int a x \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(1/x, x) + Integral(a*x*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x**2)), x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError