### 3.41 $$\int e^{\text{csch}^{-1}(a x^2)} x \, dx$$

Optimal. Leaf size=40 $\frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^4}+1}-\frac{\text{csch}^{-1}\left (a x^2\right )}{2 a}+\frac{\log (x)}{a}$

[Out]

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

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Rubi [A]  time = 0.0362239, antiderivative size = 40, 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 = {6336, 29, 335, 275, 277, 215} $\frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^4}+1}-\frac{\text{csch}^{-1}\left (a x^2\right )}{2 a}+\frac{\log (x)}{a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6336

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/
(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]

Rule 29

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

Rule 335

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 215

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

Rubi steps

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

Mathematica [A]  time = 0.031044, size = 42, normalized size = 1.05 $\frac{a x^2 \sqrt{\frac{1}{a^2 x^4}+1}+\log \left (a x^2\right )-\sinh ^{-1}\left (\frac{1}{a x^2}\right )}{2 a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.285, size = 116, normalized size = 2.9 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}-\ln \left ( 2\,{\frac{1}{{a}^{2}{x}^{2}} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ) \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\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+1/a^2/x^4)^(1/2))*x,x)

[Out]

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

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Maxima [B]  time = 0.988743, size = 96, normalized size = 2.4 \begin{align*} \frac{1}{2} \, x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - \frac{\log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right )}{4 \, a} + \frac{\log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right )}{4 \, 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+1/a^2/x^4)^(1/2))*x,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.54821, size = 205, normalized size = 5.12 \begin{align*} \frac{2 \, a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1\right ) + \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - 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+1/a^2/x^4)^(1/2))*x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.24026, size = 58, normalized size = 1.45 \begin{align*} \frac{x^{2}}{2 \sqrt{1 + \frac{1}{a^{2} x^{4}}}} + \frac{\log{\left (x \right )}}{a} - \frac{\operatorname{asinh}{\left (\frac{1}{a x^{2}} \right )}}{2 a} + \frac{1}{2 a^{2} x^{2} \sqrt{1 + \frac{1}{a^{2} x^{4}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x**2+(1+1/a**2/x**4)**(1/2))*x,x)

[Out]

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

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Giac [A]  time = 1.13163, size = 82, normalized size = 2.05 \begin{align*} \frac{{\left (a -{\left | a \right |}\right )} \log \left (\sqrt{a^{2} x^{4} + 1} + 1\right ) +{\left (a +{\left | a \right |}\right )} \log \left (\sqrt{a^{2} x^{4} + 1} - 1\right ) + 2 \, \sqrt{a^{2} x^{4} + 1}{\left | a \right |}}{4 \, a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))*x,x, algorithm="giac")

[Out]

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