### 3.45 $$\int \frac{e^{\text{csch}^{-1}(a x^2)}}{x^3} \, dx$$

Optimal. Leaf size=42 $-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{4 x^2}-\frac{1}{4 a x^4}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right )$

[Out]

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

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Rubi [A]  time = 0.039321, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {6336, 30, 335, 275, 195, 215} $-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{4 x^2}-\frac{1}{4 a x^4}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCsch[a*x^2]/x^3,x]

[Out]

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^3} \, dx &=\frac{\int \frac{1}{x^5} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^3} \, dx\\ &=-\frac{1}{4 a x^4}-\operatorname{Subst}\left (\int x \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4 a x^4}-\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{4 x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{4 x^2}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0443945, size = 24, normalized size = 0.57 $-\frac{1}{8} a \left (2 \text{csch}^{-1}\left (a x^2\right )+e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCsch[a*x^2]/x^3,x]

[Out]

-(a*(E^(2*ArcCsch[a*x^2]) + 2*ArcCsch[a*x^2]))/8

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Maple [B]  time = 0.287, size = 114, normalized size = 2.7 \begin{align*} -{\frac{1}{4\,{x}^{2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \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 ){x}^{4}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}\sqrt{{a}^{-2}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}{\frac{1}{\sqrt{{a}^{-2}}}}}-{\frac{1}{4\,{x}^{4}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^3,x)

[Out]

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

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Maxima [B]  time = 1.07214, size = 124, normalized size = 2.95 \begin{align*} -\frac{a^{2} x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1}}{4 \,{\left (a^{2} x^{4}{\left (\frac{1}{a^{2} x^{4}} + 1\right )} - 1\right )}} - \frac{1}{8} \, a \log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right ) + \frac{1}{8} \, a \log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right ) - \frac{1}{4 \, a 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^3,x, algorithm="maxima")

[Out]

-1/4*a^2*x^2*sqrt(1/(a^2*x^4) + 1)/(a^2*x^4*(1/(a^2*x^4) + 1) - 1) - 1/8*a*log(a*x^2*sqrt(1/(a^2*x^4) + 1) + 1
) + 1/8*a*log(a*x^2*sqrt(1/(a^2*x^4) + 1) - 1) - 1/4/(a*x^4)

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Fricas [B]  time = 2.51988, size = 227, normalized size = 5.4 \begin{align*} -\frac{a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1\right ) - a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - 1\right ) + 2 \, a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 2}{8 \, a 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^3,x, algorithm="fricas")

[Out]

-1/8*(a^2*x^4*log(a*x^2*sqrt((a^2*x^4 + 1)/(a^2*x^4)) + 1) - a^2*x^4*log(a*x^2*sqrt((a^2*x^4 + 1)/(a^2*x^4)) -
1) + 2*a*x^2*sqrt((a^2*x^4 + 1)/(a^2*x^4)) + 2)/(a*x^4)

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Sympy [A]  time = 4.86306, size = 39, normalized size = 0.93 \begin{align*} - \frac{a \operatorname{asinh}{\left (\frac{1}{a x^{2}} \right )}}{4} - \frac{\sqrt{1 + \frac{1}{a^{2} x^{4}}}}{4 x^{2}} - \frac{1}{4 a 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**3,x)

[Out]

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

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Giac [B]  time = 1.12455, size = 103, normalized size = 2.45 \begin{align*} -\frac{a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{4} + 1} + 1\right ) - a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{4} + 1} - 1\right ) + \frac{2 \,{\left (\sqrt{a^{2} x^{4} + 1} a^{4}{\left | a \right |} + a^{5}\right )}}{a^{2} x^{4}}}{8 \, a^{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^3,x, algorithm="giac")

[Out]

-1/8*(a^4*abs(a)*log(sqrt(a^2*x^4 + 1) + 1) - a^4*abs(a)*log(sqrt(a^2*x^4 + 1) - 1) + 2*(sqrt(a^2*x^4 + 1)*a^4
*abs(a) + a^5)/(a^2*x^4))/a^4