3.58 \(\int \frac{e^{2 \text{csch}^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=96 \[ \frac{a^3 \sqrt{\frac{1}{a^2 x^2}+1}}{8 x}-\frac{a \sqrt{\frac{1}{a^2 x^2}+1}}{12 x^3}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{3 a x^5}-\frac{1}{3 a^2 x^6}-\frac{1}{8} a^4 \text{csch}^{-1}(a x)-\frac{1}{4 x^4} \]

[Out]

-1/(3*a^2*x^6) - Sqrt[1 + 1/(a^2*x^2)]/(3*a*x^5) - 1/(4*x^4) - (a*Sqrt[1 + 1/(a^2*x^2)])/(12*x^3) + (a^3*Sqrt[
1 + 1/(a^2*x^2)])/(8*x) - (a^4*ArcCsch[a*x])/8

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Rubi [A]  time = 0.252561, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6338, 6742, 335, 279, 321, 215} \[ \frac{a^3 \sqrt{\frac{1}{a^2 x^2}+1}}{8 x}-\frac{a \sqrt{\frac{1}{a^2 x^2}+1}}{12 x^3}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{3 a x^5}-\frac{1}{3 a^2 x^6}-\frac{1}{8} a^4 \text{csch}^{-1}(a x)-\frac{1}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCsch[a*x])/x^5,x]

[Out]

-1/(3*a^2*x^6) - Sqrt[1 + 1/(a^2*x^2)]/(3*a*x^5) - 1/(4*x^4) - (a*Sqrt[1 + 1/(a^2*x^2)])/(12*x^3) + (a^3*Sqrt[
1 + 1/(a^2*x^2)])/(8*x) - (a^4*ArcCsch[a*x])/8

Rule 6338

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && Int
egerQ[n]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 279

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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 \frac{e^{2 \text{csch}^{-1}(a x)}}{x^5} \, dx &=\int \frac{\left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2}{x^5} \, dx\\ &=\int \left (\frac{2}{a^2 x^7}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a x^6}+\frac{1}{x^5}\right ) \, dx\\ &=-\frac{1}{3 a^2 x^6}-\frac{1}{4 x^4}+\frac{2 \int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^6} \, dx}{a}\\ &=-\frac{1}{3 a^2 x^6}-\frac{1}{4 x^4}-\frac{2 \operatorname{Subst}\left (\int x^4 \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{3 a^2 x^6}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{3 a x^5}-\frac{1}{4 x^4}-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a}\\ &=-\frac{1}{3 a^2 x^6}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{3 a x^5}-\frac{1}{4 x^4}-\frac{a \sqrt{1+\frac{1}{a^2 x^2}}}{12 x^3}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3 a^2 x^6}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{3 a x^5}-\frac{1}{4 x^4}-\frac{a \sqrt{1+\frac{1}{a^2 x^2}}}{12 x^3}+\frac{a^3 \sqrt{1+\frac{1}{a^2 x^2}}}{8 x}-\frac{1}{8} a^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3 a^2 x^6}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{3 a x^5}-\frac{1}{4 x^4}-\frac{a \sqrt{1+\frac{1}{a^2 x^2}}}{12 x^3}+\frac{a^3 \sqrt{1+\frac{1}{a^2 x^2}}}{8 x}-\frac{1}{8} a^4 \text{csch}^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0719423, size = 74, normalized size = 0.77 \[ \frac{\frac{\left (3 a^2 x^2+4\right ) \left (a^3 x^3 \sqrt{\frac{1}{a^2 x^2}+1}-2 a x \sqrt{\frac{1}{a^2 x^2}+1}-2\right )}{x^6}-3 a^6 \sinh ^{-1}\left (\frac{1}{a x}\right )}{24 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCsch[a*x])/x^5,x]

[Out]

(((4 + 3*a^2*x^2)*(-2 - 2*a*Sqrt[1 + 1/(a^2*x^2)]*x + a^3*Sqrt[1 + 1/(a^2*x^2)]*x^3))/x^6 - 3*a^6*ArcSinh[1/(a
*x)])/(24*a^2)

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Maple [B]  time = 0.187, size = 209, normalized size = 2.2 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{3\,{x}^{6}{a}^{2}}}-{\frac{a}{24\,{x}^{5}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( 3\,\sqrt{{a}^{-2}} \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}{x}^{4}{a}^{4}-3\,\sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{x}^{6}{a}^{4}+3\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ){x}^{6}{a}^{2}-6\, \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}\sqrt{{a}^{-2}}{x}^{2}{a}^{2}+8\, \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}\sqrt{{a}^{-2}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}{\frac{1}{\sqrt{{a}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x)

[Out]

-1/4/x^4-1/3/x^6/a^2-1/24*a*((a^2*x^2+1)/a^2/x^2)^(1/2)/x^5*(3*(1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(3/2)*x^4*a^4-3
*(1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(1/2)*x^6*a^4+3*ln(2*((1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(1/2)*a^2+1)/x/a^2)*x^6
*a^2-6*((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2)*x^2*a^2+8*((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2))/((a^2*x^2+1)/a^2
)^(1/2)/(1/a^2)^(1/2)

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Maxima [B]  time = 1.00201, size = 243, normalized size = 2.53 \begin{align*} -\frac{3 \, a^{5} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - 3 \, a^{5} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (3 \, a^{10} x^{5}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 8 \, a^{8} x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, a^{6} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}\right )}}{a^{6} x^{6}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{3} - 3 \, a^{4} x^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{2} + 3 \, a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - 1}}{48 \, a} - \frac{1}{4 \, x^{4}} - \frac{1}{3 \, a^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="maxima")

[Out]

-1/48*(3*a^5*log(a*x*sqrt(1/(a^2*x^2) + 1) + 1) - 3*a^5*log(a*x*sqrt(1/(a^2*x^2) + 1) - 1) - 2*(3*a^10*x^5*(1/
(a^2*x^2) + 1)^(5/2) - 8*a^8*x^3*(1/(a^2*x^2) + 1)^(3/2) - 3*a^6*x*sqrt(1/(a^2*x^2) + 1))/(a^6*x^6*(1/(a^2*x^2
) + 1)^3 - 3*a^4*x^4*(1/(a^2*x^2) + 1)^2 + 3*a^2*x^2*(1/(a^2*x^2) + 1) - 1))/a - 1/4/x^4 - 1/3/(a^2*x^6)

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Fricas [A]  time = 2.78773, size = 296, normalized size = 3.08 \begin{align*} -\frac{3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + 6 \, a^{2} x^{2} -{\left (3 \, a^{5} x^{5} - 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 8}{24 \, a^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="fricas")

[Out]

-1/24*(3*a^6*x^6*log(a*x*sqrt((a^2*x^2 + 1)/(a^2*x^2)) - a*x + 1) - 3*a^6*x^6*log(a*x*sqrt((a^2*x^2 + 1)/(a^2*
x^2)) - a*x - 1) + 6*a^2*x^2 - (3*a^5*x^5 - 2*a^3*x^3 - 8*a*x)*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + 8)/(a^2*x^6)

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Sympy [A]  time = 8.60466, size = 114, normalized size = 1.19 \begin{align*} - \frac{a^{4} \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{8} + \frac{a^{3}}{8 x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} + \frac{a}{24 x^{3} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 x^{4}} - \frac{5}{12 a x^{5} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{3 a^{2} x^{6}} - \frac{1}{3 a^{3} x^{7} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a**2/x**2)**(1/2))**2/x**5,x)

[Out]

-a**4*asinh(1/(a*x))/8 + a**3/(8*x*sqrt(1 + 1/(a**2*x**2))) + a/(24*x**3*sqrt(1 + 1/(a**2*x**2))) - 1/(4*x**4)
- 5/(12*a*x**5*sqrt(1 + 1/(a**2*x**2))) - 1/(3*a**2*x**6) - 1/(3*a**3*x**7*sqrt(1 + 1/(a**2*x**2)))

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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+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError