3.18 \(\int \frac{\text{sech}^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=179 \[ \frac{14 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{9 x}+\frac{2 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x}-\frac{4 a^2 \text{sech}^{-1}(a x)}{3 x}+\frac{2 \left (\frac{1-a x}{a x+1}\right )^{3/2} (a x+1)^3}{27 x^3}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x^3}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}-\frac{2 \text{sech}^{-1}(a x)}{9 x^3} \]

[Out]

(14*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(9*x) + (2*((1 - a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3)/(27*x^3) - (2
*ArcSech[a*x])/(9*x^3) - (4*a^2*ArcSech[a*x])/(3*x) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*
x^3) + (2*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*x) - ArcSech[a*x]^3/(3*x^3)

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Rubi [A]  time = 0.125027, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6285, 5373, 3311, 3296, 2637, 2633} \[ \frac{14 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{9 x}+\frac{2 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x}-\frac{4 a^2 \text{sech}^{-1}(a x)}{3 x}+\frac{2 \left (\frac{1-a x}{a x+1}\right )^{3/2} (a x+1)^3}{27 x^3}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x^3}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}-\frac{2 \text{sech}^{-1}(a x)}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a*x]^3/x^4,x]

[Out]

(14*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(9*x) + (2*((1 - a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3)/(27*x^3) - (2
*ArcSech[a*x])/(9*x^3) - (4*a^2*ArcSech[a*x])/(3*x) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*
x^3) + (2*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*x) - ArcSech[a*x]^3/(3*x^3)

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 5373

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

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\text{sech}^{-1}(a x)^3}{x^4} \, dx &=-\left (a^3 \operatorname{Subst}\left (\int x^3 \cosh ^2(x) \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}+a^3 \operatorname{Subst}\left (\int x^2 \cosh ^3(x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{2 \text{sech}^{-1}(a x)}{9 x^3}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x^3}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}+\frac{1}{9} \left (2 a^3\right ) \operatorname{Subst}\left (\int \cosh ^3(x) \, dx,x,\text{sech}^{-1}(a x)\right )+\frac{1}{3} \left (2 a^3\right ) \operatorname{Subst}\left (\int x^2 \cosh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{2 \text{sech}^{-1}(a x)}{9 x^3}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}+\frac{1}{9} \left (2 i a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\frac{i \sqrt{\frac{1-a x}{1+a x}} (1+a x)}{a x}\right )-\frac{1}{3} \left (4 a^3\right ) \operatorname{Subst}\left (\int x \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{2 a^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x)}{9 x}+\frac{2 \left (\frac{1-a x}{1+a x}\right )^{3/2} (1+a x)^3}{27 x^3}-\frac{2 \text{sech}^{-1}(a x)}{9 x^3}-\frac{4 a^2 \text{sech}^{-1}(a x)}{3 x}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}+\frac{1}{3} \left (4 a^3\right ) \operatorname{Subst}\left (\int \cosh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{14 a^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x)}{9 x}+\frac{2 \left (\frac{1-a x}{1+a x}\right )^{3/2} (1+a x)^3}{27 x^3}-\frac{2 \text{sech}^{-1}(a x)}{9 x^3}-\frac{4 a^2 \text{sech}^{-1}(a x)}{3 x}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x^3}+\frac{2 a^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{3 x}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.113122, size = 120, normalized size = 0.67 \[ \frac{2 \sqrt{\frac{1-a x}{a x+1}} \left (20 a^3 x^3+20 a^2 x^2+a x+1\right )+9 \sqrt{\frac{1-a x}{a x+1}} \left (2 a^3 x^3+2 a^2 x^2+a x+1\right ) \text{sech}^{-1}(a x)^2-6 \left (6 a^2 x^2+1\right ) \text{sech}^{-1}(a x)-9 \text{sech}^{-1}(a x)^3}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[a*x]^3/x^4,x]

[Out]

(2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x + 20*a^2*x^2 + 20*a^3*x^3) - 6*(1 + 6*a^2*x^2)*ArcSech[a*x] + 9*Sqrt[(1
- a*x)/(1 + a*x)]*(1 + a*x + 2*a^2*x^2 + 2*a^3*x^3)*ArcSech[a*x]^2 - 9*ArcSech[a*x]^3)/(27*x^3)

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Maple [A]  time = 0.229, size = 226, normalized size = 1.3 \begin{align*}{a}^{3} \left ({\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{3} \left ( ax-1 \right ) \left ( ax+1 \right ) }{3\,{x}^{3}{a}^{3}}}-{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{3}}{3\,ax}}+{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{3\,{a}^{2}{x}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{2\, \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{3}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{2\,{\rm arcsech} \left (ax\right ) \left ( ax-1 \right ) \left ( ax+1 \right ) }{9\,{x}^{3}{a}^{3}}}-{\frac{14\,{\rm arcsech} \left (ax\right )}{9\,ax}}+{\frac{2}{27\,{a}^{2}{x}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{40}{27}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(a*x)^3/x^4,x)

[Out]

a^3*(1/3*arcsech(a*x)^3/a^3/x^3*(a*x-1)*(a*x+1)-1/3*arcsech(a*x)^3/a/x+1/3*arcsech(a*x)^2/a^2/x^2*(-(a*x-1)/a/
x)^(1/2)*((a*x+1)/a/x)^(1/2)+2/3*arcsech(a*x)^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+2/9*arcsech(a*x)*(a*x
-1)/a^3/x^3*(a*x+1)-14/9/a/x*arcsech(a*x)+2/27/a^2/x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+40/27*(-(a*x-1
)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^3/x^4,x, algorithm="maxima")

[Out]

integrate(arcsech(a*x)^3/x^4, x)

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Fricas [A]  time = 1.97532, size = 406, normalized size = 2.27 \begin{align*} \frac{9 \,{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 9 \, \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{3} - 6 \,{\left (6 \, a^{2} x^{2} + 1\right )} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) + 2 \,{\left (20 \, a^{3} x^{3} + a x\right )} \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^3/x^4,x, algorithm="fricas")

[Out]

1/27*(9*(2*a^3*x^3 + a*x)*sqrt(-(a^2*x^2 - 1)/(a^2*x^2))*log((a*x*sqrt(-(a^2*x^2 - 1)/(a^2*x^2)) + 1)/(a*x))^2
- 9*log((a*x*sqrt(-(a^2*x^2 - 1)/(a^2*x^2)) + 1)/(a*x))^3 - 6*(6*a^2*x^2 + 1)*log((a*x*sqrt(-(a^2*x^2 - 1)/(a
^2*x^2)) + 1)/(a*x)) + 2*(20*a^3*x^3 + a*x)*sqrt(-(a^2*x^2 - 1)/(a^2*x^2)))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}^{3}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(a*x)**3/x**4,x)

[Out]

Integral(asech(a*x)**3/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^3/x^4,x, algorithm="giac")

[Out]

integrate(arcsech(a*x)^3/x^4, x)