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]

2/27*((-a*x+1)/(a*x+1))^(3/2)*(a*x+1)^3/x^3-2/9*arcsech(a*x)/x^3-4/3*a^2*arcsech(a*x)/x-1/3*arcsech(a*x)^3/x^3
+14/9*a^2*(a*x+1)*((-a*x+1)/(a*x+1))^(1/2)/x+1/3*(a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^(1/2)/x^3+2/3*a^2*(
a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^(1/2)/x

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Rubi [A]  time = 0.13, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 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]

Rule 2637

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

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 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 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 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])

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*}

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Mathematica [A]  time = 0.13, size = 120, normalized size = 0.67 \[ \frac {-6 \left (6 a^2 x^2+1\right ) \text {sech}^{-1}(a x)+2 \sqrt {\frac {1-a x}{a x+1}} \left (20 a^3 x^3+20 a^2 x^2+a x+1\right )+9 \sqrt {\frac {1-a x}{a x+1}} \left (2 a^3 x^3+2 a^2 x^2+a x+1\right ) \text {sech}^{-1}(a x)^2-9 \text {sech}^{-1}(a x)^3}{27 x^3} \]

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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fricas [A]  time = 0.68, size = 186, normalized size = 1.04 \[ \frac {9 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 9 \, \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{3} - 6 \, {\left (6 \, a^{2} x^{2} + 1\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) + 2 \, {\left (20 \, a^{3} x^{3} + a x\right )} \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}}}{27 \, x^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x^{4}}\,{d x} \]

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)

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maple [A]  time = 0.43, size = 192, normalized size = 1.07 \[ a^{3} \left (-\frac {\mathrm {arcsech}\left (a x \right )^{3}}{3 a^{3} x^{3}}+\frac {2 \mathrm {arcsech}\left (a x \right )^{2} \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{3}+\frac {\mathrm {arcsech}\left (a x \right )^{2} \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{3 a^{2} x^{2}}-\frac {4 \,\mathrm {arcsech}\left (a x \right )}{3 a x}+\frac {40 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{27}-\frac {2 \,\mathrm {arcsech}\left (a x \right )}{9 a^{3} x^{3}}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{27 a^{2} x^{2}}\right ) \]

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+2/3*arcsech(a*x)^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)+1/3*arcsech(a*x)^
2/a^2/x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)-4/3/a/x*arcsech(a*x)+40/27*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/
x)^(1/2)-2/9*arcsech(a*x)/a^3/x^3+2/27/a^2/x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{3}}{x^{4}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(1/(a*x))^3/x^4,x)

[Out]

int(acosh(1/(a*x))^3/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

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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