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

Optimal. Leaf size=184 \[ \frac{\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(a x)}\right )}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 a^2}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{4 a^4}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}+\frac{\text{sech}^{-1}(a x) \log \left (e^{2 \text{sech}^{-1}(a x)}+1\right )}{a^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3 \]

[Out]

(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(4*a^4) - (x^2*ArcSech[a*x])/(4*a^2) - ArcSech[a*x]^2/(2*a^4) - (Sqrt[(1
- a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(2*a^4) - (x^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2
)/(4*a^2) + (x^4*ArcSech[a*x]^3)/4 + (ArcSech[a*x]*Log[1 + E^(2*ArcSech[a*x])])/a^4 + PolyLog[2, -E^(2*ArcSech
[a*x])]/(2*a^4)

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Rubi [A]  time = 0.17673, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6285, 5418, 4186, 3767, 8, 4184, 3718, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(a x)}\right )}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 a^2}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{4 a^4}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}+\frac{\text{sech}^{-1}(a x) \log \left (e^{2 \text{sech}^{-1}(a x)}+1\right )}{a^4}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(4*a^4) - (x^2*ArcSech[a*x])/(4*a^2) - ArcSech[a*x]^2/(2*a^4) - (Sqrt[(1
- a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(2*a^4) - (x^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2
)/(4*a^2) + (x^4*ArcSech[a*x]^3)/4 + (ArcSech[a*x]*Log[1 + E^(2*ArcSech[a*x])])/a^4 + PolyLog[2, -E^(2*ArcSech
[a*x])]/(2*a^4)

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 5418

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_)^(n_.)]^(q_.), x_Symbol] :> -Simp[(
x^(m - n + 1)*Sech[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sech[a + b*x^n]^p, x],
x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]

Rule 4186

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

Rule 3767

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^3 \text{sech}^{-1}(a x)^3 \, dx &=-\frac{\operatorname{Subst}\left (\int x^3 \text{sech}^4(x) \tanh (x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^4}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3-\frac{3 \operatorname{Subst}\left (\int x^2 \text{sech}^4(x) \, dx,x,\text{sech}^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{\operatorname{Subst}\left (\int \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )}{4 a^4}-\frac{\operatorname{Subst}\left (\int x^2 \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )}{2 a^4}\\ &=-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{i \operatorname{Subst}\left (\int 1 \, dx,x,-i \sqrt{\frac{1-a x}{1+a x}} (1+a x)\right )}{4 a^4}+\frac{\operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^4}\\ &=\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{4 a^4}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\text{sech}^{-1}(a x)\right )}{a^4}\\ &=\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{4 a^4}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{\text{sech}^{-1}(a x) \log \left (1+e^{2 \text{sech}^{-1}(a x)}\right )}{a^4}-\frac{\operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^4}\\ &=\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{4 a^4}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{\text{sech}^{-1}(a x) \log \left (1+e^{2 \text{sech}^{-1}(a x)}\right )}{a^4}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \text{sech}^{-1}(a x)}\right )}{2 a^4}\\ &=\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)}{4 a^4}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{x^2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)^2}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3+\frac{\text{sech}^{-1}(a x) \log \left (1+e^{2 \text{sech}^{-1}(a x)}\right )}{a^4}+\frac{\text{Li}_2\left (-e^{2 \text{sech}^{-1}(a x)}\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.576463, size = 188, normalized size = 1.02 \[ \frac{-2 \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(a x)}\right )+a^4 x^4 \text{sech}^{-1}(a x)^3-\left (a^3 x^3 \sqrt{\frac{1-a x}{a x+1}}+a^2 x^2 \sqrt{\frac{1-a x}{a x+1}}+2 a x \sqrt{\frac{1-a x}{a x+1}}+2 \sqrt{\frac{1-a x}{a x+1}}-2\right ) \text{sech}^{-1}(a x)^2+\text{sech}^{-1}(a x) \left (4 \log \left (e^{-2 \text{sech}^{-1}(a x)}+1\right )-a^2 x^2\right )+\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{4 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - (-2 + 2*Sqrt[(1 - a*x)/(1 + a*x)] + 2*a*x*Sqrt[(1 - a*x)/(1 + a*x)] + a
^2*x^2*Sqrt[(1 - a*x)/(1 + a*x)] + a^3*x^3*Sqrt[(1 - a*x)/(1 + a*x)])*ArcSech[a*x]^2 + a^4*x^4*ArcSech[a*x]^3
+ ArcSech[a*x]*(-(a^2*x^2) + 4*Log[1 + E^(-2*ArcSech[a*x])]) - 2*PolyLog[2, -E^(-2*ArcSech[a*x])])/(4*a^4)

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Maple [A]  time = 0.349, size = 246, normalized size = 1.3 \begin{align*}{\frac{{x}^{4} \left ({\rm arcsech} \left (ax\right ) \right ) ^{3}}{4}}-{\frac{{x}^{3} \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{4\,a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{x \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{2\,{a}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{{x}^{2}{\rm arcsech} \left (ax\right )}{4\,{a}^{2}}}+{\frac{x}{4\,{a}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{2\,{a}^{4}}}-{\frac{1}{4\,{a}^{4}}}+{\frac{{\rm arcsech} \left (ax\right )}{{a}^{4}}\ln \left ( 1+ \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{2} \right ) }+{\frac{1}{2\,{a}^{4}}{\it polylog} \left ( 2,- \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arcsech(a*x)^3-1/4/a*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*arcsech(a*x)^2*x^3-1/2/a^3*(-(a*x-1)/a/x
)^(1/2)*((a*x+1)/a/x)^(1/2)*arcsech(a*x)^2*x-1/4*x^2*arcsech(a*x)/a^2+1/4/a^3*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/
x)^(1/2)*x-1/2*arcsech(a*x)^2/a^4-1/4/a^4+arcsech(a*x)*ln(1+(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2)/a^4+1/2
*polylog(2,-(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2)/a^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arcsech(a*x)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{arsech}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arcsech(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*asech(a*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arcsech(a*x)^3, x)