3.1.14 \(\int \text {sech}^{-1}(a x)^3 \, dx\) [14]

Optimal. Leaf size=111 \[ x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \text {ArcTan}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )}{a} \]

[Out]

x*arcsech(a*x)^3-6*arcsech(a*x)^2*arctan(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/a+6*I*arcsech(a*x)*polylog(2,-
I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a-6*I*arcsech(a*x)*polylog(2,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/
2)))/a-6*I*polylog(3,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a+6*I*polylog(3,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1
/a/x)^(1/2)))/a

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Rubi [A]
time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6414, 5526, 4265, 2611, 2320, 6724} \begin {gather*} -\frac {6 \text {sech}^{-1}(a x)^2 \text {ArcTan}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}+x \text {sech}^{-1}(a x)^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*ArcSech[a*x]^3 - (6*ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]])/a + ((6*I)*ArcSech[a*x]*PolyLog[2, (-I)*E^ArcSech
[a*x]])/a - ((6*I)*ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]])/a - ((6*I)*PolyLog[3, (-I)*E^ArcSech[a*x]])/a +
((6*I)*PolyLog[3, I*E^ArcSech[a*x]])/a

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4265

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

Rule 5526

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 6414

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \text {sech}^{-1}(a x)^3 \, dx &=-\frac {\text {Subst}\left (\int x^3 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {3 \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(6 i) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}-\frac {(6 i) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {(6 i) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}+\frac {(6 i) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{a}\\ &=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {Li}_3\left (-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {Li}_3\left (i e^{\text {sech}^{-1}(a x)}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 128, normalized size = 1.15 \begin {gather*} x \text {sech}^{-1}(a x)^3-\frac {3 i \left (-\text {sech}^{-1}(a x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )\right )-2 \text {sech}^{-1}(a x) \left (\text {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-\text {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )\right )-2 \left (\text {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(a x)}\right )-\text {PolyLog}\left (3,i e^{-\text {sech}^{-1}(a x)}\right )\right )\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*ArcSech[a*x]^3 - ((3*I)*(-(ArcSech[a*x]^2*(Log[1 - I/E^ArcSech[a*x]] - Log[1 + I/E^ArcSech[a*x]])) - 2*ArcSe
ch[a*x]*(PolyLog[2, (-I)/E^ArcSech[a*x]] - PolyLog[2, I/E^ArcSech[a*x]]) - 2*(PolyLog[3, (-I)/E^ArcSech[a*x]]
- PolyLog[3, I/E^ArcSech[a*x]])))/a

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \mathrm {arcsech}\left (a x \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(sqrt(a*x + 1)*sqrt(-a*x + 1) + 1)^3 - integrate((a^2*x^2*log(a)^3 + (a^2*x^2 - 1)*log(x)^3 + 3*(a^2*x^2*
log(a) + (a^2*x^2*(log(a) + 1) + (a^2*x^2 - 1)*log(x) - log(a))*sqrt(a*x + 1)*sqrt(-a*x + 1) + (a^2*x^2 - 1)*l
og(x) - log(a))*log(sqrt(a*x + 1)*sqrt(-a*x + 1) + 1)^2 - log(a)^3 + 3*(a^2*x^2*log(a) - log(a))*log(x)^2 + (a
^2*x^2*log(a)^3 + (a^2*x^2 - 1)*log(x)^3 - log(a)^3 + 3*(a^2*x^2*log(a) - log(a))*log(x)^2 + 3*(a^2*x^2*log(a)
^2 - log(a)^2)*log(x))*sqrt(a*x + 1)*sqrt(-a*x + 1) - 3*(a^2*x^2*log(a)^2 + (a^2*x^2 - 1)*log(x)^2 + (a^2*x^2*
log(a)^2 + (a^2*x^2 - 1)*log(x)^2 - log(a)^2 + 2*(a^2*x^2*log(a) - log(a))*log(x))*sqrt(a*x + 1)*sqrt(-a*x + 1
) - log(a)^2 + 2*(a^2*x^2*log(a) - log(a))*log(x))*log(sqrt(a*x + 1)*sqrt(-a*x + 1) + 1) + 3*(a^2*x^2*log(a)^2
- log(a)^2)*log(x))/(a^2*x^2 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(-a*x + 1) - 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsech(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asech}^{3}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asech(a*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsech(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(1/(a*x))^3, x)

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