3.1.1 \(\int x^4 \text {sech}^{-1}(a x)^2 \, dx\) [1]

Optimal. Leaf size=164 \[ -\frac {3 x}{20 a^4}-\frac {x^3}{30 a^2}-\frac {3 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {sech}^{-1}(a x) \text {ArcTan}\left (e^{\text {sech}^{-1}(a x)}\right )}{10 a^5}+\frac {3 i \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {3 i \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5} \]

[Out]

-3/20*x/a^4-1/30*x^3/a^2+1/5*x^5*arcsech(a*x)^2-3/10*arcsech(a*x)*arctan(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)
)/a^5+3/20*I*polylog(2,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a^5-3/20*I*polylog(2,I*(1/a/x+(1/a/x-1)^(1/
2)*(1+1/a/x)^(1/2)))/a^5-3/20*x*(a*x+1)*arcsech(a*x)*((-a*x+1)/(a*x+1))^(1/2)/a^4-1/10*x^3*(a*x+1)*arcsech(a*x
)*((-a*x+1)/(a*x+1))^(1/2)/a^2

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Rubi [A]
time = 0.08, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6420, 5526, 4270, 4265, 2317, 2438} \begin {gather*} -\frac {3 \text {sech}^{-1}(a x) \text {ArcTan}\left (e^{\text {sech}^{-1}(a x)}\right )}{10 a^5}+\frac {3 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {3 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {3 x}{20 a^4}-\frac {3 x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3}{30 a^2}-\frac {x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*ArcSech[a*x]^2,x]

[Out]

(-3*x)/(20*a^4) - x^3/(30*a^2) - (3*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(20*a^4) - (x^3*Sqrt[(
1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(10*a^2) + (x^5*ArcSech[a*x]^2)/5 - (3*ArcSech[a*x]*ArcTan[E^ArcSe
ch[a*x]])/(10*a^5) + (((3*I)/20)*PolyLog[2, (-I)*E^ArcSech[a*x]])/a^5 - (((3*I)/20)*PolyLog[2, I*E^ArcSech[a*x
]])/a^5

Rule 2317

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 2438

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]

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 4270

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

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 6420

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 x^4 \text {sech}^{-1}(a x)^2 \, dx &=-\frac {\text {Subst}\left (\int x^2 \text {sech}^5(x) \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )}{a^5}\\ &=\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {2 \text {Subst}\left (\int x \text {sech}^5(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{5 a^5}\\ &=-\frac {x^3}{30 a^2}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {Subst}\left (\int x \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{10 a^5}\\ &=-\frac {3 x}{20 a^4}-\frac {x^3}{30 a^2}-\frac {3 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {Subst}\left (\int x \text {sech}(x) \, dx,x,\text {sech}^{-1}(a x)\right )}{20 a^5}\\ &=-\frac {3 x}{20 a^4}-\frac {x^3}{30 a^2}-\frac {3 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{10 a^5}+\frac {(3 i) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{20 a^5}-\frac {(3 i) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(a x)\right )}{20 a^5}\\ &=-\frac {3 x}{20 a^4}-\frac {x^3}{30 a^2}-\frac {3 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{10 a^5}+\frac {(3 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {(3 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}\\ &=-\frac {3 x}{20 a^4}-\frac {x^3}{30 a^2}-\frac {3 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{10 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^2-\frac {3 \text {sech}^{-1}(a x) \tan ^{-1}\left (e^{\text {sech}^{-1}(a x)}\right )}{10 a^5}+\frac {3 i \text {Li}_2\left (-i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {3 i \text {Li}_2\left (i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 182, normalized size = 1.11 \begin {gather*} \frac {-9 a x-2 a^3 x^3-9 a x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)-6 a^3 x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)+12 a^5 x^5 \text {sech}^{-1}(a x)^2+9 i \text {sech}^{-1}(a x) \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-9 i \text {sech}^{-1}(a x) \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )+9 i \text {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-9 i \text {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )}{60 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcSech[a*x]^2,x]

[Out]

(-9*a*x - 2*a^3*x^3 - 9*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] - 6*a^3*x^3*Sqrt[(1 - a*x)/(1 + a
*x)]*(1 + a*x)*ArcSech[a*x] + 12*a^5*x^5*ArcSech[a*x]^2 + (9*I)*ArcSech[a*x]*Log[1 - I/E^ArcSech[a*x]] - (9*I)
*ArcSech[a*x]*Log[1 + I/E^ArcSech[a*x]] + (9*I)*PolyLog[2, (-I)/E^ArcSech[a*x]] - (9*I)*PolyLog[2, I/E^ArcSech
[a*x]])/(60*a^5)

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Maple [A]
time = 0.49, size = 280, normalized size = 1.71

method	result	size
derivativedivides	\(\frac {\frac {\left (12 a^{4} x^{4} \mathrm {arcsech}\left (a x \right )^{2}-6 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a^{3} x^{3}-9 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x -2 a^{2} x^{2}-9\right ) a x}{60}+\frac {3 i \mathrm {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}-\frac {3 i \mathrm {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}+\frac {3 i \dilog \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}-\frac {3 i \dilog \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}}{a^{5}}\)	\(280\)
default	\(\frac {\frac {\left (12 a^{4} x^{4} \mathrm {arcsech}\left (a x \right )^{2}-6 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a^{3} x^{3}-9 \,\mathrm {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x -2 a^{2} x^{2}-9\right ) a x}{60}+\frac {3 i \mathrm {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}-\frac {3 i \mathrm {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}+\frac {3 i \dilog \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}-\frac {3 i \dilog \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{20}}{a^{5}}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsech(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/60*(12*a^4*x^4*arcsech(a*x)^2-6*arcsech(a*x)*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*a^3*x^3-9*arcse
ch(a*x)*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*a*x-2*a^2*x^2-9)*a*x+3/20*I*arcsech(a*x)*ln(1+I*(1/a/x+(1/a/x
-1)^(1/2)*(1+1/a/x)^(1/2)))-3/20*I*arcsech(a*x)*ln(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))+3/20*I*dilog(1
+I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))-3/20*I*dilog(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))))

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsech(a*x)^2,x, algorithm="maxima")

[Out]

Timed out

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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(x^4*arcsech(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^4*arcsech(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \operatorname {asech}^{2}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asech(a*x)**2,x)

[Out]

Integral(x**4*asech(a*x)**2, 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(x^4*arcsech(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^4*arcsech(a*x)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acosh(1/(a*x))^2,x)

[Out]

int(x^4*acosh(1/(a*x))^2, x)

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