3.1.39 \(\int \frac {\sinh ^{-1}(a x)^4}{x^2} \, dx\) [39]

Optimal. Leaf size=120 \[ -\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-24 a \text {PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )+24 a \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right ) \]

[Out]

-arcsinh(a*x)^4/x-8*a*arcsinh(a*x)^3*arctanh(a*x+(a^2*x^2+1)^(1/2))-12*a*arcsinh(a*x)^2*polylog(2,-a*x-(a^2*x^
2+1)^(1/2))+12*a*arcsinh(a*x)^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))+24*a*arcsinh(a*x)*polylog(3,-a*x-(a^2*x^2+1)^
(1/2))-24*a*arcsinh(a*x)*polylog(3,a*x+(a^2*x^2+1)^(1/2))-24*a*polylog(4,-a*x-(a^2*x^2+1)^(1/2))+24*a*polylog(
4,a*x+(a^2*x^2+1)^(1/2))

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Rubi [A]
time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5816, 4267, 2611, 6744, 2320, 6724} \begin {gather*} -12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-24 a \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )+24 a \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^4/x^2,x]

[Out]

-(ArcSinh[a*x]^4/x) - 8*a*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 12*a*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a
*x]] + 12*a*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] + 24*a*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] - 24*a*A
rcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 24*a*PolyLog[4, -E^ArcSinh[a*x]] + 24*a*PolyLog[4, E^ArcSinh[a*x]]

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 4267

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5816

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]

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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {\sinh ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\sinh ^{-1}(a x)^4}{x}+(4 a) \int \frac {\sinh ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}+(4 a) \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-(12 a) \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(12 a) \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+(24 a) \text {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-(24 a) \text {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-(24 a) \text {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(24 a) \text {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-(24 a) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+(24 a) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {\sinh ^{-1}(a x)^4}{x}-8 a \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+12 a \sinh ^{-1}(a x)^2 \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+24 a \sinh ^{-1}(a x) \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-24 a \sinh ^{-1}(a x) \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-24 a \text {Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )+24 a \text {Li}_4\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 161, normalized size = 1.34 \begin {gather*} \frac {1}{2} a \left (\pi ^4-2 \sinh ^{-1}(a x)^4-\frac {2 \sinh ^{-1}(a x)^4}{a x}-8 \sinh ^{-1}(a x)^3 \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+8 \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+48 \sinh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )-48 \sinh ^{-1}(a x) \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,-e^{-\sinh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^4/x^2,x]

[Out]

(a*(Pi^4 - 2*ArcSinh[a*x]^4 - (2*ArcSinh[a*x]^4)/(a*x) - 8*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] + 8*ArcSi
nh[a*x]^3*Log[1 - E^ArcSinh[a*x]] + 24*ArcSinh[a*x]^2*PolyLog[2, -E^(-ArcSinh[a*x])] + 24*ArcSinh[a*x]^2*PolyL
og[2, E^ArcSinh[a*x]] + 48*ArcSinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] - 48*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[
a*x]] + 48*PolyLog[4, -E^(-ArcSinh[a*x])] + 48*PolyLog[4, E^ArcSinh[a*x]]))/2

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Maple [A]
time = 1.90, size = 214, normalized size = 1.78

method	result	size
derivativedivides	\(a \left (-\frac {\arcsinh \left (a x \right )^{4}}{a x}-4 \arcsinh \left (a x \right )^{3} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-12 \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+24 \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-24 \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-24 \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+24 \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(214\)
default	\(a \left (-\frac {\arcsinh \left (a x \right )^{4}}{a x}-4 \arcsinh \left (a x \right )^{3} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-12 \arcsinh \left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+24 \arcsinh \left (a x \right ) \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-24 \polylog \left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )+4 \arcsinh \left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-24 \arcsinh \left (a x \right ) \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+24 \polylog \left (4, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-arcsinh(a*x)^4/a/x-4*arcsinh(a*x)^3*ln(1+a*x+(a^2*x^2+1)^(1/2))-12*arcsinh(a*x)^2*polylog(2,-a*x-(a^2*x^2+
1)^(1/2))+24*arcsinh(a*x)*polylog(3,-a*x-(a^2*x^2+1)^(1/2))-24*polylog(4,-a*x-(a^2*x^2+1)^(1/2))+4*arcsinh(a*x
)^3*ln(1-a*x-(a^2*x^2+1)^(1/2))+12*arcsinh(a*x)^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))-24*arcsinh(a*x)*polylog(3,a
*x+(a^2*x^2+1)^(1/2))+24*polylog(4,a*x+(a^2*x^2+1)^(1/2)))

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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(arcsinh(a*x)^4/x^2,x, algorithm="maxima")

[Out]

-log(a*x + sqrt(a^2*x^2 + 1))^4/x + integrate(4*(a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2
+ 1))^3/(a^3*x^4 + a*x^2 + (a^2*x^3 + x)*sqrt(a^2*x^2 + 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(arcsinh(a*x)^4/x^2,x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^4/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**4/x**2,x)

[Out]

Integral(asinh(a*x)**4/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(arcsinh(a*x)^4/x^2,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^4/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^4/x^2,x)

[Out]

int(asinh(a*x)^4/x^2, x)

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