3.1.40 \(\int \frac {\sinh ^{-1}(a x)^4}{x^3} \, dx\) [40]

Optimal. Leaf size=108 \[ -2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+6 a^2 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-3 a^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right ) \]

[Out]

-2*a^2*arcsinh(a*x)^3-1/2*arcsinh(a*x)^4/x^2+6*a^2*arcsinh(a*x)^2*ln(1-(a*x+(a^2*x^2+1)^(1/2))^2)+6*a^2*arcsin
h(a*x)*polylog(2,(a*x+(a^2*x^2+1)^(1/2))^2)-3*a^2*polylog(3,(a*x+(a^2*x^2+1)^(1/2))^2)-2*a*arcsinh(a*x)^3*(a^2
*x^2+1)^(1/2)/x

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Rubi [A]
time = 0.14, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5776, 5800, 5775, 3797, 2221, 2611, 2320, 6724} \begin {gather*} 6 a^2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-3 a^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {2 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{x}-2 a^2 \sinh ^{-1}(a x)^3+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^4/x^3,x]

[Out]

-2*a^2*ArcSinh[a*x]^3 - (2*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/x - ArcSinh[a*x]^4/(2*x^2) + 6*a^2*ArcSinh[a*x]
^2*Log[1 - E^(2*ArcSinh[a*x])] + 6*a^2*ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - 3*a^2*PolyLog[3, E^(2*Arc
Sinh[a*x])]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

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 \frac {\sinh ^{-1}(a x)^4}{x^3} \, dx &=-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+(2 a) \int \frac {\sinh ^{-1}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+\left (6 a^2\right ) \int \frac {\sinh ^{-1}(a x)^2}{x} \, dx\\ &=-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+\left (6 a^2\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}-\left (12 a^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\left (12 a^2\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+6 a^2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\left (6 a^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+6 a^2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-2 a^2 \sinh ^{-1}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{x}-\frac {\sinh ^{-1}(a x)^4}{2 x^2}+6 a^2 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+6 a^2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-3 a^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 113, normalized size = 1.05 \begin {gather*} -\frac {\sinh ^{-1}(a x)^4}{2 x^2}+\frac {1}{4} a^2 \left (i \pi ^3-8 \sinh ^{-1}(a x)^3-\frac {8 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{a x}+24 \sinh ^{-1}(a x)^2 \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-12 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^4/x^3,x]

[Out]

-1/2*ArcSinh[a*x]^4/x^2 + (a^2*(I*Pi^3 - 8*ArcSinh[a*x]^3 - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(a*x) + 24*Ar
cSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + 24*ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - 12*PolyLog[3, E^(2
*ArcSinh[a*x])]))/4

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Maple [A]
time = 2.55, size = 199, normalized size = 1.84

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arcsinh \left (a x \right )^{3} \left (-4 a^{2} x^{2}+4 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}-4 \arcsinh \left (a x \right )^{3}+6 \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-12 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-12 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(199\)
default	\(a^{2} \left (-\frac {\arcsinh \left (a x \right )^{3} \left (-4 a^{2} x^{2}+4 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}-4 \arcsinh \left (a x \right )^{3}+6 \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-12 \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-12 \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2*arcsinh(a*x)^3*(-4*a^2*x^2+4*a*x*(a^2*x^2+1)^(1/2)+arcsinh(a*x))/a^2/x^2-4*arcsinh(a*x)^3+6*arcsinh(
a*x)^2*ln(1+a*x+(a^2*x^2+1)^(1/2))+12*arcsinh(a*x)*polylog(2,-a*x-(a^2*x^2+1)^(1/2))-12*polylog(3,-a*x-(a^2*x^
2+1)^(1/2))+6*arcsinh(a*x)^2*ln(1-a*x-(a^2*x^2+1)^(1/2))+12*arcsinh(a*x)*polylog(2,a*x+(a^2*x^2+1)^(1/2))-12*p
olylog(3,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^3,x, algorithm="maxima")

[Out]

-1/2*log(a*x + sqrt(a^2*x^2 + 1))^4/x^2 + integrate(2*(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^5 + a*x^3 + (a^2*x^4 + x^2)*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^3,x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^4/x^3, 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^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**4/x**3,x)

[Out]

Integral(asinh(a*x)**4/x**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^4/x^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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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^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^4/x^3,x)

[Out]

int(asinh(a*x)^4/x^3, x)

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