3.3.89 \(\int \frac {\sinh ^{-1}(a x)^2}{x^3 \sqrt {1+a^2 x^2}} \, dx\) [289]

Optimal. Leaf size=135 \[ -\frac {a \sinh ^{-1}(a x)}{x}-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x^2}+a^2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+a^2 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-a^2 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-a^2 \text {PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+a^2 \text {PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5809, 5816, 4267, 2611, 2320, 6724, 5776, 272, 65, 214} \begin {gather*} a^2 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-a^2 \sinh ^{-1}(a x) \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-a^2 \text {Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+a^2 \text {Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+a^2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac {a \sinh ^{-1}(a x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^2/(x^3*Sqrt[1 + a^2*x^2]),x]

[Out]

-((a*ArcSinh[a*x])/x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x^2) + a^2*ArcSinh[a*x]^2*ArcTanh[E^ArcSinh[a*x]
] - a^2*ArcTanh[Sqrt[1 + a^2*x^2]] + a^2*ArcSinh[a*x]*PolyLog[2, -E^ArcSinh[a*x]] - a^2*ArcSinh[a*x]*PolyLog[2
, E^ArcSinh[a*x]] - a^2*PolyLog[3, -E^ArcSinh[a*x]] + a^2*PolyLog[3, E^ArcSinh[a*x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 5809

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[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[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]

Rubi steps

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

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Mathematica [A]
time = 0.95, size = 188, normalized size = 1.39 \begin {gather*} \frac {1}{8} a^2 \left (-4 \sinh ^{-1}(a x) \coth \left (\frac {1}{2} \sinh ^{-1}(a x)\right )-\sinh ^{-1}(a x)^2 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )-4 \sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )+4 \sinh ^{-1}(a x)^2 \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right )-8 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )+8 \sinh ^{-1}(a x) \text {PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )-8 \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )+8 \text {PolyLog}\left (3,e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(a x)\right )+4 \sinh ^{-1}(a x) \tanh \left (\frac {1}{2} \sinh ^{-1}(a x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^2/(x^3*Sqrt[1 + a^2*x^2]),x]

[Out]

(a^2*(-4*ArcSinh[a*x]*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]^2*Csch[ArcSinh[a*x]/2]^2 - 4*ArcSinh[a*x]^2*Log[1 -
E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]^2*Log[1 + E^(-ArcSinh[a*x])] + 8*Log[Tanh[ArcSinh[a*x]/2]] - 8*ArcSinh[a*x
]*PolyLog[2, -E^(-ArcSinh[a*x])] + 8*ArcSinh[a*x]*PolyLog[2, E^(-ArcSinh[a*x])] - 8*PolyLog[3, -E^(-ArcSinh[a*
x])] + 8*PolyLog[3, E^(-ArcSinh[a*x])] - ArcSinh[a*x]^2*Sech[ArcSinh[a*x]/2]^2 + 4*ArcSinh[a*x]*Tanh[ArcSinh[a
*x]/2]))/8

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Maple [A]
time = 4.99, size = 233, normalized size = 1.73

method	result	size
default	\(-\frac {\arcsinh \left (a x \right ) \left (x^{2} \arcsinh \left (a x \right ) a^{2}+2 \sqrt {a^{2} x^{2}+1}\, a x +\arcsinh \left (a x \right )\right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}-\frac {a^{2} \arcsinh \left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-a^{2} \arcsinh \left (a x \right ) \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+a^{2} \polylog \left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+\frac {a^{2} \arcsinh \left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+a^{2} \arcsinh \left (a x \right ) \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-a^{2} \polylog \left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 a^{2} \arctanh \left (a x +\sqrt {a^{2} x^{2}+1}\right )\)	\(233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^2/x^3/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/(a^2*x^2+1)^(1/2)/x^2*arcsinh(a*x)*(x^2*arcsinh(a*x)*a^2+2*(a^2*x^2+1)^(1/2)*a*x+arcsinh(a*x))-1/2*a^2*ar
csinh(a*x)^2*ln(1-a*x-(a^2*x^2+1)^(1/2))-a^2*arcsinh(a*x)*polylog(2,a*x+(a^2*x^2+1)^(1/2))+a^2*polylog(3,a*x+(
a^2*x^2+1)^(1/2))+1/2*a^2*arcsinh(a*x)^2*ln(1+a*x+(a^2*x^2+1)^(1/2))+a^2*arcsinh(a*x)*polylog(2,-a*x-(a^2*x^2+
1)^(1/2))-a^2*polylog(3,-a*x-(a^2*x^2+1)^(1/2))-2*a^2*arctanh(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)^2/x^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)^2/(a^2*x^5 + x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**2/x**3/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)**2/(x**3*sqrt(a**2*x**2 + 1)), 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)^2/x^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^2/(sqrt(a^2*x^2 + 1)*x^3), 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 )}^2}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^2/(x^3*(a^2*x^2 + 1)^(1/2)),x)

[Out]

int(asinh(a*x)^2/(x^3*(a^2*x^2 + 1)^(1/2)), x)

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