\(\int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx\) [347]

Optimal result

Integrand size = 23, antiderivative size = 102 \[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=-2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \]

[Out]

-2*arcsinh(a*x)^3*arctanh(a*x+(a^2*x^2+1)^(1/2))-3*arcsinh(a*x)^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+3*arcsinh(
a*x)^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))+6*arcsinh(a*x)*polylog(3,-a*x-(a^2*x^2+1)^(1/2))-6*arcsinh(a*x)*polylo
g(3,a*x+(a^2*x^2+1)^(1/2))-6*polylog(4,-a*x-(a^2*x^2+1)^(1/2))+6*polylog(4,a*x+(a^2*x^2+1)^(1/2))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5816, 4267, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=-2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \]

[In]

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

[Out]

-2*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 3*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]] + 3*ArcSinh[a*x]^2*Po
lyLog[2, E^ArcSinh[a*x]] + 6*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] - 6*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*
x]] - 6*PolyLog[4, -E^ArcSinh[a*x]] + 6*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 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*} \text {integral}& = \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+3 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} \left (\pi ^4-2 \text {arcsinh}(a x)^4-8 \text {arcsinh}(a x)^3 \log \left (1+e^{-\text {arcsinh}(a x)}\right )+8 \text {arcsinh}(a x)^3 \log \left (1-e^{\text {arcsinh}(a x)}\right )+24 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+24 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+48 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )-48 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )\right ) \]

[In]

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

[Out]

(Pi^4 - 2*ArcSinh[a*x]^4 - 8*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] + 8*ArcSinh[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*PolyLog[2, E^ArcSinh[a*x]] + 48*Arc
Sinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] - 48*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] + 48*PolyLog[4, -E^(-Arc
Sinh[a*x])] + 48*PolyLog[4, E^ArcSinh[a*x]])/8

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.93

method	result	size
default	\(-\operatorname {arcsinh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-3 \operatorname {arcsinh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (4, -a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (4, a x +\sqrt {a^{2} x^{2}+1}\right )\)	\(197\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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