3.62 \(\int \frac {x^2}{\sinh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=81 \[ -\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)} \]

[Out]

-x/a^2/arcsinh(a*x)-3/2*x^3/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))/a^3+9/8*Chi(3*arcsinh(a*x))/a^3-1/2*x^2*(a^2*x^
2+1)^(1/2)/a/arcsinh(a*x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5667, 5774, 5669, 5448, 3301, 5657} \[ -\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSinh[a*x]^3,x]

[Out]

-(x^2*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - x/(a^2*ArcSinh[a*x]) - (3*x^3)/(2*ArcSinh[a*x]) - CoshIntegral
[ArcSinh[a*x]]/(8*a^3) + (9*CoshIntegral[3*ArcSinh[a*x]])/(8*a^3)

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5657

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

Rule 5667

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

Rule 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5774

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

Rubi steps

\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a x)^3} \, dx &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}+\frac {\int \frac {x}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2} \, dx\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)}+\frac {9}{2} \int \frac {x^2}{\sinh ^{-1}(a x)} \, dx+\frac {\int \frac {1}{\sinh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)}+\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)}+\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{a^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \sinh ^{-1}(a x)^2}-\frac {x}{a^2 \sinh ^{-1}(a x)}-\frac {3 x^3}{2 \sinh ^{-1}(a x)}-\frac {\text {Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 64, normalized size = 0.79 \[ -\frac {\frac {4 a x \left (a x \sqrt {a^2 x^2+1}+\left (3 a^2 x^2+2\right ) \sinh ^{-1}(a x)\right )}{\sinh ^{-1}(a x)^2}+\text {Chi}\left (\sinh ^{-1}(a x)\right )-9 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSinh[a*x]^3,x]

[Out]

-1/8*((4*a*x*(a*x*Sqrt[1 + a^2*x^2] + (2 + 3*a^2*x^2)*ArcSinh[a*x]))/ArcSinh[a*x]^2 + CoshIntegral[ArcSinh[a*x
]] - 9*CoshIntegral[3*ArcSinh[a*x]])/a^3

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arcsinh(a*x)^3, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arcsinh(a*x)^3, x)

________________________________________________________________________________________

maple [A]  time = 0.14, size = 81, normalized size = 1.00 \[ \frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \arcsinh \left (a x \right )^{2}}+\frac {a x}{8 \arcsinh \left (a x \right )}-\frac {\Chi \left (\arcsinh \left (a x \right )\right )}{8}-\frac {\cosh \left (3 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )^{2}}-\frac {3 \sinh \left (3 \arcsinh \left (a x \right )\right )}{8 \arcsinh \left (a x \right )}+\frac {9 \Chi \left (3 \arcsinh \left (a x \right )\right )}{8}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsinh(a*x)^3,x)

[Out]

1/a^3*(1/8/arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+1/8*a*x/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))-1/8/arcsinh(a*x)^2*cosh
(3*arcsinh(a*x))-3/8/arcsinh(a*x)*sinh(3*arcsinh(a*x))+9/8*Chi(3*arcsinh(a*x)))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{8} x^{9} + 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} + a^{2} x^{3} + {\left (a^{5} x^{6} + a^{3} x^{4}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{6} x^{7} + 5 \, a^{4} x^{5} + 2 \, a^{2} x^{3}\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (3 \, a^{8} x^{9} + 9 \, a^{6} x^{7} + 9 \, a^{4} x^{5} + 3 \, a^{2} x^{3} + {\left (3 \, a^{5} x^{6} + 4 \, a^{3} x^{4} + a x^{2}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (9 \, a^{6} x^{7} + 17 \, a^{4} x^{5} + 10 \, a^{2} x^{3} + 2 \, x\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (9 \, a^{7} x^{8} + 22 \, a^{5} x^{6} + 18 \, a^{3} x^{4} + 5 \, a x^{2}\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (3 \, a^{7} x^{8} + 7 \, a^{5} x^{6} + 5 \, a^{3} x^{4} + a x^{2}\right )} \sqrt {a^{2} x^{2} + 1}}{2 \, {\left (a^{8} x^{6} + 3 \, a^{6} x^{4} + {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{5} x^{3} + 3 \, a^{4} x^{2} + 3 \, {\left (a^{6} x^{4} + a^{4} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + a^{2} + 3 \, {\left (a^{7} x^{5} + 2 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}} + \int \frac {9 \, a^{10} x^{10} + 36 \, a^{8} x^{8} + 54 \, a^{6} x^{6} + 36 \, a^{4} x^{4} + 9 \, a^{2} x^{2} + {\left (9 \, a^{6} x^{6} + 4 \, a^{4} x^{4} - a^{2} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )}^{2} + {\left (36 \, a^{7} x^{7} + 48 \, a^{5} x^{5} + 13 \, a^{3} x^{3} - 2 \, a x\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (54 \, a^{8} x^{8} + 120 \, a^{6} x^{6} + 83 \, a^{4} x^{4} + 19 \, a^{2} x^{2} + 2\right )} {\left (a^{2} x^{2} + 1\right )} + {\left (36 \, a^{9} x^{9} + 112 \, a^{7} x^{7} + 123 \, a^{5} x^{5} + 57 \, a^{3} x^{3} + 10 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{2 \, {\left (a^{10} x^{8} + 4 \, a^{8} x^{6} + {\left (a^{2} x^{2} + 1\right )}^{2} a^{6} x^{4} + 6 \, a^{6} x^{4} + 4 \, a^{4} x^{2} + 4 \, {\left (a^{7} x^{5} + a^{5} x^{3}\right )} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{8} x^{6} + 2 \, a^{6} x^{4} + a^{4} x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + a^{2} + 4 \, {\left (a^{9} x^{7} + 3 \, a^{7} x^{5} + 3 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^8*x^9 + 3*a^6*x^7 + 3*a^4*x^5 + a^2*x^3 + (a^5*x^6 + a^3*x^4)*(a^2*x^2 + 1)^(3/2) + (3*a^6*x^7 + 5*a^4
*x^5 + 2*a^2*x^3)*(a^2*x^2 + 1) + (3*a^8*x^9 + 9*a^6*x^7 + 9*a^4*x^5 + 3*a^2*x^3 + (3*a^5*x^6 + 4*a^3*x^4 + a*
x^2)*(a^2*x^2 + 1)^(3/2) + (9*a^6*x^7 + 17*a^4*x^5 + 10*a^2*x^3 + 2*x)*(a^2*x^2 + 1) + (9*a^7*x^8 + 22*a^5*x^6
+ 18*a^3*x^4 + 5*a*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^8 + 7*a^5*x^6 + 5*a^3*x^4
+ a*x^2)*sqrt(a^2*x^2 + 1))/((a^8*x^6 + 3*a^6*x^4 + (a^2*x^2 + 1)^(3/2)*a^5*x^3 + 3*a^4*x^2 + 3*(a^6*x^4 + a^4
*x^2)*(a^2*x^2 + 1) + a^2 + 3*(a^7*x^5 + 2*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2)
+ integrate(1/2*(9*a^10*x^10 + 36*a^8*x^8 + 54*a^6*x^6 + 36*a^4*x^4 + 9*a^2*x^2 + (9*a^6*x^6 + 4*a^4*x^4 - a^
2*x^2)*(a^2*x^2 + 1)^2 + (36*a^7*x^7 + 48*a^5*x^5 + 13*a^3*x^3 - 2*a*x)*(a^2*x^2 + 1)^(3/2) + (54*a^8*x^8 + 12
0*a^6*x^6 + 83*a^4*x^4 + 19*a^2*x^2 + 2)*(a^2*x^2 + 1) + (36*a^9*x^9 + 112*a^7*x^7 + 123*a^5*x^5 + 57*a^3*x^3
+ 10*a*x)*sqrt(a^2*x^2 + 1))/((a^10*x^8 + 4*a^8*x^6 + (a^2*x^2 + 1)^2*a^6*x^4 + 6*a^6*x^4 + 4*a^4*x^2 + 4*(a^7
*x^5 + a^5*x^3)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^6 + 2*a^6*x^4 + a^4*x^2)*(a^2*x^2 + 1) + a^2 + 4*(a^9*x^7 + 3*a
^7*x^5 + 3*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/asinh(a*x)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/asinh(a*x)**3, x)

________________________________________________________________________________________