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

Optimal. Leaf size=257 \[ \frac{a^2 \text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac{a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}-\frac{a^2 \sqrt{(a+b x)^2+1}}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{\text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac{2 a \text{Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac{3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac{2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1} (a+b x)^2}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{a+b x}{b^3 \sinh ^{-1}(a+b x)}+\frac{a \sqrt{(a+b x)^2+1} (a+b x)}{b^3 \sinh ^{-1}(a+b x)^2}+\frac{a}{b^3 \sinh ^{-1}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.497086, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5865, 5803, 5655, 5774, 5657, 3301, 5667, 5669, 5448, 12, 3298, 5675} \[ \frac{a^2 \text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac{a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}-\frac{a^2 \sqrt{(a+b x)^2+1}}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{\text{Chi}\left (\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac{9 \text{Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac{2 a \text{Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac{3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac{2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1} (a+b x)^2}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{a+b x}{b^3 \sinh ^{-1}(a+b x)}+\frac{a \sqrt{(a+b x)^2+1} (a+b x)}{b^3 \sinh ^{-1}(a+b x)^2}+\frac{a}{b^3 \sinh ^{-1}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSinh[a + b*x]^3,x]

[Out]

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

Rule 5865

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

Rule 5803

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

Rule 5655

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

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]

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 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 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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3298

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

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.391158, size = 110, normalized size = 0.43 \[ \frac{-\frac{4 b x \left (b x \sqrt{a^2+2 a b x+b^2 x^2+1}+\left (2 a^2+5 a b x+3 b^2 x^2+2\right ) \sinh ^{-1}(a+b x)\right )}{\sinh ^{-1}(a+b x)^2}+\left (4 a^2-1\right ) \text{Chi}\left (\sinh ^{-1}(a+b x)\right )+9 \text{Chi}\left (3 \sinh ^{-1}(a+b x)\right )-16 a \text{Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSinh[a + b*x]^3,x]

[Out]

((-4*b*x*(b*x*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (2 + 2*a^2 + 5*a*b*x + 3*b^2*x^2)*ArcSinh[a + b*x]))/ArcSinh
[a + b*x]^2 + (-1 + 4*a^2)*CoshIntegral[ArcSinh[a + b*x]] + 9*CoshIntegral[3*ArcSinh[a + b*x]] - 16*a*SinhInte
gral[2*ArcSinh[a + b*x]])/(8*b^3)

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Maple [A]  time = 0.053, size = 215, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ( -{\frac{a \left ( 4\,{\it Shi} \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ) \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}-2\,\cosh \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ){\it Arcsinh} \left ( bx+a \right ) -\sinh \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ) \right ) }{2\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}}+{\frac{1}{8\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}\sqrt{1+ \left ( bx+a \right ) ^{2}}}+{\frac{bx+a}{8\,{\it Arcsinh} \left ( bx+a \right ) }}-{\frac{{\it Chi} \left ({\it Arcsinh} \left ( bx+a \right ) \right ) }{8}}-{\frac{\cosh \left ( 3\,{\it Arcsinh} \left ( bx+a \right ) \right ) }{8\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3\,\sinh \left ( 3\,{\it Arcsinh} \left ( bx+a \right ) \right ) }{8\,{\it Arcsinh} \left ( bx+a \right ) }}+{\frac{9\,{\it Chi} \left ( 3\,{\it Arcsinh} \left ( bx+a \right ) \right ) }{8}}+{\frac{{a}^{2}}{2\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}} \left ({\it Chi} \left ({\it Arcsinh} \left ( bx+a \right ) \right ) \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}- \left ( bx+a \right ){\it Arcsinh} \left ( bx+a \right ) -\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3*(-1/2*a*(4*Shi(2*arcsinh(b*x+a))*arcsinh(b*x+a)^2-2*cosh(2*arcsinh(b*x+a))*arcsinh(b*x+a)-sinh(2*arcsinh
(b*x+a)))/arcsinh(b*x+a)^2+1/8/arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)+1/8/arcsinh(b*x+a)*(b*x+a)-1/8*Chi(arcsinh
(b*x+a))-1/8/arcsinh(b*x+a)^2*cosh(3*arcsinh(b*x+a))-3/8/arcsinh(b*x+a)*sinh(3*arcsinh(b*x+a))+9/8*Chi(3*arcsi
nh(b*x+a))+1/2*a^2*(Chi(arcsinh(b*x+a))*arcsinh(b*x+a)^2-(b*x+a)*arcsinh(b*x+a)-(1+(b*x+a)^2)^(1/2))/arcsinh(b
*x+a)^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^8*x^9 + 7*a*b^7*x^8 + 3*(7*a^2*b^6 + b^6)*x^7 + 5*(7*a^3*b^5 + 3*a*b^5)*x^6 + (35*a^4*b^4 + 30*a^2*b^4
+ 3*b^4)*x^5 + 3*(7*a^5*b^3 + 10*a^3*b^3 + 3*a*b^3)*x^4 + (7*a^6*b^2 + 15*a^4*b^2 + 9*a^2*b^2 + b^2)*x^3 + (a
^7*b + 3*a^5*b + 3*a^3*b + a*b)*x^2 + (b^5*x^6 + 4*a*b^4*x^5 + (6*a^2*b^3 + b^3)*x^4 + 2*(2*a^3*b^2 + a*b^2)*x
^3 + (a^4*b + a^2*b)*x^2)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*b^6*x^7 + 15*a*b^5*x^6 + 5*(6*a^2*b^4 + b^4
)*x^5 + 15*(2*a^3*b^3 + a*b^3)*x^4 + (15*a^4*b^2 + 15*a^2*b^2 + 2*b^2)*x^3 + (3*a^5*b + 5*a^3*b + 2*a*b)*x^2)*
(b^2*x^2 + 2*a*b*x + a^2 + 1) + (3*b^8*x^9 + 23*a*b^7*x^8 + (77*a^2*b^6 + 9*b^6)*x^7 + 3*(49*a^3*b^5 + 17*a*b^
5)*x^6 + (175*a^4*b^4 + 120*a^2*b^4 + 9*b^4)*x^5 + (133*a^5*b^3 + 150*a^3*b^3 + 33*a*b^3)*x^4 + 3*(21*a^6*b^2
+ 35*a^4*b^2 + 15*a^2*b^2 + b^2)*x^3 + (17*a^7*b + 39*a^5*b + 27*a^3*b + 5*a*b)*x^2 + (3*b^5*x^6 + 14*a*b^4*x^
5 + 2*(13*a^2*b^3 + 2*b^3)*x^4 + 12*(2*a^3*b^2 + a*b^2)*x^3 + (11*a^4*b + 12*a^2*b + b)*x^2 + 2*(a^5 + 2*a^3 +
a)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (9*b^6*x^7 + 51*a*b^5*x^6 + (120*a^2*b^4 + 17*b^4)*x^5 + 5*(30*a^
3*b^3 + 13*a*b^3)*x^4 + (105*a^4*b^2 + 93*a^2*b^2 + 10*b^2)*x^3 + (39*a^5*b + 59*a^3*b + 20*a*b)*x^2 + 2*(3*a^
6 + 7*a^4 + 5*a^2 + 1)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*(a^8 + 3*a^6 + 3*a^4 + a^2)*x + (9*b^7*x^8 + 60*a*
b^6*x^7 + (171*a^2*b^5 + 22*b^5)*x^6 + 2*(135*a^3*b^4 + 52*a*b^4)*x^5 + (255*a^4*b^3 + 196*a^2*b^3 + 18*b^3)*x
^4 + 2*(72*a^5*b^2 + 92*a^3*b^2 + 25*a*b^2)*x^3 + (45*a^6*b + 86*a^4*b + 46*a^2*b + 5*b)*x^2 + 2*(3*a^7 + 8*a^
5 + 7*a^3 + 2*a)*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (3*b
^7*x^8 + 18*a*b^6*x^7 + (45*a^2*b^5 + 7*b^5)*x^6 + 4*(15*a^3*b^4 + 7*a*b^4)*x^5 + (45*a^4*b^3 + 42*a^2*b^3 + 5
*b^3)*x^4 + 2*(9*a^5*b^2 + 14*a^3*b^2 + 5*a*b^2)*x^3 + (3*a^6*b + 7*a^4*b + 5*a^2*b + b)*x^2)*sqrt(b^2*x^2 + 2
*a*b*x + a^2 + 1))/((b^8*x^6 + 6*a*b^7*x^5 + a^6*b^2 + 3*a^4*b^2 + 3*(5*a^2*b^6 + b^6)*x^4 + 3*a^2*b^2 + 4*(5*
a^3*b^5 + 3*a*b^5)*x^3 + 3*(5*a^4*b^4 + 6*a^2*b^4 + b^4)*x^2 + (b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)
*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 3*(b^6*x^4 + 4*a*b^5*x^3 + a^4*b^2 + a^2*b^2 + (6*a^2*b^4 + b^4)*x^2 +
2*(2*a^3*b^3 + a*b^3)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + b^2 + 6*(a^5*b^3 + 2*a^3*b^3 + a*b^3)*x + 3*(b^7*x^5
+ 5*a*b^6*x^4 + a^5*b^2 + 2*a^3*b^2 + 2*(5*a^2*b^5 + b^5)*x^3 + a*b^2 + 2*(5*a^3*b^4 + 3*a*b^4)*x^2 + (5*a^4*b
^3 + 6*a^2*b^3 + b^3)*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2
) + integrate(1/2*(9*b^10*x^10 + 82*a*b^9*x^9 + 2*a^10 + 2*(167*a^2*b^8 + 18*b^8)*x^8 + 8*a^8 + 32*(25*a^3*b^7
+ 8*a*b^7)*x^7 + 2*(623*a^4*b^6 + 394*a^2*b^6 + 27*b^6)*x^6 + 12*a^6 + 4*(329*a^5*b^5 + 342*a^3*b^5 + 69*a*b^
5)*x^5 + 4*(238*a^6*b^4 + 365*a^4*b^4 + 144*a^2*b^4 + 9*b^4)*x^4 + 8*a^4 + 16*(29*a^7*b^3 + 61*a^5*b^3 + 39*a^
3*b^3 + 7*a*b^3)*x^3 + (9*b^6*x^6 + 46*a*b^5*x^5 + 2*a^6 + 4*(24*a^2*b^4 + b^4)*x^4 + 4*a^4 + 8*(13*a^3*b^3 +
2*a*b^3)*x^3 + (61*a^4*b^2 + 24*a^2*b^2 - b^2)*x^2 + 2*a^2 + 2*(9*a^5*b + 8*a^3*b - a*b)*x)*(b^2*x^2 + 2*a*b*x
+ a^2 + 1)^2 + (145*a^8*b^2 + 396*a^6*b^2 + 366*a^4*b^2 + 124*a^2*b^2 + 9*b^2)*x^2 + (36*b^7*x^7 + 220*a*b^6*
x^6 + 8*a^7 + 8*(71*a^2*b^5 + 6*b^5)*x^5 + 20*a^5 + 16*(50*a^3*b^4 + 13*a*b^4)*x^4 + (660*a^4*b^3 + 356*a^2*b^
3 + 13*b^3)*x^3 + 16*a^3 + (316*a^5*b^2 + 300*a^3*b^2 + 39*a*b^2)*x^2 + 2*(40*a^6*b + 62*a^4*b + 21*a^2*b - b)
*x + 4*a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (54*b^8*x^8 + 384*a*b^7*x^7 + 12*a^8 + 6*(197*a^2*b^6 + 20*b^6
)*x^6 + 36*a^6 + 12*(171*a^3*b^5 + 52*a*b^5)*x^5 + (2190*a^4*b^4 + 1332*a^2*b^4 + 83*b^4)*x^4 + 38*a^4 + 4*(36
6*a^5*b^3 + 372*a^3*b^3 + 71*a*b^3)*x^3 + (594*a^6*b^2 + 912*a^4*b^2 + 357*a^2*b^2 + 19*b^2)*x^2 + 16*a^2 + 2*
(66*a^7*b + 144*a^5*b + 97*a^3*b + 19*a*b)*x + 2)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*a^2 + 2*(13*a^9*b + 44*a^7
*b + 54*a^5*b + 28*a^3*b + 5*a*b)*x + (36*b^9*x^9 + 292*a*b^8*x^8 + 8*a^9 + 4*(261*a^2*b^7 + 28*b^7)*x^7 + 28*
a^7 + 4*(539*a^3*b^6 + 172*a*b^6)*x^6 + (2828*a^4*b^5 + 1788*a^2*b^5 + 123*b^5)*x^5 + 36*a^5 + (2436*a^5*b^4 +
2540*a^3*b^4 + 519*a*b^4)*x^4 + (1372*a^6*b^3 + 2120*a^4*b^3 + 855*a^2*b^3 + 57*b^3)*x^3 + 20*a^3 + (484*a^7*
b^2 + 1032*a^5*b^2 + 681*a^3*b^2 + 133*a*b^2)*x^2 + 2*(48*a^8*b + 134*a^6*b + 129*a^4*b + 48*a^2*b + 5*b)*x +
4*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((b^10*x^8 + 8*a*b^9*x^7 + a^8*b^2 + 4*a^6*b^2 + 4*(7*a^2*b^8 + b^8)*x
^6 + 6*a^4*b^2 + 8*(7*a^3*b^7 + 3*a*b^7)*x^5 + 2*(35*a^4*b^6 + 30*a^2*b^6 + 3*b^6)*x^4 + 4*a^2*b^2 + 8*(7*a^5*
b^5 + 10*a^3*b^5 + 3*a*b^5)*x^3 + (b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2)*(b^2*x^2 + 2
*a*b*x + a^2 + 1)^2 + 4*(7*a^6*b^4 + 15*a^4*b^4 + 9*a^2*b^4 + b^4)*x^2 + 4*(b^7*x^5 + 5*a*b^6*x^4 + a^5*b^2 +
a^3*b^2 + (10*a^2*b^5 + b^5)*x^3 + (10*a^3*b^4 + 3*a*b^4)*x^2 + (5*a^4*b^3 + 3*a^2*b^3)*x)*(b^2*x^2 + 2*a*b*x
+ a^2 + 1)^(3/2) + 6*(b^8*x^6 + 6*a*b^7*x^5 + a^6*b^2 + 2*a^4*b^2 + (15*a^2*b^6 + 2*b^6)*x^4 + a^2*b^2 + 4*(5*
a^3*b^5 + 2*a*b^5)*x^3 + (15*a^4*b^4 + 12*a^2*b^4 + b^4)*x^2 + 2*(3*a^5*b^3 + 4*a^3*b^3 + a*b^3)*x)*(b^2*x^2 +
2*a*b*x + a^2 + 1) + b^2 + 8*(a^7*b^3 + 3*a^5*b^3 + 3*a^3*b^3 + a*b^3)*x + 4*(b^9*x^7 + 7*a*b^8*x^6 + a^7*b^2
+ 3*a^5*b^2 + 3*(7*a^2*b^7 + b^7)*x^5 + 3*a^3*b^2 + 5*(7*a^3*b^6 + 3*a*b^6)*x^4 + (35*a^4*b^5 + 30*a^2*b^5 +
3*b^5)*x^3 + a*b^2 + 3*(7*a^5*b^4 + 10*a^3*b^4 + 3*a*b^4)*x^2 + (7*a^6*b^3 + 15*a^4*b^3 + 9*a^2*b^3 + b^3)*x)*
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{\operatorname{arsinh}\left (b x + a\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arcsinh(b*x + a)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/asinh(a + b*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{arsinh}\left (b x + a\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arcsinh(b*x + a)^3, x)