3.25 \(\int x^2 (a+b \text {csch}^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=194 \[ -\frac {b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c^3}+\frac {b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \text {csch}^{-1}(c x)\right )}{c^2}-\frac {b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^3}+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {b^3 \text {Li}_3\left (-e^{\text {csch}^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (e^{\text {csch}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c^3} \]

[Out]

b^2*x*(a+b*arccsch(c*x))/c^2+1/3*x^3*(a+b*arccsch(c*x))^3-b*(a+b*arccsch(c*x))^2*arctanh(1/c/x+(1+1/c^2/x^2)^(
1/2))/c^3+b^3*arctanh((1+1/c^2/x^2)^(1/2))/c^3-b^2*(a+b*arccsch(c*x))*polylog(2,-1/c/x-(1+1/c^2/x^2)^(1/2))/c^
3+b^2*(a+b*arccsch(c*x))*polylog(2,1/c/x+(1+1/c^2/x^2)^(1/2))/c^3+b^3*polylog(3,-1/c/x-(1+1/c^2/x^2)^(1/2))/c^
3-b^3*polylog(3,1/c/x+(1+1/c^2/x^2)^(1/2))/c^3+1/2*b*x^2*(a+b*arccsch(c*x))^2*(1+1/c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.21, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6286, 5452, 4186, 3770, 4182, 2531, 2282, 6589} \[ -\frac {b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c^3}+\frac {b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c^3}+\frac {b^3 \text {PolyLog}\left (3,-e^{\text {csch}^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {PolyLog}\left (3,e^{\text {csch}^{-1}(c x)}\right )}{c^3}+\frac {b^2 x \left (a+b \text {csch}^{-1}(c x)\right )}{c^2}+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c}-\frac {b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^3}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {b^3 \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsch[c*x])^3,x]

[Out]

(b^2*x*(a + b*ArcCsch[c*x]))/c^2 + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x])^2)/(2*c) + (x^3*(a + b*Ar
cCsch[c*x])^3)/3 - (b*(a + b*ArcCsch[c*x])^2*ArcTanh[E^ArcCsch[c*x]])/c^3 + (b^3*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]
])/c^3 - (b^2*(a + b*ArcCsch[c*x])*PolyLog[2, -E^ArcCsch[c*x]])/c^3 + (b^2*(a + b*ArcCsch[c*x])*PolyLog[2, E^A
rcCsch[c*x]])/c^3 + (b^3*PolyLog[3, -E^ArcCsch[c*x]])/c^3 - (b^3*PolyLog[3, E^ArcCsch[c*x]])/c^3

Rule 2282

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 2531

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4182

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 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 6286

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 6589

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

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Mathematica [B]  time = 7.52, size = 548, normalized size = 2.82 \[ \frac {a^3 x^3}{3}+\frac {a^2 b x^2 \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}}{2 c}-\frac {a^2 b \log \left (x \left (\sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{2 c^3}+a^2 b x^3 \text {csch}^{-1}(c x)+\frac {a b^2 \left (2 c^3 x^3 \left (-\frac {4 \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )}{c^3 x^3}+4 \text {csch}^{-1}(c x)^2+2 \cosh \left (2 \text {csch}^{-1}(c x)\right )-\frac {3 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )}{c x}+\frac {3 \text {csch}^{-1}(c x) \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )}{c x}+2 \text {csch}^{-1}(c x) \sinh \left (2 \text {csch}^{-1}(c x)\right )+\text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right ) \sinh \left (3 \text {csch}^{-1}(c x)\right )-\text {csch}^{-1}(c x) \log \left (e^{-\text {csch}^{-1}(c x)}+1\right ) \sinh \left (3 \text {csch}^{-1}(c x)\right )-2\right )+8 \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )\right )}{8 c^3}+\frac {b^3 \left (16 c^3 x^3 \text {csch}^{-1}(c x)^3 \sinh ^4\left (\frac {1}{2} \text {csch}^{-1}(c x)\right )+48 \text {csch}^{-1}(c x) \text {Li}_2\left (-e^{-\text {csch}^{-1}(c x)}\right )-48 \text {csch}^{-1}(c x) \text {Li}_2\left (e^{-\text {csch}^{-1}(c x)}\right )+48 \text {Li}_3\left (-e^{-\text {csch}^{-1}(c x)}\right )-48 \text {Li}_3\left (e^{-\text {csch}^{-1}(c x)}\right )+\frac {\text {csch}^{-1}(c x)^3 \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c x)\right )}{c x}+6 \text {csch}^{-1}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {csch}^{-1}(c x)\right )-4 \text {csch}^{-1}(c x)^3 \coth \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )+24 \text {csch}^{-1}(c x) \coth \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )+24 \text {csch}^{-1}(c x)^2 \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )-24 \text {csch}^{-1}(c x)^2 \log \left (e^{-\text {csch}^{-1}(c x)}+1\right )+4 \text {csch}^{-1}(c x)^3 \tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )-24 \text {csch}^{-1}(c x) \tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )+6 \text {csch}^{-1}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {csch}^{-1}(c x)\right )-48 \log \left (\tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )\right )\right )}{48 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCsch[c*x])^3,x]

[Out]

(a^3*x^3)/3 + (a^2*b*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + a^2*b*x^3*ArcCsch[c*x] - (a^2*b*Log[x*(1 + Sqr
t[(1 + c^2*x^2)/(c^2*x^2)])])/(2*c^3) + (a*b^2*(8*PolyLog[2, -E^(-ArcCsch[c*x])] + 2*c^3*x^3*(-2 + 4*ArcCsch[c
*x]^2 + 2*Cosh[2*ArcCsch[c*x]] - (3*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])])/(c*x) + (3*ArcCsch[c*x]*Log[1 + E
^(-ArcCsch[c*x])])/(c*x) - (4*PolyLog[2, E^(-ArcCsch[c*x])])/(c^3*x^3) + 2*ArcCsch[c*x]*Sinh[2*ArcCsch[c*x]] +
ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])]*Sinh[3*ArcCsch[c*x]] - ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])]*Sinh[3
*ArcCsch[c*x]])))/(8*c^3) + (b^3*(24*ArcCsch[c*x]*Coth[ArcCsch[c*x]/2] - 4*ArcCsch[c*x]^3*Coth[ArcCsch[c*x]/2]
+ 6*ArcCsch[c*x]^2*Csch[ArcCsch[c*x]/2]^2 + (ArcCsch[c*x]^3*Csch[ArcCsch[c*x]/2]^4)/(c*x) + 24*ArcCsch[c*x]^2
*Log[1 - E^(-ArcCsch[c*x])] - 24*ArcCsch[c*x]^2*Log[1 + E^(-ArcCsch[c*x])] - 48*Log[Tanh[ArcCsch[c*x]/2]] + 48
*ArcCsch[c*x]*PolyLog[2, -E^(-ArcCsch[c*x])] - 48*ArcCsch[c*x]*PolyLog[2, E^(-ArcCsch[c*x])] + 48*PolyLog[3, -
E^(-ArcCsch[c*x])] - 48*PolyLog[3, E^(-ArcCsch[c*x])] + 6*ArcCsch[c*x]^2*Sech[ArcCsch[c*x]/2]^2 + 16*c^3*x^3*A
rcCsch[c*x]^3*Sinh[ArcCsch[c*x]/2]^4 - 24*ArcCsch[c*x]*Tanh[ArcCsch[c*x]/2] + 4*ArcCsch[c*x]^3*Tanh[ArcCsch[c*
x]/2]))/(48*c^3)

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fricas [F]  time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \operatorname {arcsch}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname {arcsch}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname {arcsch}\left (c x\right ) + a^{3} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x^2*arccsch(c*x)^3 + 3*a*b^2*x^2*arccsch(c*x)^2 + 3*a^2*b*x^2*arccsch(c*x) + a^3*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsch(c*x) + a)^3*x^2, x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccsch(c*x))^3,x)

[Out]

int(x^2*(a+b*arccsch(c*x))^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*log(sqrt(c^2*x^2 + 1) + 1)^3 + 1/3*a^3*x^3 + 1/4*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c
^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*a^2*
b - integrate(((b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^4 + (b^3*c^2*x^4 + b^3*x^2)*log(x)^3 + (b^3*log(c)^
3 - 3*a*b^2*log(c)^2)*x^2 + 3*((b^3*c^2*log(c) - a*b^2*c^2)*x^4 + (b^3*log(c) - a*b^2)*x^2)*log(x)^2 + (3*(b^3
*c^2*log(c) - a*b^2*c^2)*x^4 + 3*(b^3*log(c) - a*b^2)*x^2 + 3*(b^3*c^2*x^4 + b^3*x^2)*log(x) + ((b^3*c^2*(3*lo
g(c) + 1) - 3*a*b^2*c^2)*x^4 + 3*(b^3*log(c) - a*b^2)*x^2 + 3*(b^3*c^2*x^4 + b^3*x^2)*log(x))*sqrt(c^2*x^2 + 1
))*log(sqrt(c^2*x^2 + 1) + 1)^2 + 3*((b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^4 + (b^3*log(c)^2 - 2*a*b^2*log
(c))*x^2)*log(x) - 3*((b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^4 + (b^3*log(c)^2 - 2*a*b^2*log(c))*x^2 + (b^3
*c^2*x^4 + b^3*x^2)*log(x)^2 + 2*((b^3*c^2*log(c) - a*b^2*c^2)*x^4 + (b^3*log(c) - a*b^2)*x^2)*log(x) + ((b^3*
c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^4 + (b^3*log(c)^2 - 2*a*b^2*log(c))*x^2 + (b^3*c^2*x^4 + b^3*x^2)*log(x)^
2 + 2*((b^3*c^2*log(c) - a*b^2*c^2)*x^4 + (b^3*log(c) - a*b^2)*x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^
2 + 1) + 1) + ((b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^4 + (b^3*c^2*x^4 + b^3*x^2)*log(x)^3 + (b^3*log(c)^
3 - 3*a*b^2*log(c)^2)*x^2 + 3*((b^3*c^2*log(c) - a*b^2*c^2)*x^4 + (b^3*log(c) - a*b^2)*x^2)*log(x)^2 + 3*((b^3
*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^4 + (b^3*log(c)^2 - 2*a*b^2*log(c))*x^2)*log(x))*sqrt(c^2*x^2 + 1))/(c^2
*x^2 + (c^2*x^2 + 1)^(3/2) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asinh(1/(c*x)))^3,x)

[Out]

int(x^2*(a + b*asinh(1/(c*x)))^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acsch(c*x))**3,x)

[Out]

Integral(x**2*(a + b*acsch(c*x))**3, x)

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