3.1.96 \(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x} \, dx\) [96]

Optimal. Leaf size=178 \[ \frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d^2 \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \]

[Out]

1/2*b*d^2*arccsch(c*x)^2+d*e*x^2*(a+b*arccsch(c*x))+1/4*e^2*x^4*(a+b*arccsch(c*x))-b*d^2*arccsch(c*x)*ln(1-(1/
c/x+(1+1/c^2/x^2)^(1/2))^2)+b*d^2*arccsch(c*x)*ln(1/x)-d^2*(a+b*arccsch(c*x))*ln(1/x)-1/2*b*d^2*polylog(2,(1/c
/x+(1+1/c^2/x^2)^(1/2))^2)+1/6*b*(6*c^2*d-e)*e*x*(1+1/c^2/x^2)^(1/2)/c^3+1/12*b*e^2*x^3*(1+1/c^2/x^2)^(1/2)/c

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Rubi [A]
time = 0.29, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {6439, 272, 45, 5822, 6874, 464, 270, 2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e^2 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (6 c^2 d-e\right )}{6 c^3}-\frac {1}{2} b d^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \log \left (\frac {1}{x}\right ) \text {csch}^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x,x]

[Out]

(b*(6*c^2*d - e)*e*Sqrt[1 + 1/(c^2*x^2)]*x)/(6*c^3) + (b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(12*c) + (b*d^2*ArcCsc
h[c*x]^2)/2 + d*e*x^2*(a + b*ArcCsch[c*x]) + (e^2*x^4*(a + b*ArcCsch[c*x]))/4 - b*d^2*ArcCsch[c*x]*Log[1 - E^(
2*ArcCsch[c*x])] + b*d^2*ArcCsch[c*x]*Log[x^(-1)] - d^2*(a + b*ArcCsch[c*x])*Log[x^(-1)] - (b*d^2*PolyLog[2, E
^(2*ArcCsch[c*x])])/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2362

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

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

Rule 5822

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (GtQ[p, 0] || (
IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 6439

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int
[(e + d*x^2)^p*((a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ
[n, 0] && IntegersQ[m, p]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (-\frac {e \left (e+4 d x^2\right )}{4 x^4 \sqrt {1+\frac {x^2}{c^2}}}+\frac {d^2 \log (x)}{\sqrt {1+\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \text {Subst}\left (\int \frac {e+4 d x^2}{x^4 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \text {Subst}\left (\int \frac {\sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )-\frac {\left (b \left (6 c^2 d-e\right ) e\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (2 b d^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (b d^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} \left (b d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {b \left (6 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} b d^2 \text {csch}^{-1}(c x)^2+d e x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )-b d^2 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d^2 \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d^2 \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 148, normalized size = 0.83 \begin {gather*} a d e x^2+\frac {1}{4} a e^2 x^4+\frac {b d e x \left (\sqrt {1+\frac {1}{c^2 x^2}}+c x \text {csch}^{-1}(c x)\right )}{c}+\frac {b e^2 x \left (\sqrt {1+\frac {1}{c^2 x^2}} \left (-2+c^2 x^2\right )+3 c^3 x^3 \text {csch}^{-1}(c x)\right )}{12 c^3}+a d^2 \log (x)+\frac {1}{2} b d^2 \left (-\text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x,x]

[Out]

a*d*e*x^2 + (a*e^2*x^4)/4 + (b*d*e*x*(Sqrt[1 + 1/(c^2*x^2)] + c*x*ArcCsch[c*x]))/c + (b*e^2*x*(Sqrt[1 + 1/(c^2
*x^2)]*(-2 + c^2*x^2) + 3*c^3*x^3*ArcCsch[c*x]))/(12*c^3) + a*d^2*Log[x] + (b*d^2*(-(ArcCsch[c*x]*(ArcCsch[c*x
] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch[c*x])]))/2

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2*(a+b*arccsch(c*x))/x,x)

[Out]

int((e*x^2+d)^2*(a+b*arccsch(c*x))/x,x)

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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((e*x^2+d)^2*(a+b*arccsch(c*x))/x,x, algorithm="maxima")

[Out]

1/4*a*x^4*e^2 + 4*b*c^2*d^2*integrate(1/4*x*log(x)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) +
1), x) + a*d*x^2*e - b*d^2*log(c)*log(x) - 1/4*(2*log(c^2*x^2 + 1)*log(x) + dilog(-c^2*x^2))*b*d^2 + a*d^2*log
(x) + 1/2*b*d*(2*sqrt(c^2*x^2 + 1) - log(c^2*x^2 + 1))*e/c^2 - 1/8*(2*b*c^2*x^4*e^2*log(c) + 4*b*c^2*d^2*log(x
)^2 + (8*b*c^2*d*e*log(c) - b*e^2)*x^2 + 2*(b*c^2*x^4*e^2 + 4*b*c^2*d*x^2*e)*log(x) - 2*(b*c^2*x^4*e^2 + 4*b*c
^2*d*x^2*e + 4*b*c^2*d^2*log(x))*log(sqrt(c^2*x^2 + 1) + 1))/c^2 - 1/24*(3*c^2*x^2 - 2*(c^2*x^2 + 1)^(3/2) + 6
*sqrt(c^2*x^2 + 1) - 3*log(c^2*x^2 + 1) + 3)*b*e^2/c^4 + 1/8*(4*b*c^2*d*e - b*e^2)*log(c^2*x^2 + 1)/c^4

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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((e*x^2+d)^2*(a+b*arccsch(c*x))/x,x, algorithm="fricas")

[Out]

integral((a*x^4*e^2 + 2*a*d*x^2*e + a*d^2 + (b*x^4*e^2 + 2*b*d*x^2*e + b*d^2)*arccsch(c*x))/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*acsch(c*x))/x,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)**2/x, 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((e*x^2+d)^2*(a+b*arccsch(c*x))/x,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x,x)

[Out]

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x, x)

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