3.1.97 \(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x^3} \, dx\) [97]

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

[Out]

-1/4*b*c^2*d^2*arccsch(c*x)+b*d*e*arccsch(c*x)^2-1/2*d^2*(a+b*arccsch(c*x))/x^2+1/2*e^2*x^2*(a+b*arccsch(c*x))
-2*b*d*e*arccsch(c*x)*ln(1-(1/c/x+(1+1/c^2/x^2)^(1/2))^2)+2*b*d*e*arccsch(c*x)*ln(1/x)-2*d*e*(a+b*arccsch(c*x)
)*ln(1/x)-b*d*e*polylog(2,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)+1/4*b*c*d^2*(1+1/c^2/x^2)^(1/2)/x+1/2*b*e^2*x*(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 = 14, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6439, 272, 45, 5822, 12, 6874, 270, 327, 221, 2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {\frac {1}{c^2 x^2}+1}}{4 x}-\frac {1}{4} b c^2 d^2 \text {csch}^{-1}(c x)+\frac {b e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}-b d e \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )+b d e \text {csch}^{-1}(c x)^2-2 b d e \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+2 b d e \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^3,x]

[Out]

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

Rule 12

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

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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 327

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

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

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Mathematica [A]
time = 0.59, size = 187, normalized size = 1.05 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2-\frac {2 b d^2 \text {csch}^{-1}(c x)}{x^2}+\frac {2 b e^2 x \left (\sqrt {1+\frac {1}{c^2 x^2}}+c x \text {csch}^{-1}(c x)\right )}{c}-\frac {b d^2 \left (-1-c^2 x^2+c^2 x^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\right )}{c \sqrt {1+\frac {1}{c^2 x^2}} x^3}-4 b d e \text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )+8 a d e \log (x)+4 b d e \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^3,x]

[Out]

((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*d^2*ArcCsch[c*x])/x^2 + (2*b*e^2*x*(Sqrt[1 + 1/(c^2*x^2)] + c*x*ArcCsch[c
*x]))/c - (b*d^2*(-1 - c^2*x^2 + c^2*x^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]]))/(c*Sqrt[1 + 1/(c^2*x^2
)]*x^3) - 4*b*d*e*ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*Poly
Log[2, E^(-2*ArcCsch[c*x])])/4

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x^2+d)^2*(a+b*arccsch(c*x))/x^3,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^3,x, algorithm="maxima")

[Out]

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

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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^3,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^3, 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^{3}}\, 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**3,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)**2/x**3, 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^3,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)/x^3, 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^3} \,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^3,x)

[Out]

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^3, x)

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