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

Optimal. Leaf size=128 \[ -\frac{1}{2} b e \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d \text{csch}^{-1}(c x)+\frac{1}{2} b e \text{csch}^{-1}(c x)^2-b e \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b e \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]

[Out]

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

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Rubi [A]  time = 0.299569, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 13, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.684, Rules used = {6304, 14, 5789, 12, 6742, 321, 215, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{1}{2} b e \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b c d \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d \text{csch}^{-1}(c x)+\frac{1}{2} b e \text{csch}^{-1}(c x)^2-b e \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b e \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^3,x]

[Out]

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

Rule 6304

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5789

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 12

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

Rule 6742

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

Rule 321

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 215

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 2325

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSinh[(Rt[e, 2]*x)/S
qrt[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 5659

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

Rule 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.558171, size = 138, normalized size = 1.08 \[ \frac{1}{4} \left (2 b e \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\frac{2 a d}{x^2}+4 a e \log (x)-\frac{b d \left (-c^2 x^2+c^2 x^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-1\right )}{c x^3 \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{2 b d \text{csch}^{-1}(c x)}{x^2}-2 b e \text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.231, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) \left ( a+b{\rm arccsch} \left (cx\right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (4 \, c^{2} \int \frac{x^{2} \log \left (x\right )}{c^{2} x^{3} + x}\,{d x} - 2 \, c^{2} \int \frac{x \log \left (x\right )}{c^{2} x^{2} +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 1}\,{d x} -{\left (\log \left (c^{2} x^{2} + 1\right ) - 2 \, \log \left (x\right )\right )} \log \left (c\right ) + \log \left (c^{2} x^{2} + 1\right ) \log \left (c\right ) - 2 \, \log \left (x\right ) \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + 2 \, \int \frac{\log \left (x\right )}{c^{2} x^{3} + x}\,{d x}\right )} b e + \frac{1}{8} \, b d{\left (\frac{\frac{2 \, c^{4} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac{4 \, \operatorname{arcsch}\left (c x\right )}{x^{2}}\right )} + a e \log \left (x\right ) - \frac{a d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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