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

Optimal. Leaf size=309 \[ \frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.780008, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 14, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.737, Rules used = {6303, 14, 5790, 12, 6742, 90, 52, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6303

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int
[((e + d*x^2)^p*(a + b*ArcCosh[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 5790

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && 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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 2328

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :>
Dist[Sqrt[1 + (e1*e2*x^2)/(d1*d2)]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(a + b*Log[c*x^n])/Sqrt[1 + (e1*e2*x
^2)/(d1*d2)], x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0]

Rule 2326

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

Rule 4625

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

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, 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{sech}^{-1}(c x)\right )}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right ) \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}}+\frac{2 e \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}\right ) \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{(b d) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )+\frac{\left (b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (2 i b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (i b e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ \end{align*}

Mathematica [A]  time = 0.591147, size = 149, normalized size = 0.48 \[ \frac{1}{4} \left (2 b e \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )-\frac{2 a d}{x^2}+4 a e \log (x)-\frac{b d \sqrt{\frac{1-c x}{c x+1}} \left (-c^2 x^2+c^2 x^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+1\right )}{x^2 (c x-1)}-\frac{2 b d \text{sech}^{-1}(c x)}{x^2}-2 b e \text{sech}^{-1}(c x) \left (\text{sech}^{-1}(c x)+2 \log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.334, size = 170, normalized size = 0.6 \begin{align*} ae\ln \left ( cx \right ) -{\frac{ad}{2\,{x}^{2}}}+{\frac{b \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}e}{2}}+{\frac{bcd}{4\,x}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{b{c}^{2}d{\rm arcsech} \left (cx\right )}{4}}-{\frac{b{\rm arcsech} \left (cx\right )d}{2\,{x}^{2}}}-be{\rm arcsech} \left (cx\right )\ln \left ( 1+ \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) -{\frac{be}{2}{\it polylog} \left ( 2,- \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e*ln(c*x)-1/2/x^2*a*d+1/2*b*arcsech(c*x)^2*e+1/4*c*b*d/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+1/4*b*c^2*
d*arcsech(c*x)-1/2*b*arcsech(c*x)*d/x^2-b*e*arcsech(c*x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-1/2*
b*e*polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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{arsech}\left (c x\right )}{x^{2}}\right )} + b e \int \frac{\log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{x}\,{d x} + 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*arcsech(c*x))/x^3,x, algorithm="maxima")

[Out]

-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*arcsech(c*x)/x^2) + b*e*integrate(log(sqrt(1/(c*x) + 1)*sq
rt(1/(c*x) - 1) + 1/(c*x))/x, x) + 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{arsech}\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*arcsech(c*x))/x^3,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arcsech(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{asech}{\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*asech(c*x))/x**3,x)

[Out]

Integral((a + b*asech(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{arsech}\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*arcsech(c*x))/x^3,x, algorithm="giac")

[Out]

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