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

Optimal. Leaf size=296 \[ \frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-d \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.875166, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {6303, 14, 5790, 6742, 95, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-d \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(2*c) + ((I/2)*b*d*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(Sqrt[-1
+ 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (e*x^2*(a + b*ArcSech[c*x]))/2 - (b*d*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*d*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log
[x^(-1)])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - d*(a + b*ArcSech[c*x])*Log[x^(-1)] + ((I/2)*b*d*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 6742

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

Rule 95

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

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} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right ) \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e}{2 x^2}+d \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \left (-\frac{e}{2 x^2 \sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}+\frac{d \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{(b d) \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 e) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{b e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{1}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d \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}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{b d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (2 i b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d \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}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d \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 d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d \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}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d \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 d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (i b d \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 e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d \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}{2} e x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b d \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 d \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}}}-d \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{i b d \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 = 0.291598, size = 98, normalized size = 0.33 \[ \frac{1}{2} \left (b d \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )+2 a d \log (x)+a e x^2-\frac{b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}-b d \text{sech}^{-1}(c x) \left (\text{sech}^{-1}(c x)+2 \log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right )\right )+b e x^2 \text{sech}^{-1}(c x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.365, size = 166, normalized size = 0.6 \begin{align*}{\frac{ae{x}^{2}}{2}}+\ln \left ( cx \right ) ad+{\frac{b \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}d}{2}}-{\frac{bex}{2\,c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{b{\rm arcsech} \left (cx\right ){x}^{2}e}{2}}+{\frac{be}{2\,{c}^{2}}}-bd{\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{bd}{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,x)

[Out]

1/2*a*e*x^2+ln(c*x)*a*d+1/2*b*arcsech(c*x)^2*d-1/2*b/c*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*x*e+1/2*b*arcs
ech(c*x)*x^2*e+1/2/c^2*b*e-b*d*arcsech(c*x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-1/2*b*d*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}{2} \, a e x^{2} + a d \log \left (x\right ) + \int b e x \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right ) + \frac{b d \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{x}\,{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,x, algorithm="maxima")

[Out]

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

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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}, 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,x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*asech(c*x))*(d + e*x**2)/x, 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}\,{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,x, algorithm="giac")

[Out]

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