3.1.98 \(\int \frac {(d+e x^2) (a+b \text {sech}^{-1}(c x))}{x} \, dx\) [98]

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

[Out]

1/2*e*x^2*(a+b*arcsech(c*x))-d*(a+b*arcsech(c*x))*ln(1/x)+1/2*I*b*d*arccsc(c*x)^2*(1-1/c^2/x^2)^(1/2)/(-1+1/c/
x)^(1/2)/(1+1/c/x)^(1/2)-b*d*arccsc(c*x)*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)*(1-1/c^2/x^2)^(1/2)/(-1+1/c/x)^(1
/2)/(1+1/c/x)^(1/2)+b*d*arccsc(c*x)*ln(1/x)*(1-1/c^2/x^2)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1/2*I*b*d*pol
ylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)*(1-1/c^2/x^2)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-1/2*b*e*x*(-1+1/c/x
)^(1/2)*(1+1/c/x)^(1/2)/c

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Rubi [A]
time = 0.60, antiderivative size = 296, normalized size of antiderivative = 1.00, 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 = {6438, 14, 5958, 6874, 97, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -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}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(b*e*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/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 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 97

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 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 2363

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-e, 2]*(x/Sqr
t[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 2365

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

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 3798

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 4721

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

Rule 5958

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 6438

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 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 ) \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx &=-\text {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 \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.20, size = 98, normalized size = 0.33 \begin {gather*} \frac {1}{2} \left (a e x^2-\frac {b e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c^2}+b e x^2 \text {sech}^{-1}(c x)-b d \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+2 a d \log (x)+b d \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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 = 0.76, size = 166, normalized size = 0.56

method	result	size
derivativedivides	\(\frac {a e \,x^{2}}{2}+\ln \left (c x \right ) a d +\frac {b d \mathrm {arcsech}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arcsech}\left (c x \right ) e \,x^{2}}{2}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e x}{2 c}+\frac {b e}{2 c^{2}}-b d \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {b d \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\)	\(166\)
default	\(\frac {a e \,x^{2}}{2}+\ln \left (c x \right ) a d +\frac {b d \mathrm {arcsech}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arcsech}\left (c x \right ) e \,x^{2}}{2}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e x}{2 c}+\frac {b e}{2 c^{2}}-b d \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {b d \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e*x^2+ln(c*x)*a*d+1/2*b*d*arcsech(c*x)^2+1/2*b*arcsech(c*x)*e*x^2-1/2*b/c*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/
c/x)^(1/2)*e*x+1/2*b/c^2*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 = 0.00, 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)*(a+b*arcsech(c*x))/x,x, algorithm="maxima")

[Out]

1/2*a*x^2*e + a*d*log(x) + integrate(b*x*e*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 = 0.00, 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)*(a+b*arcsech(c*x))/x,x, algorithm="fricas")

[Out]

integral((a*x^2*e + a*d + (b*x^2*e + b*d)*arcsech(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 {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \end {gather*}

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 = 0.00, 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)*(a+b*arcsech(c*x))/x,x, algorithm="giac")

[Out]

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

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

[Out]

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

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