3.2.9 \(\int \frac {(d+e x^2)^2 (a+b \text {sech}^{-1}(c x))}{x^3} \, dx\) [109]

Optimal. Leaf size=373 \[ \frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {6438, 272, 45, 5958, 12, 6874, 97, 92, 54, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {2 b d 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 {2 b d 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^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{4 x}-\frac {b e^2 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)^2*(a + b*ArcSech[c*x]))/x^3,x]

[Out]

(b*c*d^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(4*x) - (b*e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(2*c) +
(I*b*d*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^2*ArcSech[c*x]
)/4 - (d^2*(a + b*ArcSech[c*x]))/(2*x^2) + (e^2*x^2*(a + b*ArcSech[c*x]))/2 - (2*b*d*e*Sqrt[1 - 1/(c^2*x^2)]*A
rcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (2*b*d*e*Sqrt[1 - 1/(c^2*x
^2)]*ArcCsc[c*x]*Log[x^(-1)])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - 2*d*e*(a + b*ArcSech[c*x])*Log[x^(-1)]
+ (I*b*d*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 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 54

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 92

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

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Mathematica [A]
time = 0.51, size = 227, normalized size = 0.61 \begin {gather*} \frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2-\frac {2 b e^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c^2}-\frac {2 b d^2 \text {sech}^{-1}(c x)}{x^2}+2 b e^2 x^2 \text {sech}^{-1}(c x)+\frac {b d^2 \sqrt {\frac {1-c x}{1+c x}} \left (\sqrt {1-c x} (1+c x)-2 i c^2 x^2 \sqrt {1+c x} \text {ArcTan}\left (c x+i \sqrt {1-c^2 x^2}\right )\right )}{x^2 \sqrt {1-c x}}-4 b d e \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+8 a d e \log (x)+4 b d e \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)^2*(a + b*ArcSech[c*x]))/x^3,x]

[Out]

((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*e^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/c^2 - (2*b*d^2*ArcSech[c*x])/x^2
+ 2*b*e^2*x^2*ArcSech[c*x] + (b*d^2*Sqrt[(1 - c*x)/(1 + c*x)]*(Sqrt[1 - c*x]*(1 + c*x) - (2*I)*c^2*x^2*Sqrt[1
+ c*x]*ArcTan[c*x + I*Sqrt[1 - c^2*x^2]]))/(x^2*Sqrt[1 - c*x]) - 4*b*d*e*ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1
+ E^(-2*ArcSech[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*PolyLog[2, -E^(-2*ArcSech[c*x])])/4

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Maple [A]
time = 0.96, size = 279, normalized size = 0.75

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} d e}{c^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b d e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b d e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}\right )\)	\(279\)
default	\(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b \mathrm {arcsech}\left (c x \right )^{2} d e}{c^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\mathrm {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\mathrm {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b d e \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b d e \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}\right )\)	\(279\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*(1/2*a/c^2*x^2*e^2-1/2*a*d^2/c^2/x^2+2*a/c^2*d*e*ln(c*x)+b/c^2*arcsech(c*x)^2*d*e+1/4*b*arcsech(c*x)*d^2+1
/4*b*d^2/c/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-1/2*b*arcsech(c*x)*d^2/c^2/x^2+1/2*b/c^2*arcsech(c*x)*x^
2*e^2-1/2*b/c^3*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*e^2*x+1/2*b/c^4*e^2-2*b/c^2*d*e*arcsech(c*x)*ln(1+(1/
c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-b/c^2*d*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 = 0.00, 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*arcsech(c*x))/x^3,x, algorithm="maxima")

[Out]

-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*arcsech(c*x)/x^2) + 1/2*a*x^2*e^2 + 2*a*d*e*log(x) - 1/2
*a*d^2/x^2 + integrate(b*x*e^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x)) + 2*b*d*e*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)^2*(a+b*arcsech(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)*arcsech(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 {asech}{\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*asech(c*x))/x**3,x)

[Out]

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

[Out]

integrate((e*x^2 + d)^2*(b*arcsech(c*x) + a)/x^3, 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 )}^2\,\left (a+b\,\mathrm {acosh}\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*acosh(1/(c*x))))/x^3,x)

[Out]

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

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