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

Optimal. Leaf size=542 \[ -\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}+\frac {b \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.36, antiderivative size = 542, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6303, 5792, 5788, 519, 377, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 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]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 5788

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

Rule 5792

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

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]

Rubi steps

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

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Mathematica [F]  time = 42.09, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (c x\right ) + a}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arcsech(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.17, size = 3326, normalized size = 6.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*c^2*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d
+2*(e*(c^2*d+e))^(1/2)-2*e))-1/2*b/d^2*sum((_R1^2*c^2*d+2*c^2*d+4*e)/(_R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln(
(_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1
=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))-1/4*b/d^2*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2
))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-b/c^2/d^3*e*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^
2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-b/c^4/d^4*e^2*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2
/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))+1/4*b*c^2/d/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(
1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-1/2*b*c^2/d/(c^2*d+e)*arcsech(c*x)^2+1/2*b/c^2/d^3*polylog(2,c^2*d
*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)-b/c^2/d^3*
arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)+2*b/c^2/d^3*arcsech(c*x)^2*e+2*b/c^4/d^4*e^2*arcsech(c*x)^2-1/2*b*(e*(c^2*d
+e))^(1/2)/d^2/(c^2*d+e)*arctanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/(c^2*d*e
+e^2)^(1/2))-5/2*b/d^2/(c^2*d+e)*arcsech(c*x)^2*e-3/4*b/d^2/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*
(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)+5/4*b/d^2*e/(c^2*d+e)*polylog(2,c^2
*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))+b*(e*(c^2*d+e))^(1/2)/d^2/(c
^2*d+e)*arcsech(c*x)^2+1/4*b*(e*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/
x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+1/2*a*c^2/d/(c^2*e*x^2+c^2*d)+a/d^2*ln(c*x)-1/2*a/d^2*ln(c^2*e
*x^2+c^2*d)-1/4*b*c^2/d/e/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+
e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-3*b/c^2/d^3/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1
/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)*e+1/4*b*c^2*(e*(c^2*d+e))^
(1/2)/d/e/(c^2*d+e)*arcsech(c*x)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))
^(1/2)-2*e))-2*b/c^4/d^4*e^2/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2
*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-1/2*b/d^2*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/
2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)-1/2*b*c^2*x^2*arcsech(c*x)*e/(c^2*e*x^2+c^2*d)/d^2+b*ar
csech(c*x)^2/d^2-2*b/c^4/d^4*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/
2)-2*e))*arcsech(c*x)*e^2+b/c^2/d^3*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+
e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)-3/2*b/d^2/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c
/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*(e*(c^2*d+e))^(1/2)+b/c^4/d^4*e^3/(c^2*d+e)*poly
log(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-b/c^4/d^4*e^2/(c^2*
d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+
e))^(1/2)+3*b/c^2/d^3/(c^2*d+e)*arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)*e-3/2*b/c^2/d^3/(c^2*d+e)*polylog(2,c^2*d*(
1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)*e+4*b/c^2/d^
3/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(
c*x)*e^2-2*b/c^4/d^4*e*arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)+5/2*b/d^2*e/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(
1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)+1/2*b*c^2/d/(c^2*d+e)*ln(1-c^2*d*(1/c
/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)+2*b/c^2/d^3/(c^2*d+e)*
polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*e^2-2*b/c^4/d^4
*e^3/(c^2*d+e)*arcsech(c*x)^2-2*b/c^2/d^3*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(
c^2*d+e))^(1/2)-2*e))*arcsech(c*x)*e+1/2*b*(e*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*arcsech(c*x)*ln(1-c^2*d*(1/c/x+(-
1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))-4*b/c^2/d^3/(c^2*d+e)*arcsech(c*x)^2*e^2
+b/c^4/d^4*e*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*(e
*(c^2*d+e))^(1/2)-1/8*b*c^2/d/e/(c^2*d+e)*polylog(2,c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2
*(e*(c^2*d+e))^(1/2)-2*e))*(e*(c^2*d+e))^(1/2)+2*b/c^4/d^4*e^2/(c^2*d+e)*arcsech(c*x)^2*(e*(c^2*d+e))^(1/2)+2*
b/c^4/d^4*e^3/(c^2*d+e)*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*
e))*arcsech(c*x)+2*b/c^4/d^4*ln(1-c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/
2)-2*e))*arcsech(c*x)*e*(e*(c^2*d+e))^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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