3.282 \(\int \frac {(a+b \coth ^{-1}(c x)) (d+e \log (f+g x^2))}{x^3} \, dx\)

Optimal. Leaf size=712 \[ -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{c x+1}\right )+b c^2 e \log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 f}+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b e g \log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 f} \]

[Out]

a*e*g*ln(x)/f+b*e*g*arccoth(c*x)*ln(2/(c*x+1))/f+b*c^2*e*arctanh(c*x)*ln(2/(c*x+1))-1/2*a*e*g*ln(g*x^2+f)/f-1/
2*b*c*(d+e*ln(g*x^2+f))/x-1/2*(a+b*arccoth(c*x))*(d+e*ln(g*x^2+f))/x^2+1/2*b*c^2*arctanh(c*x)*(d+e*ln(g*x^2+f)
)-1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))/f-1/2*b*c^2*e*arctanh(c
*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))-1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)+x*
g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f-1/2*b*c^2*e*arctanh(c*x)*ln(2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(
-f)^(1/2)+g^(1/2)))+1/2*b*e*g*polylog(2,-1/c/x)/f-1/2*b*e*g*polylog(2,1/c/x)/f-1/2*b*c^2*e*polylog(2,1-2/(c*x+
1))-1/2*b*e*g*polylog(2,1-2/(c*x+1))/f+1/4*b*c^2*e*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2
)-g^(1/2)))+1/4*b*e*g*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))/f+1/4*b*c^2*e*pol
ylog(2,1-2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))+1/4*b*e*g*polylog(2,1-2*c*((-f)^(1/2)+x*g^
(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f+b*c*e*arctan(x*g^(1/2)/f^(1/2))*g^(1/2)/f^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.11, antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 17, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5917, 325, 206, 6086, 6725, 801, 635, 205, 260, 5993, 5913, 5921, 2402, 2315, 2447, 5992, 5920} \[ \frac {1}{4} b c^2 e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )+\frac {1}{4} b c^2 e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )-\frac {1}{2} b c^2 e \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )+\frac {b e g \text {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 f}+\frac {b e g \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{4 f}+\frac {b e g \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{4 f}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )+b c^2 e \log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b e g \log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]

[Out]

(b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f + (b*e*g*ArcCoth[c*x]*Log[2/(1 + c*x)])
/f + b*c^2*e*ArcTanh[c*x]*Log[2/(1 + c*x)] - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f]
- Sqrt[g])*(1 + c*x))])/(2*f) - (b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g]
)*(1 + c*x))])/2 - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(
2*f) - (b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/2 - (a*e*g*
Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) - ((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/(2*
x^2) + (b*c^2*ArcTanh[c*x]*(d + e*Log[f + g*x^2]))/2 + (b*e*g*PolyLog[2, -(1/(c*x))])/(2*f) - (b*e*g*PolyLog[2
, 1/(c*x)])/(2*f) - (b*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/2 - (b*e*g*PolyLog[2, 1 - 2/(1 + c*x)])/(2*f) + (b*c
^2*e*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/4 + (b*e*g*PolyLog[2, 1
- (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*f) + (b*c^2*e*PolyLog[2, 1 - (2*c*(Sqrt
[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/4 + (b*e*g*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))
/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*f)

Rule 205

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 5913

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b*PolyLog[2, -(c*x)^(-1)
])/2, x] - Simp[(b*PolyLog[2, 1/(c*x)])/2, x]) /; FreeQ[{a, b, c}, x]

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 5920

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

Rule 5921

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

Rule 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 5993

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcCoth[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6086

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> Wit
h[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegra
nd[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(2 e g) \int \left (\frac {-a-b c x-b \coth ^{-1}(c x)}{2 x \left (f+g x^2\right )}+\frac {b c^2 x \tanh ^{-1}(c x)}{2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \frac {-a-b c x-b \coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx-\left (b c^2 e g\right ) \int \frac {x \tanh ^{-1}(c x)}{f+g x^2} \, dx\\ &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \left (\frac {-a-b c x}{x \left (f+g x^2\right )}-\frac {b \coth ^{-1}(c x)}{x \left (f+g x^2\right )}\right ) \, dx-\left (b c^2 e g\right ) \int \left (-\frac {\tanh ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\tanh ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} \left (b c^2 e \sqrt {g}\right ) \int \frac {\tanh ^{-1}(c x)}{\sqrt {-f}-\sqrt {g} x} \, dx-\frac {1}{2} \left (b c^2 e \sqrt {g}\right ) \int \frac {\tanh ^{-1}(c x)}{\sqrt {-f}+\sqrt {g} x} \, dx-(e g) \int \frac {-a-b c x}{x \left (f+g x^2\right )} \, dx+(b e g) \int \frac {\coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx\\ &=b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-2 \left (\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\right )+\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx+\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx-(e g) \int \left (-\frac {a}{f x}+\frac {-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx+(b e g) \int \left (\frac {\coth ^{-1}(c x)}{f x}-\frac {g x \coth ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-2 \left (\frac {1}{2} \left (b c^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )\right )-\frac {(e g) \int \frac {-b c f+a g x}{f+g x^2} \, dx}{f}+\frac {(b e g) \int \frac {\coth ^{-1}(c x)}{x} \, dx}{f}-\frac {\left (b e g^2\right ) \int \frac {x \coth ^{-1}(c x)}{f+g x^2} \, dx}{f}\\ &=\frac {a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+(b c e g) \int \frac {1}{f+g x^2} \, dx-\frac {\left (a e g^2\right ) \int \frac {x}{f+g x^2} \, dx}{f}-\frac {\left (b e g^2\right ) \int \left (-\frac {\coth ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\coth ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {\left (b e g^{3/2}\right ) \int \frac {\coth ^{-1}(c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f}-\frac {\left (b e g^{3/2}\right ) \int \frac {\coth ^{-1}(c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-2 \frac {(b c e g) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {(b c e g) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {(b c e g) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f}-2 \frac {(b e g) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 f}\\ &=\frac {b c e \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \text {Li}_2\left (-\frac {1}{c x}\right )}{2 f}-\frac {b e g \text {Li}_2\left (\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )-\frac {b e g \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 f}+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {1}{4} b c^2 e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f}\\ \end {align*}

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Mathematica [C]  time = 5.90, size = 1318, normalized size = 1.85 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]

[Out]

-1/4*(2*a*d*f - 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] - 4*a*e*g*x^2*Log[x] + 2*a*e*g*x^2*Log
[f + g*x^2] + 2*e*f*(a + b*c*x + (b - b*c^2*x^2)*ArcCoth[c*x])*Log[f + g*x^2] + b*c^2*e*f*x^2*(-4*ArcCoth[c*x]
^2 - 4*ArcCoth[c*x]*Log[1 - E^(-2*ArcCoth[c*x])] + 2*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(
c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + 2*ArcCoth[c*x]*Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f) + 2*c*
Sqrt[-f]*Sqrt[g] + g)] + 2*PolyLog[2, E^(-2*ArcCoth[c*x])] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*
f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g
)]) - b*d*(-2*(c*f*x + g*x^2*ArcCoth[c*x]^2 + ArcCoth[c*x]*(f - c^2*f*x^2 + 2*g*x^2*Log[1 + E^(-2*ArcCoth[c*x]
)]) - g*x^2*PolyLog[2, -E^(-2*ArcCoth[c*x])]) + g*x^2*(2*ArcCoth[c*x]*(-ArcCoth[c*x] + Log[1 + (E^(2*ArcCoth[c
*x])*(c^2*f + g))/(-(c^2*f) - 2*c*Sqrt[-f]*Sqrt[g] + g)] + Log[1 + (E^(2*ArcCoth[c*x])*(c^2*f + g))/(-(c^2*f)
+ 2*c*Sqrt[-f]*Sqrt[g] + g)]) + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)
] + PolyLog[2, (E^(2*ArcCoth[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g)])) + b*d*g*x^2*(2*ArcCoth[c
*x]^2 - (4*I)*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)] - 2*ArcCoth[c*x]*(ArcCoth[c*x] +
2*Log[1 + E^(-2*ArcCoth[c*x])]) + 2*(ArcCoth[c*x] - I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCot
h[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*(ArcCoth[c*x
] + I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(
c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*PolyLog[2, -E^(-2*ArcCoth[c*x])] - PolyLog[2, (c^2*f - g + 2*
Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - PolyLog[2, -((-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*A
rcCoth[c*x])*(c^2*f + g)))]) + b*e*g*x^2*(2*ArcCoth[c*x]^2 - (4*I)*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(
Sqrt[-(c^2*f*g)]*x)] - 2*ArcCoth[c*x]*(ArcCoth[c*x] + 2*Log[1 + E^(-2*ArcCoth[c*x])]) + 2*(ArcCoth[c*x] - I*Ar
cSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)
])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*(ArcCoth[c*x] + I*ArcSin[Sqrt[g/(c^2*f + g)]])*Log[(c^2*(-1 + E^(2*Ar
cCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 2*PolyLog[
2, -E^(-2*ArcCoth[c*x])] - PolyLog[2, (c^2*f - g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - Pol
yLog[2, -((-(c^2*f) + g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g)))]))/(f*x^2)

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {arcoth}\left (c x\right ) + a d + {\left (b e \operatorname {arcoth}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="fricas")

[Out]

integral((b*d*arccoth(c*x) + a*d + (b*e*arccoth(c*x) + a*e)*log(g*x^2 + f))/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x) + a)*(e*log(g*x^2 + f) + d)/x^3, x)

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maple [A]  time = 4.68, size = 936, normalized size = 1.31 \[ \frac {d b \,c^{2} \ln \left (c x +1\right )}{4}-\frac {d b \ln \left (c x +1\right )}{4 x^{2}}+\frac {a e g \ln \relax (x )}{f}-\frac {a e g \ln \left (g \,x^{2}+f \right )}{2 f}-\frac {d a}{2 x^{2}}-\frac {b c d}{2 x}+\left (-\frac {b e \ln \left (c x +1\right )}{4 x^{2}}+\frac {e \left (b \,c^{2} \ln \left (c x +1\right ) x^{2}-b \,c^{2} \ln \left (c x -1\right ) x^{2}-2 x b c +b \ln \left (c x -1\right )-2 a \right )}{4 x^{2}}\right ) \ln \left (g \,x^{2}+f \right )-\frac {b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x -1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x -1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g e b c \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{\sqrt {f g}}-\frac {d b \,c^{2} \ln \left (c x -1\right )}{4}+\frac {d b \ln \left (c x -1\right )}{4 x^{2}}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x -1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x -1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x -1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x -1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x -1\right ) \ln \left (c x \right )}{2 f}-\frac {g b e \dilog \left (c x +1\right )}{2 f}-\frac {g b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \dilog \left (\frac {c \sqrt {-f g}-\left (c x -1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \dilog \left (\frac {c \sqrt {-f g}+\left (c x -1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \dilog \left (c x \right )}{2 f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth(c*x))*(d+e*ln(g*x^2+f))/x^3,x)

[Out]

1/4*d*b*c^2*ln(c*x+1)-1/4*d*b*ln(c*x+1)/x^2+a*e*g*ln(x)/f-1/2*a*e*g*ln(g*x^2+f)/f-1/2/x^2*d*a-1/2*b*c*d/x+(-1/
4*b*e/x^2*ln(c*x+1)+1/4*e*(b*c^2*ln(c*x+1)*x^2-b*c^2*ln(c*x-1)*x^2-2*x*b*c+b*ln(c*x-1)-2*a)/x^2)*ln(g*x^2+f)-1
/2*g*b*e*dilog(c*x+1)/f-1/4*b*e*dilog((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2-1/4*b*e*dilog((c*(-
f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2-1/4*b*e*ln(c*x+1)*ln((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1
/2)+g))*c^2-1/4*b*e*ln(c*x+1)*ln((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2-1/4*g*b*e/f*dilog((c*(-f
*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))-
1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(
1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))+g*e*b*c/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/4*d*b*c^2*ln(c*x-1)+1/4*d*
b*ln(c*x-1)/x^2+1/4*b*e*ln(c*x-1)*ln((c*(-f*g)^(1/2)-(c*x-1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*ln(c*x-1)*ln
((c*(-f*g)^(1/2)+(c*x-1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2+1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)-(c*x-1)*g-g)/(c*(-f*g)
^(1/2)-g))+1/4*b*e*dilog((c*(-f*g)^(1/2)-(c*x-1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*dilog((c*(-f*g)^(1/2)+(c
*x-1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2+1/4*g*b*e/f*ln(c*x-1)*ln((c*(-f*g)^(1/2)-(c*x-1)*g-g)/(c*(-f*g)^(1/2)-g))+1
/4*g*b*e/f*ln(c*x-1)*ln((c*(-f*g)^(1/2)+(c*x-1)*g+g)/(c*(-f*g)^(1/2)+g))-1/2*g*b*e/f*ln(c*x-1)*ln(c*x)+1/4*g*b
*e/f*dilog((c*(-f*g)^(1/2)+(c*x-1)*g+g)/(c*(-f*g)^(1/2)+g))-1/2*g*b*e/f*dilog(c*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {arcoth}\left (c x\right )}{x^{2}}\right )} b d - \frac {1}{2} \, {\left (g {\left (\frac {\log \left (g x^{2} + f\right )}{f} - \frac {\log \left (x^{2}\right )}{f}\right )} + \frac {\log \left (g x^{2} + f\right )}{x^{2}}\right )} a e - \frac {1}{4} \, {\left (2 \, c^{2} g \int \frac {x^{2} \log \left (c x + 1\right )}{g x^{3} + f x}\,{d x} - 2 \, c^{2} g \int \frac {x^{2} \log \left (c x - 1\right )}{g x^{3} + f x}\,{d x} + \frac {2 i \, c g {\left (\log \left (\frac {i \, g x}{\sqrt {f g}} + 1\right ) - \log \left (-\frac {i \, g x}{\sqrt {f g}} + 1\right )\right )}}{\sqrt {f g}} - 2 \, g \int \frac {\log \left (c x + 1\right )}{g x^{3} + f x}\,{d x} + 2 \, g \int \frac {\log \left (c x - 1\right )}{g x^{3} + f x}\,{d x} + \frac {{\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )\right )} \log \left (g x^{2} + f\right )}{x^{2}}\right )} b e - \frac {a d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="maxima")

[Out]

1/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arccoth(c*x)/x^2)*b*d - 1/2*(g*(log(g*x^2 + f)/f - log(x^2)
/f) + log(g*x^2 + f)/x^2)*a*e - 1/4*(2*c^2*g*integrate(x^2*log(c*x + 1)/(g*x^3 + f*x), x) - 2*c^2*g*integrate(
x^2*log(c*x - 1)/(g*x^3 + f*x), x) + 2*I*c*g*(log(I*g*x/sqrt(f*g) + 1) - log(-I*g*x/sqrt(f*g) + 1))/sqrt(f*g)
- 2*g*integrate(log(c*x + 1)/(g*x^3 + f*x), x) + 2*g*integrate(log(c*x - 1)/(g*x^3 + f*x), x) + (2*c*x - (c^2*
x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log(c*x - 1))*log(g*x^2 + f)/x^2)*b*e - 1/2*a*d/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acoth(c*x))*(d + e*log(f + g*x^2)))/x^3,x)

[Out]

int(((a + b*acoth(c*x))*(d + e*log(f + g*x^2)))/x^3, 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*acoth(c*x))*(d+e*ln(g*x**2+f))/x**3,x)

[Out]

Timed out

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