3.1.57 \(\int \frac {\tanh ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx\) [57]

Optimal. Leaf size=545 \[ \frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {\sqrt {d} \log (1-a-b x) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (1+a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{\sqrt {-c}-a \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (1-a-b x)}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {-c} (1+a+b x)}{(1+a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}} \]

[Out]

1/2*(-b*x-a+1)*ln(-b*x-a+1)/b/c+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c+1/4*ln(-b*x-a+1)*ln(-b*(-x*(-c)^(1/2)+d^(1/2))/(
(1-a)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*ln(b*x+a+1)*ln(-b*(x*(-c)^(1/2)+d^(1/2))/((1+a)*(-c)^(1/2)
-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*ln(-b*x-a+1)*ln(b*(x*(-c)^(1/2)+d^(1/2))/((1-a)*(-c)^(1/2)+b*d^(1/2)))*d^(
1/2)/(-c)^(3/2)-1/4*ln(b*x+a+1)*ln(b*(-x*(-c)^(1/2)+d^(1/2))/((1+a)*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+
1/4*polylog(2,(-b*x-a+1)*(-c)^(1/2)/((-c)^(1/2)-a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/4*polylog(2,(b*x
+a+1)*(-c)^(1/2)/((1+a)*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*polylog(2,(-b*x-a+1)*(-c)^(1/2)/((1-a)*(
-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/4*polylog(2,(b*x+a+1)*(-c)^(1/2)/((1+a)*(-c)^(1/2)+b*d^(1/2)))*d^(1
/2)/(-c)^(3/2)

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Rubi [A]
time = 0.65, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6250, 2456, 2436, 2332, 2441, 2440, 2438} \begin {gather*} \frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{-\sqrt {-c} a+\sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+1)}{\sqrt {-c} (1-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+1)}{\sqrt {-c} (a+1)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (-a-b x+1) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(1-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(1-a) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(a+1) \sqrt {-c}+b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+1) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/(c + d/x^2),x]

[Out]

((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (Sqrt[d]*Log[1 - a - b*x
]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x
]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*L
og[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 - a - b*x]*L
og[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*
(1 - a - b*x))/(Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b
*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*S
qrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqr
t[d])])/(4*(-c)^(3/2))

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2440

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

Rule 2441

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

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 6250

Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Dist[1/2, Int[Log[1 + c + d*x]/(e + f*x
^n), x], x] - Dist[1/2, Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 20.08, size = 1456, normalized size = 2.67 \begin {gather*} \frac {(a+b x) \tanh ^{-1}(a+b x)-\log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )}{b c}+\frac {\sqrt {d} \left (2 i \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )-2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )-2 i \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )+2 i a^2 \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )-2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+b \sqrt {d} \sqrt {\frac {(-1+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+a b \sqrt {d} \sqrt {\frac {(-1+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2+b \sqrt {d} \sqrt {\frac {(1+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2-a b \sqrt {d} \sqrt {\frac {(1+a)^2 c+b^2 d}{b^2 d}} e^{-i \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )^2-4 \left (-1+a^2\right ) \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \tanh ^{-1}(a+b x)+2 \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (1-e^{-2 i \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )-2 \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )+2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )+2 \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )-2 a^2 \sqrt {c} \text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right ) \log \left (-\sin \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )\right )-i \left (-1+a^2\right ) \sqrt {c} \text {PolyLog}\left (2,e^{-2 i \left (\text {ArcTan}\left (\frac {(-1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )+i \left (-1+a^2\right ) \sqrt {c} \text {PolyLog}\left (2,e^{-2 i \left (\text {ArcTan}\left (\frac {(1+a) \sqrt {c}}{b \sqrt {d}}\right )+\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}\right )\right )}{4 \left (-1+a^2\right ) c^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*x)*ArcTanh[a + b*x] - Log[1/Sqrt[1 - (a + b*x)^2]])/(b*c) + (Sqrt[d]*((2*I)*Sqrt[c]*ArcTan[((-1 + a)*S
qrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - (2*I)*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Ar
cTan[(Sqrt[c]*x)/Sqrt[d]] - (2*I)*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] +
(2*I)*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*ArcTan[(Sqrt[c]*x)/Sqrt[d]] - 2*b*Sqrt[d]*ArcTan[(Sqrt
[c]*x)/Sqrt[d]]^2 + (b*Sqrt[d]*Sqrt[((-1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan
[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]) + (a*b*Sqrt[d]*Sqrt[((-1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[
d]]^2)/E^(I*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]) + (b*Sqrt[d]*Sqrt[((1 + a)^2*c + b^2*d)/(b^2*d)]*ArcTan[(S
qrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]) - (a*b*Sqrt[d]*Sqrt[((1 + a)^2*c + b^2*d)/(b
^2*d)]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]^2)/E^(I*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]) - 4*(-1 + a^2)*Sqrt[c]*ArcTa
n[(Sqrt[c]*x)/Sqrt[d]]*ArcTanh[a + b*x] + 2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(
ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[
c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] +
2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt
[c]*x)/Sqrt[d]]))] - 2*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b
*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I
)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] + 2*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt
[c])/(b*Sqrt[d])]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] -
2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[
c]*x)/Sqrt[d]]))] + 2*a^2*Sqrt[c]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[1 - E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*S
qrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]))] - 2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[(
(-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + 2*a^2*Sqrt[c]*ArcTan[((-1 + a)*Sqrt[c])/(b*Sq
rt[d])]*Log[-Sin[ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] + 2*Sqrt[c]*ArcTan[((1
+ a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sqrt[c]*x)/Sqrt[d]]]] - 2
*a^2*Sqrt[c]*ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]*Log[-Sin[ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcTan[(Sq
rt[c]*x)/Sqrt[d]]]] - I*(-1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((-1 + a)*Sqrt[c])/(b*Sqrt[d])] + ArcT
an[(Sqrt[c]*x)/Sqrt[d]]))] + I*(-1 + a^2)*Sqrt[c]*PolyLog[2, E^((-2*I)*(ArcTan[((1 + a)*Sqrt[c])/(b*Sqrt[d])]
+ ArcTan[(Sqrt[c]*x)/Sqrt[d]]))]))/(4*(-1 + a^2)*c^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.90, size = 10288, normalized size = 18.88

method	result	size
risch	\(-\frac {\ln \left (-b x -a +1\right ) x}{2 c}-\frac {\ln \left (-b x -a +1\right ) a}{2 b c}+\frac {\ln \left (-b x -a +1\right )}{2 b c}-\frac {1}{b c}-\frac {d \ln \left (-b x -a +1\right ) \ln \left (\frac {b \sqrt {-d c}-c \left (-b x -a +1\right )-a c +c}{b \sqrt {-d c}-a c +c}\right )}{4 c \sqrt {-d c}}+\frac {d \ln \left (-b x -a +1\right ) \ln \left (\frac {b \sqrt {-d c}+c \left (-b x -a +1\right )+a c -c}{b \sqrt {-d c}+a c -c}\right )}{4 c \sqrt {-d c}}-\frac {d \dilog \left (\frac {b \sqrt {-d c}-c \left (-b x -a +1\right )-a c +c}{b \sqrt {-d c}-a c +c}\right )}{4 c \sqrt {-d c}}+\frac {d \dilog \left (\frac {b \sqrt {-d c}+c \left (-b x -a +1\right )+a c -c}{b \sqrt {-d c}+a c -c}\right )}{4 c \sqrt {-d c}}+\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-d c}-c \left (b x +a +1\right )+a c +c}{b \sqrt {-d c}+a c +c}\right )}{4 c \sqrt {-d c}}+\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-d c}+c \left (b x +a +1\right )-a c -c}{b \sqrt {-d c}-a c -c}\right )}{4 c \sqrt {-d c}}-\frac {d \dilog \left (\frac {b \sqrt {-d c}-c \left (b x +a +1\right )+a c +c}{b \sqrt {-d c}+a c +c}\right )}{4 c \sqrt {-d c}}+\frac {d \dilog \left (\frac {b \sqrt {-d c}+c \left (b x +a +1\right )-a c -c}{b \sqrt {-d c}-a c -c}\right )}{4 c \sqrt {-d c}}\)	\(581\)
derivativedivides	\(\text {Expression too large to display}\)	\(10288\)
default	\(\text {Expression too large to display}\)	\(10288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 651, normalized size = 1.19 \begin {gather*} -{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \operatorname {artanh}\left (b x + a\right ) + \frac {2 \, {\left (a + 1\right )} c \log \left (b x + a + 1\right ) - 2 \, {\left (a - 1\right )} c \log \left (b x + a - 1\right ) + {\left (b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, {\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (\frac {{\left (a - 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (-\frac {{\left (a - 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a + i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a + i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right ) - i \, b {\rm Li}_2\left (\frac {{\left (a + 1\right )} b c x + b^{2} d + {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} + b^{2} d - {\left (a^{2} + 2 \, a + 1\right )} c}\right ) + i \, b {\rm Li}_2\left (-\frac {{\left (a + 1\right )} b c x + b^{2} d - {\left (i \, b^{2} x + {\left (-i \, a - i\right )} b\right )} \sqrt {c} \sqrt {d}}{2 \, {\left (-i \, a - i\right )} b \sqrt {c} \sqrt {d} - b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - {\left (b \arctan \left (\frac {{\left (b^{2} x + {\left (a + 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}, \frac {{\left (a + 1\right )} b c x + {\left (a^{2} + 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} + 2 \, a + 1\right )} c}\right ) - b \arctan \left (\frac {{\left (b^{2} x + {\left (a - 1\right )} b\right )} \sqrt {c} \sqrt {d}}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}, \frac {{\left (a - 1\right )} b c x + {\left (a^{2} - 2 \, a + 1\right )} c}{b^{2} d + {\left (a^{2} - 2 \, a + 1\right )} c}\right )\right )} \log \left (c x^{2} + d\right )\right )} \sqrt {c} \sqrt {d}}{4 \, b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c) - x/c)*arctanh(b*x + a) + 1/4*(2*(a + 1)*c*log(b*x + a + 1) - 2*(a - 1
)*c*log(b*x + a - 1) + (b*arctan(sqrt(c)*x/sqrt(d))*log((b^2*c*x^2 + 2*(a + 1)*b*c*x + (a^2 + 2*a + 1)*c)/(b^2
*d + (a^2 + 2*a + 1)*c)) - b*arctan(sqrt(c)*x/sqrt(d))*log((b^2*c*x^2 + 2*(a - 1)*b*c*x + (a^2 - 2*a + 1)*c)/(
b^2*d + (a^2 - 2*a + 1)*c)) + I*b*dilog(((a - 1)*b*c*x + b^2*d + (I*b^2*x + (-I*a + I)*b)*sqrt(c)*sqrt(d))/(2*
(-I*a + I)*b*sqrt(c)*sqrt(d) + b^2*d - (a^2 - 2*a + 1)*c)) - I*b*dilog(-((a - 1)*b*c*x + b^2*d - (I*b^2*x + (-
I*a + I)*b)*sqrt(c)*sqrt(d))/(2*(-I*a + I)*b*sqrt(c)*sqrt(d) - b^2*d + (a^2 - 2*a + 1)*c)) - I*b*dilog(((a + 1
)*b*c*x + b^2*d + (I*b^2*x + (-I*a - I)*b)*sqrt(c)*sqrt(d))/(2*(-I*a - I)*b*sqrt(c)*sqrt(d) + b^2*d - (a^2 + 2
*a + 1)*c)) + I*b*dilog(-((a + 1)*b*c*x + b^2*d - (I*b^2*x + (-I*a - I)*b)*sqrt(c)*sqrt(d))/(2*(-I*a - I)*b*sq
rt(c)*sqrt(d) - b^2*d + (a^2 + 2*a + 1)*c)) - (b*arctan2((b^2*x + (a + 1)*b)*sqrt(c)*sqrt(d)/(b^2*d + (a^2 + 2
*a + 1)*c), ((a + 1)*b*c*x + (a^2 + 2*a + 1)*c)/(b^2*d + (a^2 + 2*a + 1)*c)) - b*arctan2((b^2*x + (a - 1)*b)*s
qrt(c)*sqrt(d)/(b^2*d + (a^2 - 2*a + 1)*c), ((a - 1)*b*c*x + (a^2 - 2*a + 1)*c)/(b^2*d + (a^2 - 2*a + 1)*c)))*
log(c*x^2 + d))*sqrt(c)*sqrt(d))/(b*c^2)

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

[Out]

integral(x^2*arctanh(b*x + a)/(c*x^2 + d), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a + b*x)/(c + d/x^2),x)

[Out]

int(atanh(a + b*x)/(c + d/x^2), x)

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