3.274 \(\int (a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2)) \, dx\)

Optimal. Leaf size=104 \[ x \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{e \left (a+b \coth ^{-1}(c x)\right )^2}{b c}-2 a e x+\frac{b \left (e \log \left (1-c^2 x^2\right )+d\right )^2}{4 c e}-\frac{b e \log \left (1-c^2 x^2\right )}{c}-2 b e x \coth ^{-1}(c x) \]

[Out]

-2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (e*(a + b*ArcCoth[c*x])^2)/(b*c) - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCo
th[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*(d + e*Log[1 - c^2*x^2])^2)/(4*c*e)

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Rubi [A]  time = 0.198226, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6074, 2475, 2390, 2301, 5981, 5911, 260, 5949} \[ x \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac{e \left (a+b \coth ^{-1}(c x)\right )^2}{b c}-2 a e x+\frac{b \left (e \log \left (1-c^2 x^2\right )+d\right )^2}{4 c e}-\frac{b e \log \left (1-c^2 x^2\right )}{c}-2 b e x \coth ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

-2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (e*(a + b*ArcCoth[c*x])^2)/(b*c) - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCo
th[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*(d + e*Log[1 - c^2*x^2])^2)/(4*c*e)

Rule 6074

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

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2390

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

Rule 2301

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

Rule 5981

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCoth[c*x])
^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 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 5949

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

Rubi steps

\begin{align*} \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx &=x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-(b c) \int \frac{x \left (d+e \log \left (1-c^2 x^2\right )\right )}{1-c^2 x^2} \, dx+\left (2 c^2 e\right ) \int \frac{x^2 \left (a+b \coth ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+e \log \left (1-c^2 x\right )}{1-c^2 x} \, dx,x,x^2\right )-(2 e) \int \left (a+b \coth ^{-1}(c x)\right ) \, dx+(2 e) \int \frac{a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-2 a e x+\frac{e \left (a+b \coth ^{-1}(c x)\right )^2}{b c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \operatorname{Subst}\left (\int \frac{d+e \log (x)}{x} \, dx,x,1-c^2 x^2\right )}{2 c}-(2 b e) \int \coth ^{-1}(c x) \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{e \left (a+b \coth ^{-1}(c x)\right )^2}{b c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{4 c e}+(2 b c e) \int \frac{x}{1-c^2 x^2} \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{e \left (a+b \coth ^{-1}(c x)\right )^2}{b c}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac{b \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{4 c e}\\ \end{align*}

Mathematica [A]  time = 0.0166919, size = 144, normalized size = 1.38 \[ a e x \log \left (1-c^2 x^2\right )+\frac{2 a e \tanh ^{-1}(c x)}{c}+a d x-2 a e x+\frac{b d \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \log ^2\left (1-c^2 x^2\right )}{4 c}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+b e x \log \left (1-c^2 x^2\right ) \coth ^{-1}(c x)+b d x \coth ^{-1}(c x)+\frac{b e \coth ^{-1}(c x)^2}{c}-2 b e x \coth ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

a*d*x - 2*a*e*x + b*d*x*ArcCoth[c*x] - 2*b*e*x*ArcCoth[c*x] + (b*e*ArcCoth[c*x]^2)/c + (2*a*e*ArcTanh[c*x])/c
+ (b*d*Log[1 - c^2*x^2])/(2*c) - (b*e*Log[1 - c^2*x^2])/c + a*e*x*Log[1 - c^2*x^2] + b*e*x*ArcCoth[c*x]*Log[1
- c^2*x^2] + (b*e*Log[1 - c^2*x^2]^2)/(4*c)

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Maple [C]  time = 0.566, size = 2210, normalized size = 21.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1)),x)

[Out]

1/c*b*e*ln((c*x+1)/(c*x-1)-1)^2-1/c*b*ln((c*x+1)/(c*x-1)-1)*d+2/c*b*ln((c*x+1)/(c*x-1)-1)*e+1/c*b*arccoth(c*x)
*d-2/c*b*e*arccoth(c*x)+b*arccoth(c*x)*x*d+ln((c*x-1)/(c*x+1))*(-arccoth(c*x)*x*c+ln((c*x+1)/(c*x-1)-1)-arccot
h(c*x))*b*e/c-2/c*b*ln(2)*ln((c*x+1)/(c*x-1)-1)*e+2/c*b*arccoth(c*x)*e*ln(2)-2*b*ln((c*x+1)/(c*x-1)-1)*arccoth
(c*x)*x*e+2*b*arccoth(c*x)*ln(2)*x*e-2*a*x*e-2*b*e*x*arccoth(c*x)+a*d*x+a*e*x*ln(-c^2*x^2+1)-a*e/c*ln(c*x-1)+a
*e/c*ln(c*x+1)+1/2*I/c*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x+1)/(c*x-1))^3-1/2*I/c*b*Pi*ln((c*x+1)/(c*x-1)-
1)*e*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3-1/2*I/c*b*Pi*ln((c*x+1)/(c*x-1)-1)*e*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x
-1)-1)^2)^3-1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1))^3+1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/
(c*x-1)-1)^2)^3+1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/2*I/c*b*csgn(I*(
c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I/((c*x+1)/(c*x-1)-1)^2)*e*ln((c*x+1)/(c*x-1)-1)*Pi-1/2*I/c*b*arc
coth(c*x)*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))+I/c*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)
/(c*x-1)-1)^2)^2*e*ln((c*x+1)/(c*x-1)-1)*Pi-I/c*b*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*Pi*e*arccoth
(c*x)-1/2*I*b*Pi*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1))^3*x*e+1/2*I*b*Pi*arccoth(c*x)*csgn(I*((c*x+1)/(c*x-1)-1)
^2)^3*x*e+1/2*I*b*Pi*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3*x*e-I*b*Pi*arccoth(c*x)*csgn
(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x*e+1/2*I/c*b*csgn(I*(c*x+1)/(c*x-1))*csgn(I*(c*x+1)/(c*x-1)/((c*x
+1)/(c*x-1)-1)^2)*csgn(I/((c*x+1)/(c*x-1)-1)^2)*e*ln((c*x+1)/(c*x-1)-1)*Pi-1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I/
((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))-1/2*I*b*Pi*arccot
h(c*x)*csgn(I*(c*x+1)/(c*x-1))*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*x*e
+I/c*b*arccoth(c*x)*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2+I*b*Pi*arccoth(c*x)*csgn(I/
((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*x*e+I/c*b*csgn(I*((c*x+1)/(c*x-1)-1))*csgn(I*((c*x+1)/(c*x-
1)-1)^2)^2*e*ln((c*x+1)/(c*x-1)-1)*Pi+1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2
)^2*csgn(I*(c*x+1)/(c*x-1))-I/c*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1)
)+1/2*I/c*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2+1/2*I/c*b*arccoth(c*
x)*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2-1/2*I*b*Pi*arccoth(c*x)*
csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*x*e+1/2*I*b*Pi*arccoth(c*x)*csgn(I*(c*x+1)/(c*x-1))*
csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x*e-I*b*Pi*arccoth(c*x)*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I
*((c*x+1)/(c*x-1)-1))*x*e+1/2*I*b*Pi*arccoth(c*x)*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2*
x*e+1/2*I*b*Pi*arccoth(c*x)*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x*e+
1/2*I/c*b*csgn(I*(c*x+1)/(c*x-1))*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*e*ln((c*x+1)/(c*x-1)-1)*Pi-I/c*b*csgn(I*(c
*x+1)/(c*x-1))^2*csgn(I/((c*x-1)/(c*x+1))^(1/2))*e*ln((c*x+1)/(c*x-1)-1)*Pi-1/2*I/c*b*csgn(I*(c*x+1)/(c*x-1))*
csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*e*ln((c*x+1)/(c*x-1)-1)*Pi-1/2*I/c*b*csgn(I*((c*x+1)/(c*x-1)-1
))^2*csgn(I*((c*x+1)/(c*x-1)-1)^2)*e*ln((c*x+1)/(c*x-1)-1)*Pi+I/c*b*Pi*e*arccoth(c*x)+I*b*Pi*arccoth(c*x)*x*e-
I/c*b*e*ln((c*x+1)/(c*x-1)-1)*Pi

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Maxima [C]  time = 1.14086, size = 240, normalized size = 2.31 \begin{align*} -{\left (c^{2}{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )} - x \log \left (-c^{2} x^{2} + 1\right )\right )} b e \operatorname{arcoth}\left (c x\right ) -{\left (c^{2}{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )} - x \log \left (-c^{2} x^{2} + 1\right )\right )} a e + a d x + \frac{{\left (2 \, c x \operatorname{arcoth}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} + \frac{{\left ({\left (i \, \pi + 2 \, \log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) +{\left (i \, \pi - 2\right )} \log \left (c x - 1\right )\right )} b e}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxima")

[Out]

-(c^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3) - x*log(-c^2*x^2 + 1))*b*e*arccoth(c*x) - (c^2*(2*x/c^2
- log(c*x + 1)/c^3 + log(c*x - 1)/c^3) - x*log(-c^2*x^2 + 1))*a*e + a*d*x + 1/2*(2*c*x*arccoth(c*x) + log(-c^2
*x^2 + 1))*b*d/c + 1/2*((I*pi + 2*log(c*x - 1) - 2)*log(c*x + 1) + (I*pi - 2)*log(c*x - 1))*b*e/c

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Fricas [A]  time = 1.69862, size = 304, normalized size = 2.92 \begin{align*} \frac{b e \log \left (-c^{2} x^{2} + 1\right )^{2} + b e \log \left (\frac{c x + 1}{c x - 1}\right )^{2} + 4 \,{\left (a c d - 2 \, a c e\right )} x + 2 \,{\left (2 \, a c e x + b d - 2 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 2 \,{\left (b c e x \log \left (-c^{2} x^{2} + 1\right ) + 2 \, a e +{\left (b c d - 2 \, b c e\right )} x\right )} \log \left (\frac{c x + 1}{c x - 1}\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*e*log(-c^2*x^2 + 1)^2 + b*e*log((c*x + 1)/(c*x - 1))^2 + 4*(a*c*d - 2*a*c*e)*x + 2*(2*a*c*e*x + b*d - 2
*b*e)*log(-c^2*x^2 + 1) + 2*(b*c*e*x*log(-c^2*x^2 + 1) + 2*a*e + (b*c*d - 2*b*c*e)*x)*log((c*x + 1)/(c*x - 1))
)/c

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Sympy [A]  time = 9.84375, size = 155, normalized size = 1.49 \begin{align*} \begin{cases} a d x + a e x \log{\left (- c^{2} x^{2} + 1 \right )} - 2 a e x + \frac{2 a e \operatorname{acoth}{\left (c x \right )}}{c} + b d x \operatorname{acoth}{\left (c x \right )} + b e x \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )} - 2 b e x \operatorname{acoth}{\left (c x \right )} + \frac{b d \log{\left (- c^{2} x^{2} + 1 \right )}}{2 c} + \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )}^{2}}{4 c} - \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )}}{c} + \frac{b e \operatorname{acoth}^{2}{\left (c x \right )}}{c} & \text{for}\: c \neq 0 \\d x \left (a + \frac{i \pi b}{2}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acoth(c*x))*(d+e*ln(-c**2*x**2+1)),x)

[Out]

Piecewise((a*d*x + a*e*x*log(-c**2*x**2 + 1) - 2*a*e*x + 2*a*e*acoth(c*x)/c + b*d*x*acoth(c*x) + b*e*x*log(-c*
*2*x**2 + 1)*acoth(c*x) - 2*b*e*x*acoth(c*x) + b*d*log(-c**2*x**2 + 1)/(2*c) + b*e*log(-c**2*x**2 + 1)**2/(4*c
) - b*e*log(-c**2*x**2 + 1)/c + b*e*acoth(c*x)**2/c, Ne(c, 0)), (d*x*(a + I*pi*b/2), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x) + a)*(e*log(-c^2*x^2 + 1) + d), x)