\(\int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx\) [54]

Optimal result

Integrand size = 16, antiderivative size = 287 \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\frac {\text {arctanh}(a+b x) \log \left (\frac {2 b \left (\sqrt {-c}-\sqrt {d} x\right )}{\left (b \sqrt {-c}-(1-a) \sqrt {d}\right ) (1+a+b x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arctanh}(a+b x) \log \left (\frac {2 b \left (\sqrt {-c}+\sqrt {d} x\right )}{\left (b \sqrt {-c}+(1-a) \sqrt {d}\right ) (1+a+b x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt {-c}-\sqrt {d} x\right )}{\left (b \sqrt {-c}-(1-a) \sqrt {d}\right ) (1+a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt {-c}+\sqrt {d} x\right )}{\left (b \sqrt {-c}+(1-a) \sqrt {d}\right ) (1+a+b x)}\right )}{4 \sqrt {-c} \sqrt {d}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.20 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.27 \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\frac {-\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(-1+a) \sqrt {d}}\right )+\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )+\log (1-a-b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(-1+a) \sqrt {d}}\right )-\log (1+a+b x) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (-1+a+b x)}{b \sqrt {-c}-(-1+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (-1+a+b x)}{b \sqrt {-c}+(-1+a) \sqrt {d}}\right )-\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}-(1+a) \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (1+a+b x)}{b \sqrt {-c}+(1+a) \sqrt {d}}\right )}{4 \sqrt {-c} \sqrt {d}} \] Input:

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

Output:

(-(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (-1 + a)* 
Sqrt[d])]) + Log[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + 
 (1 + a)*Sqrt[d])] + Log[1 - a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sq 
rt[-c] - (-1 + a)*Sqrt[d])] - Log[1 + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]* 
x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])] + PolyLog[2, -((Sqrt[d]*(-1 + a + b*x) 
)/(b*Sqrt[-c] - (-1 + a)*Sqrt[d]))] - PolyLog[2, (Sqrt[d]*(-1 + a + b*x))/ 
(b*Sqrt[-c] + (-1 + a)*Sqrt[d])] - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b 
*Sqrt[-c] - (1 + a)*Sqrt[d]))] + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqr 
t[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])
 

Rubi [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6665, 2856, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx\)

\(\Big \downarrow \) 6665

\(\displaystyle \frac {1}{2} \int \frac {\log (a+b x+1)}{d x^2+c}dx-\frac {1}{2} \int \frac {\log (-a-b x+1)}{d x^2+c}dx\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {1}{2} \int \left (\frac {\sqrt {-c} \log (a+b x+1)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (a+b x+1)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx-\frac {1}{2} \int \left (\frac {\sqrt {-c} \log (-a-b x+1)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \log (-a-b x+1)}{2 c \left (\sqrt {d} x+\sqrt {-c}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (-a-b x+1)}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (-a-b x+1)}{\sqrt {d} (1-a)+b \sqrt {-c}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}-(1-a) \sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{(1-a) \sqrt {d}+b \sqrt {-c}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )+\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} (a+b x+1)}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} (a+b x+1)}{\sqrt {d} (a+1)+b \sqrt {-c}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{(a+1) \sqrt {d}+b \sqrt {-c}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(a+1) \sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}\right )\)

Input:

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

Output:

(-1/2*(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (1 - 
a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) + (Log[1 - a - b*x]*Log[(b*(Sqrt[-c] + Sq 
rt[d]*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[ 
2, -((Sqrt[d]*(1 - a - b*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d]))]/(2*Sqrt[-c]* 
Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a - b*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d 
])]/(2*Sqrt[-c]*Sqrt[d]))/2 + ((Log[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d 
]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - (Log[1 + a + 
 b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])])/(2*S 
qrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + 
 a)*Sqrt[d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/( 
b*Sqrt[-c] + (1 + a)*Sqrt[d])]/(2*Sqrt[-c]*Sqrt[d]))/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[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] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 6665

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

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.55

method	result	size
risch	\(\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (-b x -a +1\right ) d -a d +d}{b \sqrt {-c d}-a d +d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (-b x -a +1\right ) d +a d -d}{b \sqrt {-c d}+a d -d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (-b x -a +1\right ) d -a d +d}{b \sqrt {-c d}-a d +d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (-b x -a +1\right ) d +a d -d}{b \sqrt {-c d}+a d -d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) d +a d +d}{b \sqrt {-c d}+a d +d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) d -a d -d}{b \sqrt {-c d}-a d -d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}-\left (b x +a +1\right ) d +a d +d}{b \sqrt {-c d}+a d +d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+\left (b x +a +1\right ) d -a d -d}{b \sqrt {-c d}-a d -d}\right )}{4 \sqrt {-c d}}\)	\(444\)
derivativedivides	\(\text {Expression too large to display}\)	\(1289\)
default	\(\text {Expression too large to display}\)	\(1289\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{2} + c} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.06 \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{2} + c} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\int \frac {\mathrm {atanh}\left (a+b\,x\right )}{d\,x^2+c} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^2} \, dx=\int \frac {\mathit {atanh} \left (b x +a \right )}{d \,x^{2}+c}d x \] Input:

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

Output:

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