\(\int (d+e x) (a+b \text {arctanh}(c x^2))^2 \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 1085 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \] Output:

-1/2*b^2*d*polylog(2,1-2*c^(1/2)*(1+(-c)^(1/2)*x)/((-c)^(1/2)+c^(1/2))/(1+ 
c^(1/2)*x))/c^(1/2)-1/2*b^2*d*polylog(2,1+2*c^(1/2)*(1-(-c)^(1/2)*x)/((-c) 
^(1/2)-c^(1/2))/(1+c^(1/2)*x))/c^(1/2)+b^2*d*polylog(2,1-2/(1+c^(1/2)*x))/ 
c^(1/2)+b^2*d*polylog(2,1-2/(1-c^(1/2)*x))/c^(1/2)-b^2*d*arctanh(c^(1/2)*x 
)^2/c^(1/2)-1/2*b^2*e*polylog(2,1-2/(-c*x^2+1))/c+1/4*b^2*d*x*ln(c*x^2+1)^ 
2+1/4*b^2*d*x*ln(-c*x^2+1)^2-1/2*b^2*d*x*ln(-c*x^2+1)*ln(c*x^2+1)-2*b^2*d* 
arctanh(c^(1/2)*x)*ln(2/(1+c^(1/2)*x))/c^(1/2)+2*b^2*d*arctanh(c^(1/2)*x)* 
ln(2/(1-c^(1/2)*x))/c^(1/2)-2*a*b*d*arctanh(c^(1/2)*x)/c^(1/2)+a^2*d*x+2*b 
^2*d*arctan(c^(1/2)*x)*ln(2/(1+I*c^(1/2)*x))/c^(1/2)-2*b^2*d*arctan(c^(1/2 
)*x)*ln(2/(1-I*c^(1/2)*x))/c^(1/2)+2*a*b*d*arctan(c^(1/2)*x)/c^(1/2)-1/2*I 
*b^2*d*polylog(2,1+(-1+I)*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(1/2)-1/2*I*b^2 
*d*polylog(2,1-(1+I)*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(1/2)+b^2*d*arctan(c 
^(1/2)*x)*ln(c*x^2+1)/c^(1/2)+b^2*d*arctanh(c^(1/2)*x)*ln(-c*x^2+1)/c^(1/2 
)-b^2*d*arctan(c^(1/2)*x)*ln(-c*x^2+1)/c^(1/2)+I*b^2*d*polylog(2,1-2/(1+I* 
c^(1/2)*x))/c^(1/2)+I*b^2*d*polylog(2,1-2/(1-I*c^(1/2)*x))/c^(1/2)+b^2*d*a 
rctanh(c^(1/2)*x)*ln(2*c^(1/2)*(1+(-c)^(1/2)*x)/((-c)^(1/2)+c^(1/2))/(1+c^ 
(1/2)*x))/c^(1/2)+b^2*d*arctanh(c^(1/2)*x)*ln(-2*c^(1/2)*(1-(-c)^(1/2)*x)/ 
((-c)^(1/2)-c^(1/2))/(1+c^(1/2)*x))/c^(1/2)+b^2*d*arctan(c^(1/2)*x)*ln((1+ 
I)*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(1/2)+b^2*d*arctan(c^(1/2)*x)*ln((1-I) 
*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(1/2)+I*b^2*d*arctan(c^(1/2)*x)^2/c^(...
 

Mathematica [A] (warning: unable to verify)

Time = 2.08 (sec) , antiderivative size = 684, normalized size of antiderivative = 0.63 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a^2*c*d*x^2 + a^2*c*e*x^3 + 4*a*b*c*d*x^2*ArcTanh[c*x^2] + 4*a*b*d*Sqrt 
[c*x^2]*(ArcTan[Sqrt[c*x^2]] - ArcTanh[Sqrt[c*x^2]]) + b^2*e*x*ArcTanh[c*x 
^2]*((-1 + c*x^2)*ArcTanh[c*x^2] - 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a*b 
*e*x*(2*c*x^2*ArcTanh[c*x^2] + Log[1 - c^2*x^4]) + b^2*e*x*PolyLog[2, -E^( 
-2*ArcTanh[c*x^2])] - b^2*d*Sqrt[c*x^2]*((2*I)*ArcTan[Sqrt[c*x^2]]^2 - 4*A 
rcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] - 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 - 2*Arc 
Tan[Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] - 2*ArcTanh[c*x^2] 
*Log[1 - Sqrt[c*x^2]] + Log[2]*Log[1 - Sqrt[c*x^2]] - Log[1 - Sqrt[c*x^2]] 
^2/2 + Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] + 2*ArcTan 
h[c*x^2]*Log[1 + Sqrt[c*x^2]] - Log[2]*Log[1 + Sqrt[c*x^2]] - Log[((1 + I) 
 - (1 - I)*Sqrt[c*x^2])/2]*Log[1 + Sqrt[c*x^2]] - Log[(-1/2 - I/2)*(I + Sq 
rt[c*x^2])]*Log[1 + Sqrt[c*x^2]] + Log[1 + Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt 
[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] + (I/2)*PolyLog[2, -E^((4* 
I)*ArcTan[Sqrt[c*x^2]])] - PolyLog[2, (1 - Sqrt[c*x^2])/2] + PolyLog[2, (- 
1/2 - I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c*x^2] 
)] + PolyLog[2, (1 + Sqrt[c*x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c* 
x^2])] - PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c*x^2])]))/(2*c*x)
 

Rubi [A] (verified)

Time = 2.09 (sec) , antiderivative size = 1085, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6488, 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.

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

\(\Big \downarrow \) 6488

\(\displaystyle \int \left (d \left (a+b \text {arctanh}\left (c x^2\right )\right )^2+e x \left (a+b \text {arctanh}\left (c x^2\right )\right )^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle d x a^2+\frac {2 b d \arctan \left (\sqrt {c} x\right ) a}{\sqrt {c}}-\frac {2 b d \text {arctanh}\left (\sqrt {c} x\right ) a}{\sqrt {c}}-b d x \log \left (1-c x^2\right ) a+b d x \log \left (c x^2+1\right ) a+\frac {i b^2 d \arctan \left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {b^2 d \text {arctanh}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{2} e x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2+\frac {e \left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 c}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (c x^2+1\right )+\frac {2 b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {2 b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{\sqrt {c}}+\frac {b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{\sqrt {c}}+\frac {b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b e \left (a+b \text {arctanh}\left (c x^2\right )\right ) \log \left (\frac {2}{1-c x^2}\right )}{c}-\frac {b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \arctan \left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {b^2 d \text {arctanh}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (c x^2+1\right )+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {b^2 d \operatorname {PolyLog}\left (2,\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right )}{2 \sqrt {c}}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{2 \sqrt {c}}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}-\frac {b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{2 c}\)

Input:

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

Output:

a^2*d*x + (2*a*b*d*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*d*ArcTan[Sqrt[c]*x] 
^2)/Sqrt[c] - (2*a*b*d*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c 
]*x]^2)/Sqrt[c] + (e*(a + b*ArcTanh[c*x^2])^2)/(2*c) + (e*x^2*(a + b*ArcTa 
nh[c*x^2])^2)/2 + (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt 
[c] - (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2* 
d*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt 
[c] + (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^ 
2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTanh[Sq 
rt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt 
[c]*x))])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c] 
*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*d*ArcTan[Sqrt 
[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (b*e*(a 
 + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c - a*b*d*x*Log[1 - c*x^2] - (b^2 
*d*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*L 
og[1 - c*x^2])/Sqrt[c] + (b^2*d*x*Log[1 - c*x^2]^2)/4 + a*b*d*x*Log[1 + c* 
x^2] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*d*ArcTanh[S 
qrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*d*x*Log[1 - c*x^2]*Log[1 + c*x^2] 
)/2 + (b^2*d*x*Log[1 + c*x^2]^2)/4 + (b^2*d*PolyLog[2, 1 - 2/(1 - Sqrt[c]* 
x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] - ((I 
/2)*b^2*d*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/...
 

Defintions of rubi rules used

rule 2009

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

rule 6488

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x^n])^p, (d + e*x)^m, x 
], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 0]
 

Maple [F]

\[\int \left (e x +d \right ) {\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*atanh(c*x**2))**2*(d + e*x), x)
 

Maxima [F]

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

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

Output:

1/2*a^2*e*x^2 + (c*(2*arctan(sqrt(c)*x)/c^(3/2) + log((c*x - sqrt(c))/(c*x 
 + sqrt(c)))/c^(3/2)) + 2*x*arctanh(c*x^2))*a*b*d + a^2*d*x + 1/2*(2*c*x^2 
*arctanh(c*x^2) + log(-c^2*x^4 + 1))*a*b*e/c + 1/8*(b^2*e*x^2 + 2*b^2*d*x) 
*log(-c*x^2 + 1)^2 - integrate(-1/4*((b^2*c*e*x^3 + b^2*c*d*x^2 - b^2*e*x 
- b^2*d)*log(c*x^2 + 1)^2 - 2*(b^2*c*e*x^3 + 2*b^2*c*d*x^2 + (b^2*c*e*x^3 
+ b^2*c*d*x^2 - b^2*e*x - b^2*d)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^2 - 
 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {4 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) a b d +4 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) a b d +4 \mathit {atanh} \left (c \,x^{2}\right ) a b c d x +2 \mathit {atanh} \left (c \,x^{2}\right ) a b c e \,x^{2}-2 \mathit {atanh} \left (c \,x^{2}\right ) a b e +4 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) a b d -2 \sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) a b d +2 \left (\int \mathit {atanh} \left (c \,x^{2}\right )^{2}d x \right ) b^{2} c d +2 \left (\int \mathit {atanh} \left (c \,x^{2}\right )^{2} x d x \right ) b^{2} c e +2 \,\mathrm {log}\left (c \,x^{2}+1\right ) a b e +2 a^{2} c d x +a^{2} c e \,x^{2}}{2 c} \] Input:

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

Output:

(4*sqrt(c)*atan((c*x)/sqrt(c))*a*b*d + 4*sqrt(c)*atanh(c*x**2)*a*b*d + 4*a 
tanh(c*x**2)*a*b*c*d*x + 2*atanh(c*x**2)*a*b*c*e*x**2 - 2*atanh(c*x**2)*a* 
b*e + 4*sqrt(c)*log(sqrt(c)*x - 1)*a*b*d - 2*sqrt(c)*log(c*x**2 + 1)*a*b*d 
 + 2*int(atanh(c*x**2)**2,x)*b**2*c*d + 2*int(atanh(c*x**2)**2*x,x)*b**2*c 
*e + 2*log(c*x**2 + 1)*a*b*e + 2*a**2*c*d*x + a**2*c*e*x**2)/(2*c)