3.1.29 \(\int (d+e x) (a+b \text {arctanh}(c x^2))^2 \, dx\) [29]

3.1.29.1 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

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

3.1.29.2 Mathematica [A] (verified)

Time = 2.27 (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=\frac {2 a^2 c d x^2+a^2 c e x^3+4 a b c d x^2 \text {arctanh}\left (c x^2\right )+4 a b d \sqrt {c x^2} \left (\arctan \left (\sqrt {c x^2}\right )-\text {arctanh}\left (\sqrt {c x^2}\right )\right )+b^2 e x \text {arctanh}\left (c x^2\right ) \left (\left (-1+c x^2\right ) \text {arctanh}\left (c x^2\right )-2 \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+a b e x \left (2 c x^2 \text {arctanh}\left (c x^2\right )+\log \left (1-c^2 x^4\right )\right )+b^2 e x \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )-b^2 d \sqrt {c x^2} \left (2 i \arctan \left (\sqrt {c x^2}\right )^2-4 \arctan \left (\sqrt {c x^2}\right ) \text {arctanh}\left (c x^2\right )-2 \sqrt {c x^2} \text {arctanh}\left (c x^2\right )^2-2 \arctan \left (\sqrt {c x^2}\right ) \log \left (1+e^{4 i \arctan \left (\sqrt {c x^2}\right )}\right )-2 \text {arctanh}\left (c x^2\right ) \log \left (1-\sqrt {c x^2}\right )+\log (2) \log \left (1-\sqrt {c x^2}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {c x^2}\right )\right )+2 \text {arctanh}\left (c x^2\right ) \log \left (1+\sqrt {c x^2}\right )-\log (2) \log \left (1+\sqrt {c x^2}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {c x^2}\right )\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{4 i \arctan \left (\sqrt {c x^2}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {c x^2}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )\right )}{2 c x} \]

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)
 

3.1.29.3 Rubi [A] (verified)

Time = 2.10 (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)])/...
 

3.1.29.3.1 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]
 

3.1.29.4 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)
 

3.1.29.5 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)
 

3.1.29.6 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)
 

3.1.29.7 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)
 

3.1.29.8 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)
 

3.1.29.9 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)