\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{\sqrt {d x}} \, dx\) [91]

Optimal result

Integrand size = 20, antiderivative size = 6127 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 11.69 (sec) , antiderivative size = 5216, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6466, 6458, 6438, 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.

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

\(\Big \downarrow \) 6466

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

\(\Big \downarrow \) 6458

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

\(\Big \downarrow \) 6438

\(\displaystyle \frac {2 \sqrt {x} \int \left (a^2-b \log \left (1-c x^2\right ) a+b \log \left (c x^2+1\right ) a+\frac {1}{4} b^2 \log ^2\left (1-c x^2\right )+\frac {1}{4} b^2 \log ^2\left (c x^2+1\right )-\frac {1}{2} b^2 \log \left (1-c x^2\right ) \log \left (c x^2+1\right )\right )d\sqrt {x}}{\sqrt {d x}}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*Sqrt[x]*(a^2*Sqrt[x] - (Sqrt[2]*a*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] 
)/c^(1/4) + (Sqrt[2]*a*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/c^(1/4) + (I 
*b^2*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/(-c)^(1/4) - (2*a*b*ArcTan[c^(1/4)*Sqrt 
[x]])/c^(1/4) + (I*b^2*ArcTan[c^(1/4)*Sqrt[x]]^2)/c^(1/4) - (b^2*ArcTanh[( 
-c)^(1/4)*Sqrt[x]]^2)/(-c)^(1/4) - (2*a*b*ArcTanh[c^(1/4)*Sqrt[x]])/c^(1/4 
) - (b^2*ArcTanh[c^(1/4)*Sqrt[x]]^2)/c^(1/4) + (2*b^2*ArcTanh[(-c)^(1/4)*S 
qrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(-c)^(1/4) - (2*b^2*ArcTan[(-c)^( 
1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(-c)^(1/4) + (b^2*ArcTan[ 
(-c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*S 
qrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(-c)^(1/4) + (b^ 
2*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x] 
))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(-c)^(1/ 
4) - (b^2*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]) 
)/(1 - I*(-c)^(1/4)*Sqrt[x])])/(-c)^(1/4) + (2*b^2*ArcTan[(-c)^(1/4)*Sqrt[ 
x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(-c)^(1/4) - (2*b^2*ArcTanh[(-c)^(1 
/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(-c)^(1/4) - (b^2*ArcTanh[(- 
c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt 
[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(-c)^(1/4) - (b^2*Ar 
cTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x])) 
/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(-c)^(1/4)...
 

Defintions of rubi rules used

rule 2009

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

rule 6438

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandI 
ntegrand[(a + b*(Log[1 + c*x^n]/2) - b*(Log[1 - c*x^n]/2))^p, x], x] /; Fre 
eQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]
 

rule 6458

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> 
With[{k = Denominator[m]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcT 
anh[c*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1 
] && IGtQ[n, 0] && FractionQ[m]
 

rule 6466

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_)*(x_))^(m_), x_Sym 
bol] :> Simp[d^IntPart[m]*((d*x)^FracPart[m]/x^FracPart[m])   Int[x^m*(a + 
b*ArcTanh[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] & 
& (EqQ[p, 1] || RationalQ[m, n])
 

Maple [F]

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

Input:

int((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x)
 

Output:

int((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x)
 

Fricas [F]

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

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+b*atanh(c*x**2))**2/(d*x)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="maxima")
 

Output:

-1/2*a^2*c*((-I*(log(I*c^(1/4)*sqrt(x) + 1) - log(-I*c^(1/4)*sqrt(x) + 1)) 
/c^(1/4) - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqrt(x) + c^(1/4)))/c^ 
(1/4))/(c*sqrt(d)) - 4*sqrt(x)/(c*sqrt(d))) + b^2*c*integrate(1/4*x^(3/2)* 
log(c*x^2 + 1)^2/(c*sqrt(d)*x^2 - sqrt(d)), x) - 2*b^2*c*integrate(1/4*x^( 
3/2)*log(c*x^2 + 1)*log(-c*x^2 + 1)/(c*sqrt(d)*x^2 - sqrt(d)), x) + 4*a*b* 
c*integrate(1/4*x^(3/2)*log(c*x^2 + 1)/(c*sqrt(d)*x^2 - sqrt(d)), x) - 4*a 
*b*c*integrate(1/4*x^(3/2)*log(-c*x^2 + 1)/(c*sqrt(d)*x^2 - sqrt(d)), x) - 
 8*b^2*c*integrate(1/4*x^(3/2)*log(-c*x^2 + 1)/(c*sqrt(d)*x^2 - sqrt(d)), 
x) + 1/2*b^2*sqrt(x)*log(-c*x^2 + 1)^2/sqrt(d) - b^2*integrate(1/4*log(c*x 
^2 + 1)^2/((c*sqrt(d)*x^2 - sqrt(d))*sqrt(x)), x) + 2*b^2*integrate(1/4*lo 
g(c*x^2 + 1)*log(-c*x^2 + 1)/((c*sqrt(d)*x^2 - sqrt(d))*sqrt(x)), x) - 4*a 
*b*integrate(1/4*log(c*x^2 + 1)/((c*sqrt(d)*x^2 - sqrt(d))*sqrt(x)), x) + 
4*a*b*integrate(1/4*log(-c*x^2 + 1)/((c*sqrt(d)*x^2 - sqrt(d))*sqrt(x)), x 
) + 1/2*a^2*(-I*(log(I*c^(1/4)*sqrt(x) + 1) - log(-I*c^(1/4)*sqrt(x) + 1)) 
/c^(1/4) - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqrt(x) + c^(1/4)))/c^ 
(1/4))/sqrt(d)
 

Giac [F]

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

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((a + b*atanh(c*x^2))^2/(d*x)^(1/2),x)
 

Output:

int((a + b*atanh(c*x^2))^2/(d*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\frac {-2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) a b +2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) a b -4 c^{\frac {3}{4}} \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}}}\right ) a b +2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atanh} \left (c \,x^{2}\right ) a b +4 \sqrt {x}\, \mathit {atanh} \left (c \,x^{2}\right ) a b c +c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) a b +c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) a b -2 c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x +1\right ) a b +c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, x +1\right ) a b -2 c^{\frac {3}{4}} \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) a b +2 c^{\frac {3}{4}} \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) a b +2 \sqrt {x}\, a^{2} c +\left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )^{2}}{\sqrt {x}}d x \right ) b^{2} c}{\sqrt {d}\, c} \] Input:

int((a+b*atanh(c*x^2))^2/(d*x)^(1/2),x)
 

Output:

( - 2*c**(3/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/ 
4)*sqrt(2)))*a*b + 2*c**(3/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) + 2*sqrt(x)*s 
qrt(c))/(c**(1/4)*sqrt(2)))*a*b - 4*c**(3/4)*atan((sqrt(x)*sqrt(c))/c**(1/ 
4))*a*b + 2*c**(3/4)*sqrt(2)*atanh(c*x**2)*a*b + 4*sqrt(x)*atanh(c*x**2)*a 
*b*c + c**(3/4)*sqrt(2)*log(c**(1/4) + sqrt(x)*sqrt(c))*a*b + c**(3/4)*sqr 
t(2)*log( - c**(1/4) + sqrt(x)*sqrt(c))*a*b - 2*c**(3/4)*sqrt(2)*log( - sq 
rt(x)*c**(1/4)*sqrt(2) + sqrt(c)*x + 1)*a*b + c**(3/4)*sqrt(2)*log(sqrt(c) 
*x + 1)*a*b - 2*c**(3/4)*log(c**(1/4) + sqrt(x)*sqrt(c))*a*b + 2*c**(3/4)* 
log( - c**(1/4) + sqrt(x)*sqrt(c))*a*b + 2*sqrt(x)*a**2*c + int(atanh(c*x* 
*2)**2/sqrt(x),x)*b**2*c)/(sqrt(d)*c)