\(\int \sqrt {d x} (a+b \text {arctanh}(c x^2))^2 \, dx\) [90]

Optimal result

Integrand size = 20, antiderivative size = 6274 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \] Output:

4/3*b^2*(d*x)^(1/2)*arctan(c^(1/4)*x^(1/2))*ln(2/(1-I*c^(1/4)*x^(1/2)))/c^ 
(3/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)*arctan(c^(1/4)*x^(1/2))*ln(c*x^2+1)/c^(3 
/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)*arctanh((-c)^(1/4)*x^(1/2))*ln(c*x^2+1)/(- 
c)^(3/4)/x^(1/2)-2/3*b^2*(d*x)^(1/2)*arctan((-c)^(1/4)*x^(1/2))*ln(c*x^2+1 
)/(-c)^(3/4)/x^(1/2)-2/3*b^2*(d*x)^(1/2)*arctanh(c^(1/4)*x^(1/2))*ln(c*x^2 
+1)/c^(3/4)/x^(1/2)+2/3*b*(d*x)^(1/2)*arctan(c^(1/4)*x^(1/2))*(2*a-b*ln(-c 
*x^2+1))/c^(3/4)/x^(1/2)-2/3*b^2*(d*x)^(1/2)*arctanh((-c)^(1/4)*x^(1/2))*l 
n(-c*x^2+1)/(-c)^(3/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)*arctan((-c)^(1/4)*x^(1/ 
2))*ln(-c*x^2+1)/(-c)^(3/4)/x^(1/2)-2/3*b*(d*x)^(1/2)*arctanh(c^(1/4)*x^(1 
/2))*(2*a-b*ln(-c*x^2+1))/c^(3/4)/x^(1/2)-2/3*b^2*(d*x)^(1/2)*arctanh((-c) 
^(1/4)*x^(1/2))*ln(2*(-c)^(1/4)*(1+(-(-c)^(1/2))^(1/2)*x^(1/2))/((-(-c)^(1 
/2))^(1/2)+(-c)^(1/4))/(1+(-c)^(1/4)*x^(1/2)))/(-c)^(3/4)/x^(1/2)-2/3*b^2* 
(d*x)^(1/2)*arctanh((-c)^(1/4)*x^(1/2))*ln(-2*(-c)^(1/4)*(1-(-(-c)^(1/2))^ 
(1/2)*x^(1/2))/((-(-c)^(1/2))^(1/2)-(-c)^(1/4))/(1+(-c)^(1/4)*x^(1/2)))/(- 
c)^(3/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)*arctanh((-c)^(1/4)*x^(1/2))*ln(2*(-c) 
^(1/4)*(1+(-c^(1/2))^(1/2)*x^(1/2))/((-c^(1/2))^(1/2)+(-c)^(1/4))/(1+(-c)^ 
(1/4)*x^(1/2)))/(-c)^(3/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)*arctanh((-c)^(1/4)* 
x^(1/2))*ln(-2*(-c)^(1/4)*(1-(-c^(1/2))^(1/2)*x^(1/2))/((-c^(1/2))^(1/2)-( 
-c)^(1/4))/(1+(-c)^(1/4)*x^(1/2)))/(-c)^(3/4)/x^(1/2)+2/3*b^2*(d*x)^(1/2)* 
arctanh((-c)^(1/4)*x^(1/2))*ln(2*(-c)^(1/4)*(1+c^(1/4)*x^(1/2))/((-c)^(...
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 12.42 (sec) , antiderivative size = 5360, 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, 6456, 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.

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

\(\Big \downarrow \) 6466

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

\(\Big \downarrow \) 6458

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {d x} \left (-\frac {i \arctan \left (\sqrt [4]{-c} \sqrt {x}\right )^2 b^2}{3 (-c)^{3/4}}-\frac {i \arctan \left (\sqrt [4]{c} \sqrt {x}\right )^2 b^2}{3 c^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{-c} \sqrt {x}\right )^2 b^2}{3 (-c)^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{c} \sqrt {x}\right )^2 b^2}{3 c^{3/4}}+\frac {1}{12} x^{3/2} \log ^2\left (c x^2+1\right ) b^2+\frac {2 \text {arctanh}\left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (\frac {2}{1-\sqrt [4]{-c} \sqrt {x}}\right ) b^2}{3 (-c)^{3/4}}+\frac {2 \arctan \left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (\frac {2}{1-i \sqrt [4]{-c} \sqrt {x}}\right ) b^2}{3 (-c)^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}-\frac {2 \arctan \left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (\frac {2}{i \sqrt [4]{-c} \sqrt {x}+1}\right ) b^2}{3 (-c)^{3/4}}-\frac {2 \text {arctanh}\left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (\frac {2}{\sqrt [4]{-c} \sqrt {x}+1}\right ) b^2}{3 (-c)^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {2 \text {arctanh}\left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (\frac {2}{1-\sqrt [4]{c} \sqrt {x}}\right ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {2 \arctan \left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (\frac {2}{1-i \sqrt [4]{c} \sqrt {x}}\right ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}-\frac {2 \arctan \left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (\frac {2}{i \sqrt [4]{c} \sqrt {x}+1}\right ) b^2}{3 c^{3/4}}-\frac {2 \text {arctanh}\left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (\frac {2}{\sqrt [4]{c} \sqrt {x}+1}\right ) b^2}{3 c^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 c^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{3 c^{3/4}}+\frac {2}{9} x^{3/2} \log \left (1-c x^2\right ) b^2+\frac {\arctan \left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (1-c x^2\right ) b^2}{3 (-c)^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (1-c x^2\right ) b^2}{3 (-c)^{3/4}}-\frac {\arctan \left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (c x^2+1\right ) b^2}{3 (-c)^{3/4}}+\frac {\arctan \left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (c x^2+1\right ) b^2}{3 c^{3/4}}+\frac {\text {arctanh}\left (\sqrt [4]{-c} \sqrt {x}\right ) \log \left (c x^2+1\right ) b^2}{3 (-c)^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{c} \sqrt {x}\right ) \log \left (c x^2+1\right ) b^2}{3 c^{3/4}}-\frac {1}{6} x^{3/2} \log \left (1-c x^2\right ) \log \left (c x^2+1\right ) b^2+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-c} \sqrt {x}}\right ) b^2}{3 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i \sqrt [4]{-c} \sqrt {x}}\right ) b^2}{3 (-c)^{3/4}}+\frac {i \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 ) b^2}{6 (-c)^{3/4}}+\frac {i \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 ) b^2}{6 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt [4]{-c} \sqrt {x}\right )}{1-i \sqrt [4]{-c} \sqrt {x}}\right ) b^2}{6 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i \sqrt [4]{-c} \sqrt {x}+1}\right ) b^2}{3 (-c)^{3/4}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-c} \sqrt {x}+1}\right ) b^2}{3 (-c)^{3/4}}+\frac {\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 ) b^2}{6 (-c)^{3/4}}+\frac {\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 ) b^2}{6 (-c)^{3/4}}-\frac {\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 ) b^2}{6 (-c)^{3/4}}-\frac {\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 ) b^2}{6 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt [4]{-c} \sqrt {x}+1\right )}{1-i \sqrt [4]{-c} \sqrt {x}}\right ) b^2}{6 (-c)^{3/4}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{c} \sqrt {x}}\right ) b^2}{3 c^{3/4}}+\frac {i \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 ) b^2}{6 (-c)^{3/4}}-\frac {\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 ) b^2}{6 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i \sqrt [4]{c} \sqrt {x}}\right ) b^2}{3 c^{3/4}}+\frac {i \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 ) b^2}{6 c^{3/4}}+\frac {i \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 ) b^2}{6 c^{3/4}}+\frac {i \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 ) b^2}{6 c^{3/4}}+\frac {i \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 ) b^2}{6 c^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt [4]{c} \sqrt {x}\right )}{1-i \sqrt [4]{c} \sqrt {x}}\right ) b^2}{6 c^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i \sqrt [4]{c} \sqrt {x}+1}\right ) b^2}{3 c^{3/4}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{c} \sqrt {x}+1}\right ) b^2}{3 c^{3/4}}-\frac {\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 ) b^2}{6 c^{3/4}}-\frac {\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 ) b^2}{6 c^{3/4}}+\frac {\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 ) b^2}{6 c^{3/4}}+\frac {\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 ) b^2}{6 c^{3/4}}-\frac {\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 ) b^2}{6 c^{3/4}}-\frac {\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 ) b^2}{6 c^{3/4}}+\frac {i \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 ) b^2}{6 (-c)^{3/4}}-\frac {\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 ) b^2}{6 (-c)^{3/4}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt [4]{c} \sqrt {x}+1\right )}{1-i \sqrt [4]{c} \sqrt {x}}\right ) b^2}{6 c^{3/4}}-\frac {4}{9} a x^{3/2} b-\frac {\sqrt {2} a \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right ) b}{3 c^{3/4}}+\frac {\sqrt {2} a \arctan \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right ) b}{3 c^{3/4}}+\frac {a \log \left (\sqrt {c} x-\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right ) b}{3 \sqrt {2} c^{3/4}}-\frac {a \log \left (\sqrt {c} x+\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right ) b}{3 \sqrt {2} c^{3/4}}+\frac {2}{9} x^{3/2} \left (2 a-b \log \left (1-c x^2\right )\right ) b+\frac {\arctan \left (\sqrt [4]{c} \sqrt {x}\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b}{3 c^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{c} \sqrt {x}\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b}{3 c^{3/4}}+\frac {1}{3} a x^{3/2} \log \left (c x^2+1\right ) b+\frac {1}{12} x^{3/2} \left (2 a-b \log \left (1-c x^2\right )\right )^2\right )}{\sqrt {x}}\)

Input:

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

Output:

(2*Sqrt[d*x]*((-4*a*b*x^(3/2))/9 - (Sqrt[2]*a*b*ArcTan[1 - Sqrt[2]*c^(1/4) 
*Sqrt[x]])/(3*c^(3/4)) + (Sqrt[2]*a*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]]) 
/(3*c^(3/4)) - ((I/3)*b^2*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/(-c)^(3/4) - ((I/3 
)*b^2*ArcTan[c^(1/4)*Sqrt[x]]^2)/c^(3/4) - (b^2*ArcTanh[(-c)^(1/4)*Sqrt[x] 
]^2)/(3*(-c)^(3/4)) - (b^2*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(3*c^(3/4)) + (2*b^ 
2*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/ 
4)) + (2*b^2*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])]) 
/(3*(-c)^(3/4)) - (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]))])/(3*(-c)^(3/4)) - (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]))])/(3*(-c)^(3/4)) + (b^2*ArcTan[(-c)^(1/4)*Sqrt[x]] 
*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(- 
c)^(3/4)) - (2*b^2*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt 
[x])])/(3*(-c)^(3/4)) - (2*b^2*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c) 
^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)) - (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]))])/(3*(-c)^(3/4)) - (b^2*ArcTanh[(-c)^(1/4)*Sqr 
t[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + 
 (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)) + (b^2*ArcTanh[...
 

Defintions of rubi rules used

rule 2009

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

rule 6456

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/6*b^2*sqrt(d)*x^(3/2)*log(-c*x^2 + 1)^2 + 1/6*a^2*c*sqrt(d)*(4*x^(3/2)/c 
 - 3*(I*(log(I*c^(1/4)*sqrt(x) + 1) - log(-I*c^(1/4)*sqrt(x) + 1))/c^(3/4) 
 - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqrt(x) + c^(1/4)))/c^(3/4))/c 
) + 3*b^2*c*sqrt(d)*integrate(1/12*x^(5/2)*log(c*x^2 + 1)^2/(c*x^2 - 1), x 
) - 6*b^2*c*sqrt(d)*integrate(1/12*x^(5/2)*log(c*x^2 + 1)*log(-c*x^2 + 1)/ 
(c*x^2 - 1), x) + 12*a*b*c*sqrt(d)*integrate(1/12*x^(5/2)*log(c*x^2 + 1)/( 
c*x^2 - 1), x) - 12*a*b*c*sqrt(d)*integrate(1/12*x^(5/2)*log(-c*x^2 + 1)/( 
c*x^2 - 1), x) - 8*b^2*c*sqrt(d)*integrate(1/12*x^(5/2)*log(-c*x^2 + 1)/(c 
*x^2 - 1), x) + 1/2*a^2*sqrt(d)*(I*(log(I*c^(1/4)*sqrt(x) + 1) - log(-I*c^ 
(1/4)*sqrt(x) + 1))/c^(3/4) - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqr 
t(x) + c^(1/4)))/c^(3/4)) - 3*b^2*sqrt(d)*integrate(1/12*sqrt(x)*log(c*x^2 
 + 1)^2/(c*x^2 - 1), x) + 6*b^2*sqrt(d)*integrate(1/12*sqrt(x)*log(c*x^2 + 
 1)*log(-c*x^2 + 1)/(c*x^2 - 1), x) - 12*a*b*sqrt(d)*integrate(1/12*sqrt(x 
)*log(c*x^2 + 1)/(c*x^2 - 1), x) + 12*a*b*sqrt(d)*integrate(1/12*sqrt(x)*l 
og(-c*x^2 + 1)/(c*x^2 - 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*( - 2*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c) 
)/(c**(1/4)*sqrt(2)))*a*b + 2*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) + 2* 
sqrt(x)*sqrt(c))/(c**(1/4)*sqrt(2)))*a*b + 4*c**(1/4)*atan((sqrt(x)*sqrt(c 
))/c**(1/4))*a*b - 2*c**(1/4)*sqrt(2)*atanh(c*x**2)*a*b + 4*sqrt(x)*atanh( 
c*x**2)*a*b*c*x - c**(1/4)*sqrt(2)*log(c**(1/4) + sqrt(x)*sqrt(c))*a*b - c 
**(1/4)*sqrt(2)*log( - c**(1/4) + sqrt(x)*sqrt(c))*a*b + 2*c**(1/4)*sqrt(2 
)*log( - sqrt(x)*c**(1/4)*sqrt(2) + sqrt(c)*x + 1)*a*b - c**(1/4)*sqrt(2)* 
log(sqrt(c)*x + 1)*a*b - 2*c**(1/4)*log(c**(1/4) + sqrt(x)*sqrt(c))*a*b + 
2*c**(1/4)*log( - c**(1/4) + sqrt(x)*sqrt(c))*a*b + 2*sqrt(x)*a**2*c*x + 3 
*int(sqrt(x)*atanh(c*x**2)**2,x)*b**2*c))/(3*c)