Integrand size = 20, antiderivative size = 6177 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx =\text {Too large to display} \]
2*a^2*x/(d*x)^(1/2)-b^2*polylog(2,1-2*(-c)^(1/4)*(1+x^(1/2)*(-c^(1/2))^(1/ 2))/(1+(-c)^(1/4)*x^(1/2))/((-c)^(1/4)+(-c^(1/2))^(1/2)))*x^(1/2)/(-c)^(1/ 4)/(d*x)^(1/2)+b^2*polylog(2,1-2*c^(1/4)*(1+x^(1/2)*(-c^(1/2))^(1/2))/(1+c ^(1/4)*x^(1/2))/(c^(1/4)+(-c^(1/2))^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-b^ 2*x*ln(-c*x^2+1)*ln(c*x^2+1)/(d*x)^(1/2)+I*b^2*polylog(2,1-(1+I)*(1-(-c)^( 1/4)*x^(1/2))/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+I*b ^2*polylog(2,1+(-1+I)*(1+(-c)^(1/4)*x^(1/2))/(1-I*(-c)^(1/4)*x^(1/2)))*x^( 1/2)/(-c)^(1/4)/(d*x)^(1/2)+I*b^2*polylog(2,1-(1+I)*(1-c^(1/4)*x^(1/2))/(1 -I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+I*b^2*polylog(2,1+(-1+I)* (1+c^(1/4)*x^(1/2))/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*a *b*arctan(1+c^(1/4)*2^(1/2)*x^(1/2))*2^(1/2)*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2 *I*b^2*arctan((-c)^(1/4)*x^(1/2))^2*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*I*b^2 *arctan(c^(1/4)*x^(1/2))^2*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*I*b^2*polylog(2,1 -2/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*I*b^2*polylo g(2,1-2/(1+I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*I*b^2*p olylog(2,1-2/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*I*b^2*po lylog(2,1-2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+1/2*b^2*x*l n(-c*x^2+1)^2/(d*x)^(1/2)+1/2*b^2*x*ln(c*x^2+1)^2/(d*x)^(1/2)-b^2*polylog( 2,1-2*(-c)^(1/4)*(1-c^(1/4)*x^(1/2))/((-c)^(1/4)-c^(1/4))/(1+(-c)^(1/4)*x^ (1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-b^2*polylog(2,1+2*c^(1/4)*(1-(-c...
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx \]
Time = 9.88 (sec) , antiderivative size = 5216, normalized size of antiderivative = 0.84, 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}}\) |
(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)...
3.1.91.3.1 Defintions of rubi rules used
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]
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]
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])
\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}}{\sqrt {d x}}d x\]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{\sqrt {d x}}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
-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)
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{\sqrt {d\,x}} \,d x \]