Integrand size = 20, antiderivative size = 6327 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \]
-8/9*a*b*x*(d*x)^(1/2)+4/9*b^2*x*ln(-c*x^2+1)*(d*x)^(1/2)+4/9*b*x*(2*a-b*l n(-c*x^2+1))*(d*x)^(1/2)+1/6*b^2*x*ln(c*x^2+1)^2*(d*x)^(1/2)+1/3*I*b^2*pol ylog(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)))*(d*x)^(1/2)/(-c)^(3/4)/x^(1/2)+1/3*I*b^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))) *(d*x)^(1/2)/c^(3/4)/x^(1/2)+1/3*I*b^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)))*(d*x)^(1/2)/c^(3/4 )/x^(1/2)+1/3*I*b^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)))*(d*x)^(1/2)/(-c)^(3/4)/x^(1/2)+1/3 *I*b^2*polylog(2,1+2*c^(1/4)*(1-x^(1/2)*(-(-c)^(1/2))^(1/2))/(1-I*c^(1/4)* x^(1/2))/(-c^(1/4)+I*(-(-c)^(1/2))^(1/2)))*(d*x)^(1/2)/c^(3/4)/x^(1/2)+1/3 *I*b^2*polylog(2,1-2*c^(1/4)*(1+x^(1/2)*(-(-c)^(1/2))^(1/2))/(1-I*c^(1/4)* x^(1/2))/(c^(1/4)+I*(-(-c)^(1/2))^(1/2)))*(d*x)^(1/2)/c^(3/4)/x^(1/2)+1/3* I*b^2*polylog(2,1+2*(-c)^(1/4)*(1-x^(1/2)*(-c^(1/2))^(1/2))/(1-I*(-c)^(1/4 )*x^(1/2))/(-(-c)^(1/4)+I*(-c^(1/2))^(1/2)))*(d*x)^(1/2)/(-c)^(3/4)/x^(1/2 )+1/6*x*(2*a-b*ln(-c*x^2+1))^2*(d*x)^(1/2)-1/3*I*b^2*polylog(2,1-(1+I)*(1- c^(1/4)*x^(1/2))/(1-I*c^(1/4)*x^(1/2)))*(d*x)^(1/2)/c^(3/4)/x^(1/2)-2/3*I* b^2*polylog(2,1-2/(1+I*c^(1/4)*x^(1/2)))*(d*x)^(1/2)/c^(3/4)/x^(1/2)-1/3*I *b^2*polylog(2,1+(-1+I)*(1+c^(1/4)*x^(1/2))/(1-I*c^(1/4)*x^(1/2)))*(d*x)^( 1/2)/c^(3/4)/x^(1/2)-2/3*I*b^2*arctan((-c)^(1/4)*x^(1/2))^2*(d*x)^(1/2)...
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx \]
Time = 11.56 (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}}\) |
(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[...
3.1.90.3.1 Defintions of rubi rules used
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]
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])
\[\int \sqrt {d x}\, {\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}d x\]
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { \sqrt {d x} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int \sqrt {d x} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}\, dx \]
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { \sqrt {d x} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
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)
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { \sqrt {d x} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int \sqrt {d\,x}\,{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2 \,d x \]