\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{(d x)^{3/2}} \, dx\) [92]

Optimal result

Integrand size = 20, antiderivative size = 6281 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/2*b^2*ln(c*x^2+1)^2/d/(d*x)^(1/2)+2*2^(1/2)*a*b*c^(1/4)*x^(1/2)*arctan( 
-1+2^(1/2)*c^(1/4)*x^(1/2))/d/(d*x)^(1/2)-2*2^(1/2)*a*b*c^(1/4)*x^(1/2)*ar 
ctanh(2^(1/2)*c^(1/4)*x^(1/2)/(1+c^(1/2)*x))/d/(d*x)^(1/2)+2*2^(1/2)*a*b*c 
^(1/4)*x^(1/2)*arctan(1+2^(1/2)*c^(1/4)*x^(1/2))/d/(d*x)^(1/2)-2*b^2*(-c)^ 
(1/4)*x^(1/2)*polylog(2,1-2/(1+(-c)^(1/4)*x^(1/2)))/d/(d*x)^(1/2)-2*b^2*(- 
c)^(1/4)*x^(1/2)*polylog(2,1-2/(1-(-c)^(1/4)*x^(1/2)))/d/(d*x)^(1/2)-2*b^2 
*c^(1/4)*x^(1/2)*polylog(2,1-2/(1+c^(1/4)*x^(1/2)))/d/(d*x)^(1/2)-2*b^2*c^ 
(1/4)*x^(1/2)*polylog(2,1-2/(1-c^(1/4)*x^(1/2)))/d/(d*x)^(1/2)+2*b^2*(-c)^ 
(1/4)*x^(1/2)*arctanh((-c)^(1/4)*x^(1/2))^2/d/(d*x)^(1/2)+2*b^2*c^(1/4)*x^ 
(1/2)*arctanh(c^(1/4)*x^(1/2))^2/d/(d*x)^(1/2)-b^2*c^(1/4)*x^(1/2)*polylog 
(2,1-2*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)))/d/(d*x)^(1/2)-b^2*c^(1/4)*x^(1/2)*polylog(2,1+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) 
))/d/(d*x)^(1/2)+b^2*c^(1/4)*x^(1/2)*polylog(2,1-2*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)))/d/(d*x 
)^(1/2)+b^2*c^(1/4)*x^(1/2)*polylog(2,1+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)))/d/(d*x)^(1/2)+b 
^2*(-c)^(1/4)*x^(1/2)*polylog(2,1-2*(-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)))/d/(d*x)^(1/2)+b^2* 
(-c)^(1/4)*x^(1/2)*polylog(2,1+2*(-c)^(1/4)*(1-(-c^(1/2))^(1/2)*x^(1/2)...
 

Mathematica [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx \] Input:

Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2),x]
 

Output:

Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2), x]
 

Rubi [A] (verified)

Time = 11.44 (sec) , antiderivative size = 5171, normalized size of antiderivative = 0.82, 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.

\(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6466

\(\displaystyle \frac {\sqrt {x} \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^{3/2}}dx}{d \sqrt {d x}}\)

\(\Big \downarrow \) 6458

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2),x]
 

Output:

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

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]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}}{\left (d x \right )^{\frac {3}{2}}}d x\]

Input:

int((a+b*arctanh(c*x^2))^2/(d*x)^(3/2),x)
 

Output:

int((a+b*arctanh(c*x^2))^2/(d*x)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(3/2),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^ 
2*x^2), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{\left (d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*atanh(c*x**2))**2/(d*x)**(3/2),x)
 

Output:

Integral((a + b*atanh(c*x**2))**2/(d*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(3/2),x, algorithm="maxima")
 

Output:

b^2*c*integrate(1/4*x^(3/2)*log(c*x^2 + 1)^2/(c*d^(3/2)*x^3 - d^(3/2)*x), 
x) - 2*b^2*c*integrate(1/4*x^(3/2)*log(c*x^2 + 1)*log(-c*x^2 + 1)/(c*d^(3/ 
2)*x^3 - d^(3/2)*x), x) + 4*a*b*c*integrate(1/4*x^(3/2)*log(c*x^2 + 1)/(c* 
d^(3/2)*x^3 - d^(3/2)*x), x) - 4*a*b*c*integrate(1/4*x^(3/2)*log(-c*x^2 + 
1)/(c*d^(3/2)*x^3 - d^(3/2)*x), x) + 8*b^2*c*integrate(1/4*x^(3/2)*log(-c* 
x^2 + 1)/(c*d^(3/2)*x^3 - d^(3/2)*x), x) + 1/2*a^2*(c*(I*(log(I*c^(1/4)*sq 
rt(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))/d^(3/2) - 4/(d^(3/2)*sqrt( 
x))) - b^2*integrate(1/4*log(c*x^2 + 1)^2/((c*d^(3/2)*x^3 - d^(3/2)*x)*sqr 
t(x)), x) + 2*b^2*integrate(1/4*log(c*x^2 + 1)*log(-c*x^2 + 1)/((c*d^(3/2) 
*x^3 - d^(3/2)*x)*sqrt(x)), x) - 4*a*b*integrate(1/4*log(c*x^2 + 1)/((c*d^ 
(3/2)*x^3 - d^(3/2)*x)*sqrt(x)), x) + 4*a*b*integrate(1/4*log(-c*x^2 + 1)/ 
((c*d^(3/2)*x^3 - d^(3/2)*x)*sqrt(x)), x) - 1/2*a^2*c*(I*(log(I*c^(1/4)*sq 
rt(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))/d^(3/2) - 1/2*b^2*log(-c*x 
^2 + 1)^2/(d^(3/2)*sqrt(x))
 

Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^2) + a)^2/(d*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*atanh(c*x^2))^2/(d*x)^(3/2),x)
 

Output:

int((a + b*atanh(c*x^2))^2/(d*x)^(3/2), x)
 

Reduce [F]

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

int((a+b*atanh(c*x^2))^2/(d*x)^(3/2),x)
 

Output:

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