3.1.93 \(\int \frac {(a+b \text {arctanh}(c x^2))^2}{(d x)^{5/2}} \, dx\) [93]

3.1.93.1 Optimal result

Integrand size = 20, antiderivative size = 6520 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx =\text {Too large to display} \]

output

-1/6*(2*a-b*ln(-c*x^2+1))^2/d^2/x/(d*x)^(1/2)+2/3*b^2*(-c)^(3/4)*arctan((- 
c)^(1/4)*x^(1/2))*ln((1-I)*(1+(-c)^(1/4)*x^(1/2))/(1-I*(-c)^(1/4)*x^(1/2)) 
)*x^(1/2)/d^2/(d*x)^(1/2)-4/3*b^2*c^(3/4)*arctanh(c^(1/4)*x^(1/2))*ln(2/(1 
-c^(1/4)*x^(1/2)))*x^(1/2)/d^2/(d*x)^(1/2)-2/3*b^2*(-c)^(3/4)*arctan((-c)^ 
(1/4)*x^(1/2))*ln(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)))*x^(1/2)/d^2/(d*x)^(1/2)-2/3*b^2*(-c)^(3/4)*arcta 
nh((-c)^(1/4)*x^(1/2))*ln(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)/d^2/(d*x)^(1/2)+4/3*b^2*c^(3/4)*arct 
an(c^(1/4)*x^(1/2))*ln(2/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/d^2/(d*x)^(1/2)-2/ 
3*b^2*c^(3/4)*arctan(c^(1/4)*x^(1/2))*ln(-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)))*x^(1/2)/d^2/(d*x)^(1/2)-2/3 
*b^2*c^(3/4)*arctan(c^(1/4)*x^(1/2))*ln(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)))*x^(1/2)/d^2/(d*x)^(1/2)+2/3*b 
^2*c^(3/4)*arctan(c^(1/4)*x^(1/2))*ln((1+I)*(1-c^(1/4)*x^(1/2))/(1-I*c^(1/ 
4)*x^(1/2)))*x^(1/2)/d^2/(d*x)^(1/2)-4/3*b^2*c^(3/4)*arctan(c^(1/4)*x^(1/2 
))*ln(2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/d^2/(d*x)^(1/2)+4/3*b^2*c^(3/4)*arc 
tanh(c^(1/4)*x^(1/2))*ln(2/(1+c^(1/4)*x^(1/2)))*x^(1/2)/d^2/(d*x)^(1/2)-2/ 
3*b^2*c^(3/4)*arctanh(c^(1/4)*x^(1/2))*ln(-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)/d^2/(d*x)^(1/2)-2/3*b^ 
2*c^(3/4)*arctanh(c^(1/4)*x^(1/2))*ln(2*c^(1/4)*(1+(-c)^(1/4)*x^(1/2))/...
 

3.1.93.2 Mathematica [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx \]

input

Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2),x]
 

output

Integrate[(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2), x]
 

3.1.93.3 Rubi [A] (warning: unable to verify)

Time = 9.76 (sec) , antiderivative size = 5305, normalized size of antiderivative = 0.81, 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.

\(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6466

\(\displaystyle \frac {\sqrt {x} \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^{5/2}}dx}{d^2 \sqrt {d x}}\)

\(\Big \downarrow \) 6458

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

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

input

Int[(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2),x]
 

output

(2*Sqrt[x]*(-1/3*(Sqrt[2]*a*b*c^(3/4)*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]]) 
 + (Sqrt[2]*a*b*c^(3/4)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/3 - (I/3)*b^2 
*(-c)^(3/4)*ArcTan[(-c)^(1/4)*Sqrt[x]]^2 - (I/3)*b^2*c^(3/4)*ArcTan[c^(1/4 
)*Sqrt[x]]^2 + (b^2*(-c)^(3/4)*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/3 + (b^2*c^( 
3/4)*ArcTanh[c^(1/4)*Sqrt[x]]^2)/3 - (2*b^2*(-c)^(3/4)*ArcTanh[(-c)^(1/4)* 
Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/3 + (2*b^2*(-c)^(3/4)*ArcTan[(-c 
)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/3 - (b^2*(-c)^(3/4)*Ar 
cTan[(-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 - (b^2*(- 
c)^(3/4)*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]))])/3 
 + (b^2*(-c)^(3/4)*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4) 
*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/3 - (2*b^2*(-c)^(3/4)*ArcTan[(-c)^ 
(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/3 + (2*b^2*(-c)^(3/4)*Ar 
cTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/3 + (b^2*(-c)^( 
3/4)*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*S 
qrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/3 + ( 
b^2*(-c)^(3/4)*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sq 
rt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]) 
)])/3 - (b^2*(-c)^(3/4)*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*...
 

3.1.93.3.1 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])
 

3.1.93.4 Maple [F]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}^{2}}{\left (d x \right )^{\frac {5}{2}}}d x\]

input

int((a+b*arctanh(c*x^2))^2/(d*x)^(5/2),x)
 

output

int((a+b*arctanh(c*x^2))^2/(d*x)^(5/2),x)
 

3.1.93.5 Fricas [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(5/2),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^ 
3*x^3), x)
 

3.1.93.6 Sympy [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{\left (d x\right )^{\frac {5}{2}}}\, dx \]

input

integrate((a+b*atanh(c*x**2))**2/(d*x)**(5/2),x)
 

output

Integral((a + b*atanh(c*x**2))**2/(d*x)**(5/2), x)
 

3.1.93.7 Maxima [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(5/2),x, algorithm="maxima")
 

output

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

3.1.93.8 Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]

input

integrate((a+b*arctanh(c*x^2))^2/(d*x)^(5/2),x, algorithm="giac")
 

output

integrate((b*arctanh(c*x^2) + a)^2/(d*x)^(5/2), x)
 

3.1.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]

input

int((a + b*atanh(c*x^2))^2/(d*x)^(5/2),x)
 

output

int((a + b*atanh(c*x^2))^2/(d*x)^(5/2), x)