3.91 \(\int \frac {(a+b \tanh ^{-1}(c x^2))^2}{\sqrt {d x}} \, dx\)
Optimal. Leaf size=6177 \[ \text {result too large to display} \]
[Out]
2*a^2*x/(d*x)^(1/2)+2*b^2*arctanh(c^(1/4)*x^(1/2))*ln(-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)+2*b^2*arctan(c^(1/4)*x^(1/2))*ln(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)))*x^(1/2)/c^(1/4)/(d*x)^(
1/2)-2*b^2*arctanh((-c)^(1/4)*x^(1/2))*ln(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)+2*b^2*arctanh(c^(1/4)*x^(1/2))*ln(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)+
2*b^2*arctan((-c)^(1/4)*x^(1/2))*ln(-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)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*arctanh((-c)^(1/4)*x^(1/2))*ln(-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)-2*b^2*arctanh(c^(1/4)*x^(1/2))*ln(-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)+2*b^2*arctan((-c)^(1/4)*x^(1/2))*ln(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)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*a
rctanh((-c)^(1/4)*x^(1/2))*ln(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)-2*b^2*arctanh(c^(1/4
)*x^(1/2))^2*x^(1/2)/c^(1/4)/(d*x)^(1/2)-2*b^2*arctanh((-c)^(1/4)*x^(1/2))^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*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*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)-2*
a*b*x*ln(-c*x^2+1)/(d*x)^(1/2)+2*a*b*x*ln(c*x^2+1)/(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-
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*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)+2*b^2*po
lylog(2,1-2/(1-(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*polylog(2,1-2/(1+(-c)^(1/4)*x^(1/2)))
*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*polylog(2,1-2/(1-c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-b^2*polyl
og(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)+2*b^2*polylog(2,1-2/(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)^(
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)^(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)+1/2*b^2*x*ln(-c*x
^2+1)^2/(d*x)^(1/2)+1/2*b^2*x*ln(c*x^2+1)^2/(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*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)-a*b*ln(1+x*
c^(1/2)-c^(1/4)*2^(1/2)*x^(1/2))*2^(1/2)*x^(1/2)/c^(1/4)/(d*x)^(1/2)+a*b*ln(1+x*c^(1/2)+c^(1/4)*2^(1/2)*x^(1/2
))*2^(1/2)*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)-4*b^2*arctan(c^(1/4)*x^(1/2))*ln(2/
(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/c^(1/4)/(d*x)^(1/2
)-2*b^2*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)/c^(1/4)/(d*x)^(1/2
)+4*b^2*arctan(c^(1/4)*x^(1/2))*ln(2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-4*b^2*arctanh(c^(1/4)*
x^(1/2))*ln(2/(1+c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/c^(1/4)/(d*x
)^(1/2)+2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)-2*b^2*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)/c^(1/4)/(d*x)^(1/2)+2*b^2*arctan(c^(1/4)*x^(1/2
))*ln(-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)))*x^(1/
2)/c^(1/4)/(d*x)^(1/2)-2*b^2*arctanh((-c)^(1/4)*x^(1/2))*ln(-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)-2*b^2*arctanh(c^(1/4)*x^(
1/2))*ln(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)+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*polylo
g(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(2,1-2*(-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)/(-c)^(1/4)/(d*x)^(1/2)-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)))*x^(1/2)/c^(1/4)/(d
*x)^(1/2)-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)))*x^(
1/2)/c^(1/4)/(d*x)^(1/2)-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)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-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)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-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)))*x^(1/2)/c^(1/4)/(d*x)^(1/
2)-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)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-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)))*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*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*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*polylog(2,1-2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-4*a*b*arctan(c^(1/4)*x^(1/2))*x^(1
/2)/c^(1/4)/(d*x)^(1/2)-4*a*b*arctanh(c^(1/4)*x^(1/2))*x^(1/2)/c^(1/4)/(d*x)^(1/2)-2*b^2*arctan((-c)^(1/4)*x^(
1/2))*ln(-c*x^2+1)*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*arctan(c^(1/4)*x^(1/2))*ln(-c*x^2+1)*x^(1/2)/c^(1/4)/(
d*x)^(1/2)-2*b^2*arctanh((-c)^(1/4)*x^(1/2))*ln(-c*x^2+1)*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)+2*b^2*arctanh(c^(1/4)
*x^(1/2))*ln(-c*x^2+1)*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*arctan((-c)^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/(-c)^(
1/4)/(d*x)^(1/2)-2*b^2*arctan(c^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*arctanh((-c)^(1/4
)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-2*b^2*arctanh(c^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/c^(1/
4)/(d*x)^(1/2)+4*b^2*arctanh((-c)^(1/4)*x^(1/2))*ln(2/(1-(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-4
*b^2*arctan((-c)^(1/4)*x^(1/2))*ln(2/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)+4*b^
2*arctan((-c)^(1/4)*x^(1/2))*ln(2/(1+I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-4*b^2*arctanh((-c)^
(1/4)*x^(1/2))*ln(2/(1+(-c)^(1/4)*x^(1/2)))*x^(1/2)/(-c)^(1/4)/(d*x)^(1/2)-2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)+4*b^2*arctanh(c^(1/4)*x
^(1/2))*ln(2/(1-c^(1/4)*x^(1/2)))*x^(1/2)/c^(1/4)/(d*x)^(1/2)+2*b^2*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)/(-c)^(1/4)/(d*x)^(1/2)
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Rubi [F] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x],x]
[Out]
Defer[Int][(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x], x]
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx\\ \end {align*}
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Mathematica [F] time = 62.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x],x]
[Out]
Integrate[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x], x]
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{2}\right ) + a^{2}\right )} \sqrt {d x}}{d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="fricas")
[Out]
integral((b^2*arctanh(c*x^2)^2 + 2*a*b*arctanh(c*x^2) + a^2)*sqrt(d*x)/(d*x), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\sqrt {d x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="giac")
[Out]
integrate((b*arctanh(c*x^2) + a)^2/sqrt(d*x), x)
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x)
[Out]
int((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a^{2} c {\left (\frac {-\frac {i \, {\left (\log \left (i \, c^{\frac {1}{4}} \sqrt {x} + 1\right ) - \log \left (-i \, c^{\frac {1}{4}} \sqrt {x} + 1\right )\right )}}{c^{\frac {1}{4}}} - \frac {\log \left (\frac {\sqrt {c} \sqrt {x} - c^{\frac {1}{4}}}{\sqrt {c} \sqrt {x} + c^{\frac {1}{4}}}\right )}{c^{\frac {1}{4}}}}{c \sqrt {d}} - \frac {4 \, \sqrt {x}}{c \sqrt {d}}\right )} + b^{2} c \int \frac {x^{\frac {3}{2}} \log \left (c x^{2} + 1\right )^{2}}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )}}\,{d x} - 2 \, b^{2} c \int \frac {x^{\frac {3}{2}} \log \left (c x^{2} + 1\right ) \log \left (-c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )}}\,{d x} + 4 \, a b c \int \frac {x^{\frac {3}{2}} \log \left (c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )}}\,{d x} - 4 \, a b c \int \frac {x^{\frac {3}{2}} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )}}\,{d x} - 8 \, b^{2} c \int \frac {x^{\frac {3}{2}} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )}}\,{d x} + \frac {b^{2} \sqrt {x} \log \left (-c x^{2} + 1\right )^{2}}{2 \, \sqrt {d}} - b^{2} \int \frac {\log \left (c x^{2} + 1\right )^{2}}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )} \sqrt {x}}\,{d x} + 2 \, b^{2} \int \frac {\log \left (c x^{2} + 1\right ) \log \left (-c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )} \sqrt {x}}\,{d x} - 4 \, a b \int \frac {\log \left (c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )} \sqrt {x}}\,{d x} + 4 \, a b \int \frac {\log \left (-c x^{2} + 1\right )}{4 \, {\left (c \sqrt {d} x^{2} - \sqrt {d}\right )} \sqrt {x}}\,{d x} + \frac {a^{2} {\left (-\frac {i \, {\left (\log \left (i \, c^{\frac {1}{4}} \sqrt {x} + 1\right ) - \log \left (-i \, c^{\frac {1}{4}} \sqrt {x} + 1\right )\right )}}{c^{\frac {1}{4}}} - \frac {\log \left (\frac {\sqrt {c} \sqrt {x} - c^{\frac {1}{4}}}{\sqrt {c} \sqrt {x} + c^{\frac {1}{4}}}\right )}{c^{\frac {1}{4}}}\right )}}{2 \, \sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))^2/(d*x)^(1/2),x, algorithm="maxima")
[Out]
-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)*l
og(-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*l
og(c*x^2 + 1)^2/((c*sqrt(d)*x^2 - sqrt(d))*sqrt(x)), x) + 2*b^2*integrate(1/4*log(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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{\sqrt {d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*atanh(c*x^2))^2/(d*x)^(1/2),x)
[Out]
int((a + b*atanh(c*x^2))^2/(d*x)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(c*x**2))**2/(d*x)**(1/2),x)
[Out]
Integral((a + b*atanh(c*x**2))**2/sqrt(d*x), x)
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