3.1.91 \(\int \frac {(a+b \tanh ^{-1}(c x^2))^2}{\sqrt {d x}} \, dx\) [91]
Optimal. Leaf size=6177 \[ \text {result too large to display} \]
[Out]
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^2*x/(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(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*arc
tan((-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)+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)+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)-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)+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)+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*arc
tanh((-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)+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*po
lylog(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)-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*arc
tan(c^(1/4)*x^(1/2))*ln(c*x^2+1)*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)+2*b^2*pol
ylog(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)+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)-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)-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)+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)+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)-b^2*x*ln(-c*x^2+1)*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)-b^2*polyl
og(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)
________________________________________________________________________________________
Rubi [A]
time = 9.23, antiderivative size = 6177, normalized size of antiderivative = 1.00, number of steps
used = 241, number of rules used = 33, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.650, Rules used = {6051, 6043,
6023, 2498, 327, 218, 212, 209, 2500, 2526, 2521, 2520, 12, 266, 6857, 6131, 6055, 2449,
2352, 6139, 6057, 2497, 5048, 4966, 5040, 4964, 217, 1179, 642, 1176, 631, 210, 2636}
\begin {gather*} \text {Too large to display} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[(a + b*ArcTanh[c*x^2])^2/Sqrt[d*x],x]
[Out]
(2*a^2*x)/Sqrt[d*x] - (2*Sqrt[2]*a*b*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + (2*Sqr
t[2]*a*b*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + ((2*I)*b^2*Sqrt[x]*ArcTan[(-c)^(1/
4)*Sqrt[x]]^2)/((-c)^(1/4)*Sqrt[d*x]) - (4*a*b*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + ((2*I)*b
^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/((-c
)^(1/4)*Sqrt[d*x]) - (4*a*b*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[c^(
1/4)*Sqrt[x]]^2)/(c^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x
])])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-
c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcT
an[(-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)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqr
t[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*L
og[2/(1 + (-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*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
)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((S
qrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) - (2*b^2*
Sqrt[x]*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)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) +
(2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(
1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(
1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^
2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*S
qrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan
[c^(1/4)*Sqrt[x]]*Log[(-2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))
])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(
-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*L
og[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTan[c^(1/
4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[
2/(1 + c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(-2*c^(1/4)*(1 - S
qrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sq
rt[x]*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)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*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)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*A
rcTanh[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)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(-2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt
[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sq
rt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt
[d*x]) + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c
^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[
(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/...
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))
Rule 218
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !Gt
Q[a/b, 0]
Rule 266
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 327
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]
Rule 631
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1176
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1179
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 2352
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]
Rule 2449
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Rule 2497
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]
Rule 2498
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]
Rule 2500
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[x^n*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
Rule 2520
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
Rule 2521
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
Rule 2526
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]
Rule 2636
Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[x*Log[w]*(D[v, x]/v), x],
x] - Int[SimplifyIntegrand[x*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]
Rule 4964
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]
Rule 4966
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]
Rule 5040
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Rule 5048
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a,
0])
Rule 6023
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(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]
Rule 6043
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k,
Subst[Int[x^(k*(m + 1) - 1)*(a + b*ArcTanh[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 6051
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_)*(x_))^(m_), x_Symbol] :> Dist[d^IntPart[m]*((d*x)^Fr
acPart[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])
Rule 6055
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]
Rule 6057
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
Rule 6131
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Rule 6139
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[
a, 0])
Rule 6857
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
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*}
________________________________________________________________________________________
Mathematica [F]
time = 45.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt {d x}} \, dx \end {gather*}
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]
________________________________________________________________________________________
Maple [F]
time = 0.05, 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)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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)
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{\sqrt {d x}}\, dx \end {gather*}
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)
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{\sqrt {d\,x}} \,d x \end {gather*}
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)
________________________________________________________________________________________