∫ (a+b tanh−1(cx2))2 ____________________________

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3.1.92 \(\int \frac {(a+b \tanh ^{-1}(c x^2))^2}{(d x)^{3/2}} \, dx\) [92]

Optimal. Leaf size=6334 \[ \text {result too large to display} \]

[Out]

-2*a*b*ln(c*x^2+1)/d/(d*x)^(1/2)+b^2*ln(-c*x^2+1)*ln(c*x^2+1)/d/(d*x)^(1/2)-1/2*(2*a-b*ln(-c*x^2+1))^2/d/(d*x)

^(1/2)+2*a*b*c^(1/4)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2))*2^(1/2)*x^(1/2)/d/(d*x)^(1/2)+a*b*c^(1/4)*ln(1+x*c^(1/2

)-c^(1/4)*2^(1/2)*x^(1/2))*2^(1/2)*x^(1/2)/d/(d*x)^(1/2)-a*b*c^(1/4)*ln(1+x*c^(1/2)+c^(1/4)*2^(1/2)*x^(1/2))*2

^(1/2)*x^(1/2)/d/(d*x)^(1/2)+2*a*b*c^(1/4)*arctan(-1+c^(1/4)*2^(1/2)*x^(1/2))*2^(1/2)*x^(1/2)/d/(d*x)^(1/2)+2*

I*b^2*(-c)^(1/4)*polylog(2,1-2/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+2*I*b^2*(-c)^(1/4)*polylog(2,1-

2/(1+I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+2*I*b^2*c^(1/4)*polylog(2,1-2/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)

/d/(d*x)^(1/2)+2*I*b^2*c^(1/4)*polylog(2,1-2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+2*I*b^2*(-c)^(1/4)*a

rctan((-c)^(1/4)*x^(1/2))^2*x^(1/2)/d/(d*x)^(1/2)+2*I*b^2*c^(1/4)*arctan(c^(1/4)*x^(1/2))^2*x^(1/2)/d/(d*x)^(1

/2)+2*b^2*(-c)^(1/4)*arctan((-c)^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/d/(d*x)^(1/2)-2*b^2*c^(1/4)*arctan(c^(1/4)

*x^(1/2))*ln(c*x^2+1)*x^(1/2)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*arctanh((-c)^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/d

/(d*x)^(1/2)+2*b^2*(-c)^(1/4)*arctanh((-c)^(1/4)*x^(1/2))^2*x^(1/2)/d/(d*x)^(1/2)+2*b^2*c^(1/4)*arctanh(c^(1/4

)*x^(1/2))^2*x^(1/2)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*polylog(2,1-2/(1-(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2

)-2*b^2*(-c)^(1/4)*polylog(2,1-2/(1+(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)-2*b^2*c^(1/4)*polylog(2,1-2/(1-

c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-2*b^2*c^(1/4)*polylog(2,1-2/(1+c^(1/4)*x^(1/2)))*x^(1

/2)/d/(d*x)^(1/2)+b^2*c^(1/4)*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)/d/(d*x)^(1/2)+b^2*c^(1/4)*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)/d/(d*x)^(1/2)+b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)+b^2*c^(1/4)

*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)/d/(d*x)^(1/2)-b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)+b^2*c^(1/4)*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)/d/(d*x)^(1/2)+b^2*(-c)^(1/4)*polylo

g(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)

/d/(d*x)^(1/2)-b^2*c^(1/4)*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)/d/(d*x)^(1/2)+b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-b^2*c^(1/4)*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)/d/(d*x)^(1/2)+2*b^2*c^(1/4)

*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)/d/(d*x)^(1/2)+2*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*arct

anh((-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)/d/(d*x)^(1/2)+2*b^2*c^(1/4)*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)/d/(d*x)^(1/2)+I*b^2*(-c)^(1/4)*polylog(2,1-(1+I)

*(1-(-c)^(1/4)*x^(1/2))/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+I*b^2*(-c)^(1/4)*polylog(2,1+(-1+I)*(1

+(-c)^(1/4)*x^(1/2))/(1-I*(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+I*b^2*c^(1/4)*polylog(2,1-(1+I)*(1-c^(1/4

)*x^(1/2))/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+I*b^2*c^(1/4)*polylog(2,1+(-1+I)*(1+c^(1/4)*x^(1/2))/(

1-I*c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)-I*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*c^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*c^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*(-c

)^(1/4)*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)/

d/(d*x)^(1/2)-I*b^2*c^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*c^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-I*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-1/2*b^2*ln(c*x^2+1)^2/d/(d*x)^(1/2)+2*b^2*c^(1/4)*ar

ctanh(c^(1/4)*x^(1/2))*ln(c*x^2+1)*x^(1/2)/d/(d*x)^(1/2)-4*b^2*(-c)^(1/4)*arctanh((-c)^(1/4)*x^(1/2))*ln(2/(1-

(-c)^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)-4*b^2*(-c)^(1/4)*arctan((-c)^(1/4)*x^(1/2))*ln(2/(1-I*(-c)^(1/4)*x^

(1/2)))*x^(1/2)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*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/(d*x)^(1/2)+4*b^2*(-c)^(1/4)*arctan((-c)^(1/4)*x^(1/2))*ln(2/(1+I*(-c)^(1/4)*x

^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+4*b^2*(-c)^(1/4)*arctanh((-c)^(1/4)*x^(1/2))*ln(2/(1+(-c)^(1/4)*x^(1/2)))*x^(1/

2)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*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/(d*x)^(1/2)-4*b^2*c^(1/4)*arctanh(c^(1/4)*x^(1/2))*ln(2/(1-c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x

)^(1/2)+2*b^2*(-c)^(1/4)*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/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*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/(d*x)^(1/2)-4*b^2*c^(1/4)*arctan(c^

(1/4)*x^(1/2))*ln(2/(1-I*c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+2*b^2*c^(1/4)*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/(d*x)^(1/2)+2*b^2*c^(1/4

)*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/(d*x)^(1/2)-2*b^2*c^(1/4)*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/(d*x)^(1/2)+4*b^2*c^(1/4)*arctan(c^(1/4)*x^(1/2))*ln(2/(1+I*c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)+

4*b^2*c^(1/4)*arctanh(c^(1/4)*x^(1/2))*ln(2/(1+c^(1/4)*x^(1/2)))*x^(1/2)/d/(d*x)^(1/2)-2*b^2*c^(1/4)*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/(d*x)

^(1/2)-2*b^2*c^(1/4)*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/(d*x)^(1/2)+2*b^2*(-c)^(1/4)*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/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*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/(d*x)^(

1/2)-2*b^2*c^(1/4)*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/(d*x)

^(1/2)+2*b^2*c^(1/4)*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)/d/(d*x)^(1/2)+2*b^2*(-c)^(1/4)*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)/d

/(d*x)^(1/2)-2*b^2*c^(1/4)*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)/d/(d*x)^(1/2)+2*b^2*c^(1/4)*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)/d/(d*x)^(1/2

)+2*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*arctan((-c)^(1/4)*x^(1/2))*ln

(-c*x^2+1)*x^(1/2)/d/(d*x)^(1/2)+2*b^2*(-c)^(1/4)*arctanh((-c)^(1/4)*x^(1/2))*ln(-c*x^2+1)*x^(1/2)/d/(d*x)^(1/

2)-2*b*c^(1/4)*arctan(c^(1/4)*x^(1/2))*(2*a-b*ln(-c*x^2+1))*x^(1/2)/d/(d*x)^(1/2)+2*b*c^(1/4)*arctanh(c^(1/4)*

x^(1/2))*(2*a-b*ln(-c*x^2+1))*x^(1/2)/d/(d*x)^(1/2)-2*b^2*c^(1/4)*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)/d/(d*x)^(1/2)+2*b^2*(-c)^

(1/4)*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)/d/(d*x)^(1/2)-2*b^2*(-c)^(1/4)*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)/d/(d*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 9.67, antiderivative size = 6334, normalized size of antiderivative = 1.00, number of steps used = 197, number of rules used = 33, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.650, Rules used = {6051, 6043, 6041, 2507, 2526, 212, 2520, 12, 266, 6857, 6131, 6055, 2449, 2352, 6139, 6057, 2497, 209, 5048, 4966, 5040, 4964, 2505, 304, 6874, 303, 1176, 631, 210, 1179, 642, 30, 2637} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Sqrt[2]*a*b*c^(1/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(d*Sqrt[d*x]) + (2*Sqrt[2]*a*b*c^(1/4)*Sq

rt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(d*Sqrt[d*x]) + ((2*I)*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqr

t[x]]^2)/(d*Sqrt[d*x]) + ((2*I)*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/

4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(

d*Sqrt[d*x]) - (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[

d*x]) - (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x])

+ (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sq

rt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(

1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x

]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Lo

g[2/(1 + (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d

*x]) + (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-

c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])]

)/(d*Sqrt[d*x]) - (4*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x])

+ (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) -

(4*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)

*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]))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*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]))])/(d*Sqrt[d*x]) + (2*b^2*c^

(1/4)*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]))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*A

rcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*c^

(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*c^(1/4)*Sqrt[x]*A

rcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*S

qrt[x]]*Log[(-2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(

d*Sqrt[d*x]) - (2*b^2*c^(1/4)*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]))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sq

rt[x]]*Log[(-2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*S

qrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqr

t[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*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]))])/(d*Sqrt[d*x]) - (

2*b^2*c^(1/4)*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]))])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)

^(1/4)*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]))])/(d*Sqrt[d*x]) - (2*b^...

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},

Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free

Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,

b]]))

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}

, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !

GtQ[a/b, 0]

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 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +

1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/

(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)

^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*

((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]

&& IntegerQ[n] && NeQ[m, -1]

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 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 2637

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt

egrand[z*Log[w]*(D[v, x]/v), x], x] - Int[SimplifyIntegrand[z*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFr

eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[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 6041

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] && Integer

Q[m]

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx\\ \end {align*}

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Mathematica [F]
time = 48.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{\left (d x \right )^{\frac {3}{2}}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(3/2),x, algorithm="maxima")

[Out]

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)*l

og(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

)*sqrt(x) + 1) - log(-I*c^(1/4)*sqrt(x) + 1))/c^(3/4) - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqrt(x) + c^(

1/4)))/c^(3/4))/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)*sqrt(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)*sqrt(x) + 1) - log(-I*c^(1/4)*

sqrt(x) + 1))/c^(3/4) - log((sqrt(c)*sqrt(x) - c^(1/4))/(sqrt(c)*sqrt(x) + c^(1/4)))/c^(3/4))/d^(3/2) - 1/2*b^

2*log(-c*x^2 + 1)^2/(d^(3/2)*sqrt(x))

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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)^(3/2),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^2*x^2), x)

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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}}{\left (d x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(3/2),x, algorithm="giac")

[Out]

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

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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}{{\left (d\,x\right )}^{3/2}} \,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)^(3/2),x)

[Out]

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

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