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

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

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

[Out]

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

n(-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)*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))*l

n(-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)*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)+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)*arctan((

-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)*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

)*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))*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))^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)*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)-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)

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

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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 = 99.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{3/2}} \, dx \]

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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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{2}\right ) + a^{2}\right )} \sqrt {d x}}{d^{2} x^{2}}, x\right ) \]

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

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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maple [F] time = 0.47, 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 \[ b^{2} c \int \frac {x^{\frac {3}{2}} \log \left (c x^{2} + 1\right )^{2}}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\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 d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )}}\,{d x} + 4 \, a b c \int \frac {x^{\frac {3}{2}} \log \left (c x^{2} + 1\right )}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )}}\,{d x} - 4 \, a b c \int \frac {x^{\frac {3}{2}} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )}}\,{d x} + 8 \, b^{2} c \int \frac {x^{\frac {3}{2}} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )}}\,{d x} + \frac {1}{2} \, a^{2} {\left (\frac {c {\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 {3}{4}}} - \frac {\log \left (\frac {\sqrt {c} \sqrt {x} - c^{\frac {1}{4}}}{\sqrt {c} \sqrt {x} + c^{\frac {1}{4}}}\right )}{c^{\frac {3}{4}}}\right )}}{d^{\frac {3}{2}}} - \frac {4}{d^{\frac {3}{2}} \sqrt {x}}\right )} - b^{2} \int \frac {\log \left (c x^{2} + 1\right )^{2}}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\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 d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )} \sqrt {x}}\,{d x} - 4 \, a b \int \frac {\log \left (c x^{2} + 1\right )}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )} \sqrt {x}}\,{d x} + 4 \, a b \int \frac {\log \left (-c x^{2} + 1\right )}{4 \, {\left (c d^{\frac {3}{2}} x^{3} - d^{\frac {3}{2}} x\right )} \sqrt {x}}\,{d x} - \frac {a^{2} c {\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 {3}{4}}} - \frac {\log \left (\frac {\sqrt {c} \sqrt {x} - c^{\frac {1}{4}}}{\sqrt {c} \sqrt {x} + c^{\frac {1}{4}}}\right )}{c^{\frac {3}{4}}}\right )}}{2 \, d^{\frac {3}{2}}} - \frac {b^{2} \log \left (-c x^{2} + 1\right )^{2}}{2 \, d^{\frac {3}{2}} \sqrt {x}} \]

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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \]

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

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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