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3.93 \(\int \frac{(a+b \tanh ^{-1}(c x^2))^2}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=6520 \[ \text{result too large to display} \]

[Out]

(-2*Sqrt[2]*a*b*c^(3/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*d^2*Sqrt[d*x]) + (2*Sqrt[2]*a*b*c^(3/4
)*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-
c)^(1/4)*Sqrt[x]]^2)/(d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(d^2*Sqrt[d*x
]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTa
nh[c^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) - (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-
c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)
^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x])
 - (2*b^2*(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[
(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4
*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b
^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (2*b^2*
(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*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)*Sq
rt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(-2*(-c)^(1/4)*(1 - Sq
rt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(
-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqr
t[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4*b^2*c^(3/4)*S
qrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*Ar
cTan[(-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]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (4*b^2*c^(3/4)*Sqrt[x
]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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)*Sq
rt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*Arc
Tan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4*b^2*
c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b^2*c^(3/4)*Sqr
t[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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)*Sqr
t[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*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*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*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)*Sqr
t[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sq
rt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*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]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*Arc
Tan[c^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (Sqrt[2
]*a*b*c^(3/4)*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*d^2*Sqrt[d*x]) + (Sqrt[2]*a*b*c^(3/4)*S
qrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)
^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[
1 - c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*d^2*
Sqrt[d*x]) + (2*b*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*d^2*Sqrt[d*x]) - (2*a
- b*Log[1 - c*x^2])^2/(6*d^2*x*Sqrt[d*x]) - (2*a*b*Log[1 + c*x^2])/(3*d^2*x*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqr
t[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt
[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])
/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) + (b^2*
Log[1 - c*x^2]*Log[1 + c*x^2])/(3*d^2*x*Sqrt[d*x]) - (b^2*Log[1 + c*x^2]^2)/(6*d^2*x*Sqrt[d*x]) - (2*b^2*(-c)^
(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*(-c)^(3/4)*Sqrt[x
]*PolyLog[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) + ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 +
 (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
^2*Sqrt[d*x]) + ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*S
qrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d^2*Sqrt[d*x]) - ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*Poly
Log[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*(-
c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*Pol
yLog[2, 1 - 2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1
/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x
]) - (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (
-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4
)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) +
 (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1
/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1
+ (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/
(1 - c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (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^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt
[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3
*d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) +
((I/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (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^2*Sqrt[d*x]) + ((I/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sq
rt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d^2*Sqrt[d*x]) + ((I/3)*b^2
*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (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^2*Sqrt[d*x]) + ((I/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (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^2*Sqrt[d*x]) - ((I/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1
- ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) - (((2*I)/3)*b^2*c^(3/4)*Sqrt[x]*P
olyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + c^(1/
4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x])
)/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1
- (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[
d*x]) - (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4
))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt
[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*P
olyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqr
t[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1
+ c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + ((I/3)*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (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^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x
]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d
^2*Sqrt[d*x]) - ((I/3)*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[
x])])/(d^2*Sqrt[d*x])

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Rubi [F]  time = 0.030478, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx &=\int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx\\ \end{align*}

Mathematica [F]  time = 50.7384, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{(d x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.275, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2} \left ( dx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2}}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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