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3.90 \(\int \sqrt{d x} (a+b \tanh ^{-1}(c x^2))^2 \, dx\)

Optimal. Leaf size=6327 \[ \text{result too large to display} \]

[Out]

(-8*a*b*x*Sqrt[d*x])/9 - (2*Sqrt[2]*a*b*Sqrt[d*x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*c^(3/4)*Sqrt[x]) + (
2*Sqrt[2]*a*b*Sqrt[d*x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*c^(3/4)*Sqrt[x]) - (((2*I)/3)*b^2*Sqrt[d*x]*Ar
cTan[(-c)^(1/4)*Sqrt[x]]^2)/((-c)^(3/4)*Sqrt[x]) - (((2*I)/3)*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(c^(3/4
)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh
[c^(1/4)*Sqrt[x]]^2)/(3*c^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*
Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x
])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sq
rt[d*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*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(
(1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTa
n[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTanh[(-c
)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-S
qrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2
*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*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*(-c
)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(
-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqr
t[x])])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/
4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*
c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Lo
g[(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*c^(3/4)*Sqrt[x]
) + (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*
c^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]
) - (4*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[
d*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*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d
*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*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*A
rcTanh[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*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*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*(-c)^
(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]
*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (Sqrt[2]*a*b*Sqrt[d*x]*Lo
g[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*c^(3/4)*Sqrt[x]) - (Sqrt[2]*a*b*Sqrt[d*x]*Log[1 + Sqrt[2]*c^(1/
4)*Sqrt[x] + Sqrt[c]*x])/(3*c^(3/4)*Sqrt[x]) + (4*b^2*x*Sqrt[d*x]*Log[1 - c*x^2])/9 + (2*b^2*Sqrt[d*x]*ArcTan[
(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[
1 - c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b*x*Sqrt[d*x]*(2*a - b*Log[1 - c*x^2]))/9 + (2*b*Sqrt[d*x]*ArcTan[c^(1
/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/4)*Sqrt[x]) - (2*b*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*(2*a - b*
Log[1 - c*x^2]))/(3*c^(3/4)*Sqrt[x]) + (x*Sqrt[d*x]*(2*a - b*Log[1 - c*x^2])^2)/6 + (2*a*b*x*Sqrt[d*x]*Log[1 +
 c*x^2])/3 - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[
d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]
]*Log[1 + c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*c^(3/4
)*Sqrt[x]) - (b^2*x*Sqrt[d*x]*Log[1 - c*x^2]*Log[1 + c*x^2])/3 + (b^2*x*Sqrt[d*x]*Log[1 + c*x^2]^2)/6 + (2*b^2
*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (((2*I)/3)*b^2*Sqrt[d*x]*PolyL
og[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(3/4)*Sqrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/((-c)^(3/4)*S
qrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/((-c)^(3/4)*Sqrt[x]) - ((I/3)*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 + I)
*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(3/4)*Sqrt[x]) - (((2*I)/3)*b^2*Sqrt[d*x]*PolyLo
g[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/((-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + (-c)^(1/
4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*x]*PolyLo
g[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*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*(-c)^(3
/4)*Sqrt[x]) - ((I/3)*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x
])])/((-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + ((
I/3)*b^2*Sqrt[d*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]))])/((-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*(-c)^(3/4)*Sqrt[x]) - (((2*I)/3)*b^2*Sqrt[d*x]*PolyLog[2,
1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(c^(3/4)*Sqrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/(c^(3/4)*Sqrt[x]) + ((I/3)*b^2*S
qrt[d*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]))])/(c^(3/4)*Sqrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/(c^(3/4)*Sqrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/(c^(3/4)*Sqrt[x]) - ((I/
3)*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(3/4)*Sqrt[x]) -
(((2*I)/3)*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyL
og[2, 1 - 2/(1 + c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*c^(3/
4)*Sqrt[x]) - (b^2*Sqrt[d*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*c^(3/4)*Sqrt[x]) + ((I/3)*b^2*Sqrt[d*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]))])/((-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*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*(-c)^(3/4)*Sqr
t[x]) - ((I/3)*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(3/4)
*Sqrt[x])

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Rubi [F]  time = 0.0256037, 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 = {} \[ \int \sqrt{d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 61.9463, size = 0, normalized size = 0. \[ \int \sqrt{d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.313, size = 0, normalized size = 0. \begin{align*} \int \sqrt{dx} \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*(a+b*arctanh(c*x^2))^2,x)

[Out]

int((d*x)^(1/2)*(a+b*arctanh(c*x^2))^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((d*x)^(1/2)*(a+b*arctanh(c*x^2))^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 ({\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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*(a+b*arctanh(c*x^2))^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), 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((d*x)**(1/2)*(a+b*atanh(c*x**2))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)*(a+b*arctanh(c*x^2))^2,x, algorithm="giac")

[Out]

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

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