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

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

Optimal. Leaf size=6177 \[ \text{result too large to display} \]

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

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

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

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

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

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

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

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

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

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

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

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

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

+ (I*b^2*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*S

qrt[d*x]) + ((2*I)*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*S

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

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

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

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

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

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

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

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

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

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

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

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

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

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Rubi [F] time = 0.025645, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt{d x}} \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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 = 59.3506, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{\sqrt{d x}} \, dx \]

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]

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Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}

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)

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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)^(1/2),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 x}, 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)^(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)

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

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)

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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}}{\sqrt{d x}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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

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