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

Optimal. Leaf size=170 \[ a x+\frac{b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )}{4 c^{2/3}}-\frac{b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )}{4 c^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )}{2 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )}{2 c^{2/3}}-\frac{b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )}{c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right ) \]

[Out]

a*x - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[
x])/Sqrt[3]])/(2*c^(2/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/c^(2/3) + b*x*ArcTanh[c*x^(3/2)] + (b*Log[1 - c^(1/3)
*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3))

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Rubi [A]  time = 0.287018, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6091, 329, 296, 634, 618, 204, 628, 206} \[ a x+\frac{b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )}{4 c^{2/3}}-\frac{b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )}{4 c^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )}{2 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )}{2 c^{2/3}}-\frac{b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )}{c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTanh[c*x^(3/2)],x]

[Out]

a*x - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[
x])/Sqrt[3]])/(2*c^(2/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/c^(2/3) + b*x*ArcTanh[c*x^(3/2)] + (b*Log[1 - c^(1/3)
*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3))

Rule 6091

Int[ArcTanh[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTanh[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x]
, x] /; FreeQ[{c, n}, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 296

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt
[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[(2*k*m*Pi)/n] - s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi
)/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k*m*Pi)/n] + s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x
+ s^2*x^2), x]; (2*r^(m + 2)*Int[1/(r^2 - s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k,
1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^{3/2}\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^{3/2}\right )-\frac{1}{2} (3 b c) \int \frac{x^{3/2}}{1-c^2 x^3} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^{3/2}\right )-(3 b c) \operatorname{Subst}\left (\int \frac{x^4}{1-c^2 x^6} \, dx,x,\sqrt{x}\right )\\ &=a x+b x \tanh ^{-1}\left (c x^{3/2}\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{\sqrt [3]{c}}-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{\sqrt [3]{c}}-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{\sqrt [3]{c}}\\ &=a x-\frac{b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )}{c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{4 c^{2/3}}-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{4 c^{2/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt [3]{c}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt [3]{c}}\\ &=a x-\frac{b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )}{c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right )+\frac{b \log \left (1-\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )}{4 c^{2/3}}-\frac{b \log \left (1+\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )}{4 c^{2/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt{x}\right )}{2 c^{2/3}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt{x}\right )}{2 c^{2/3}}\\ &=a x-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )}{2 c^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1+2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )}{2 c^{2/3}}-\frac{b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )}{c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right )+\frac{b \log \left (1-\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )}{4 c^{2/3}}-\frac{b \log \left (1+\sqrt [3]{c} \sqrt{x}+c^{2/3} x\right )}{4 c^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.100893, size = 141, normalized size = 0.83 \[ a x-\frac{b \left (-\log \left (c^{2/3} x-\sqrt [3]{c} \sqrt{x}+1\right )+\log \left (c^{2/3} x+\sqrt [3]{c} \sqrt{x}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} \sqrt{x}}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} \sqrt{x}+1}{\sqrt{3}}\right )+4 \tanh ^{-1}\left (\sqrt [3]{c} \sqrt{x}\right )\right )}{4 c^{2/3}}+b x \tanh ^{-1}\left (c x^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTanh[c*x^(3/2)],x]

[Out]

a*x + b*x*ArcTanh[c*x^(3/2)] - (b*(2*Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2
*c^(1/3)*Sqrt[x])/Sqrt[3]] + 4*ArcTanh[c^(1/3)*Sqrt[x]] - Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x] + Log[1 + c^(1/
3)*Sqrt[x] + c^(2/3)*x]))/(4*c^(2/3))

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Maple [A]  time = 0.032, size = 179, normalized size = 1.1 \begin{align*} ax+bx{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) +{\frac{b}{2\,c}\ln \left ( \sqrt{x}-\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{b}{4\,c}\ln \left ( x+\sqrt [3]{{c}^{-1}}\sqrt{x}+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{b\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt{x}}{\sqrt [3]{{c}^{-1}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{b}{2\,c}\ln \left ( \sqrt{x}+\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{b}{4\,c}\ln \left ( x-\sqrt [3]{{c}^{-1}}\sqrt{x}+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{b\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt{x}}{\sqrt [3]{{c}^{-1}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*x*arctanh(c*x^(3/2))+1/2*b/c/(1/c)^(1/3)*ln(x^(1/2)-(1/c)^(1/3))-1/4*b/c/(1/c)^(1/3)*ln(x+(1/c)^(1/3)*x^
(1/2)+(1/c)^(2/3))+1/2*b*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)+1))-1/2*b/c/(1/c)^(1/
3)*ln(x^(1/2)+(1/c)^(1/3))+1/4*b/c/(1/c)^(1/3)*ln(x-(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+1/2*b*3^(1/2)/c/(1/c)^(1/
3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)-1))

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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^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 10.7687, size = 4289, normalized size = 25.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arctanh(c*x^(3/2)),x, algorithm="fricas")

[Out]

a*x - 1/2*sqrt(3)*sqrt(((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 - 4*((1
/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b + 4*b^2)*arctan(1/24*(4*sqrt(3)*s
qrt(((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*b^2*c*sqrt(x) + 4*b^4*c*sq
rt(x) + 4*b^4*x + 4*b^4 - 2*(2*b^3*c*sqrt(x) + b^3)*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*
sqrt(3) + 1) + 2*b))*sqrt(((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 - 4*
((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b + 4*b^2)*c - sqrt(3)*(((1/2)^(
1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*c^2 - 4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*
b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c^2 + 4*b^2*c^2 + 8*b^2*c*sqrt(x))*sqrt(((1/2)^(1/3)*(b^3 -
(c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 - 4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^
2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b + 4*b^2))/b^3) + 1/8*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*
(I*sqrt(3) + 1) - 4*b)*log(((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*b^2
*c*sqrt(x) + 4*b^4*c*sqrt(x) + 4*b^4*x + 4*b^4 - 2*(2*b^3*c*sqrt(x) + b^3)*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c
^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)) + 1/4*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(
I*sqrt(3) + 1) + 2*b)*log(-4*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^2
*b^2*c*sqrt(x) - 4*b^4*c*sqrt(x) + 4*b^4*x + 4*b^4 - 4*(2*b^3*c*sqrt(x) - b^3)*((-1/16*b^3 + 1/16*(c^2 - 1)*b^
3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)) - 1/4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)
*(I*sqrt(3) + 1) + 2*b)*log(-1/4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)
^2*c + ((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c - b^2*c + b^2*sqrt(x)
) - 1/2*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)*log(((-1/16*b^3 + 1/16
*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^2*c + 2*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1
/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)*b*c + b^2*c + b^2*sqrt(x)) + 1/2*(b*x - b)*log(-(c^2*x^3 + 2*c*x^(3/2)
+ 1)/(c^2*x^3 - 1)) - sqrt(3*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^
2 + 6*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)*b + 3*b^2)*arctan(1/3*((
(-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^2*c^2 + 2*((-1/16*b^3 + 1/16*(c
^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)*b*c^2 + b^2*c^2 - 2*b^2*c*sqrt(x) + sqrt(-4*((-1/16
*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^2*b^2*c*sqrt(x) - 4*b^4*c*sqrt(x) + 4
*b^4*x + 4*b^4 - 4*(2*b^3*c*sqrt(x) - b^3)*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(
3) + 1) - b))*c)*sqrt(3*((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)^2 + 6*
((-1/16*b^3 + 1/16*(c^2 - 1)*b^3/c^2 + 1/16*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - b)*b + 3*b^2)/b^3)

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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**(3/2)),x)

[Out]

Timed out

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Giac [A]  time = 1.19899, size = 251, normalized size = 1.48 \begin{align*} \frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{2}} + \frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} - \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{2}} - \frac{{\left | c \right |}^{\frac{1}{3}} \log \left (x + \frac{\sqrt{x}}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}} + \frac{{\left | c \right |}^{\frac{1}{3}} \log \left (x - \frac{\sqrt{x}}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \log \left (\sqrt{x} + \frac{1}{{\left | c \right |}^{\frac{1}{3}}}\right )}{c^{2}} + \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \log \left ({\left | \sqrt{x} - \frac{1}{{\left | c \right |}^{\frac{1}{3}}} \right |}\right )}{c^{2}}\right )} + 2 \, x \log \left (-\frac{c x^{\frac{3}{2}} + 1}{c x^{\frac{3}{2}} - 1}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arctanh(c*x^(3/2)),x, algorithm="giac")

[Out]

1/4*(c*(2*sqrt(3)*abs(c)^(1/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + 1/abs(c)^(1/3))*abs(c)^(1/3))/c^2 + 2*sqrt(3)*a
bs(c)^(1/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) - 1/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - abs(c)^(1/3)*log(x + sqrt(x)/a
bs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 + abs(c)^(1/3)*log(x - sqrt(x)/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 2*abs(c)
^(1/3)*log(sqrt(x) + 1/abs(c)^(1/3))/c^2 + 2*abs(c)^(1/3)*log(abs(sqrt(x) - 1/abs(c)^(1/3)))/c^2) + 2*x*log(-(
c*x^(3/2) + 1)/(c*x^(3/2) - 1)))*b + a*x