3.10 \(\int (b \coth (c+d x))^{2/3} \, dx\)

Optimal. Leaf size=218 \[ -\frac{b^{2/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac{b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 d}-\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 d}+\frac{b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.293867, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac{b^{2/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac{b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 d}-\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 d}+\frac{b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x])^(2/3),x]

[Out]

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

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

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

Mathematica [A]  time = 0.177229, size = 149, normalized size = 0.68 \[ \frac{(b \coth (c+d x))^{2/3} \left (-\log \left (\coth ^{\frac{2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )+\log \left (\coth ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt{3}}\right )+4 \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{4 d \coth ^{\frac{2}{3}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x])^(2/3),x]

[Out]

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

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Maple [A]  time = 0.016, size = 193, normalized size = 0.9 \begin{align*}{\frac{1}{2\,d}{b}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}+\sqrt [3]{b} \right ) }-{\frac{1}{4\,d}{b}^{{\frac{2}{3}}}\ln \left ({b}^{{\frac{2}{3}}}-\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ) }-{\frac{\sqrt{3}}{2\,d}{b}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}}-1 \right ) } \right ) }-{\frac{1}{2\,d}{b}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}-\sqrt [3]{b} \right ) }+{\frac{1}{4\,d}{b}^{{\frac{2}{3}}}\ln \left ({b}^{{\frac{2}{3}}}+\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ) }-{\frac{\sqrt{3}}{2\,d}{b}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(d*x+c))^(2/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth \left (d x + c\right )\right )^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*coth(d*x + c))^(2/3), x)

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Fricas [A]  time = 2.0473, size = 871, normalized size = 4. \begin{align*} -\frac{2 \, \sqrt{3} \left (-b^{2}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} b - 2 \, \sqrt{3} \left (-b^{2}\right )^{\frac{1}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}}}{3 \, b}\right ) + 2 \, \sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} b - 2 \, \sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}}}{3 \, b}\right ) + \left (-b^{2}\right )^{\frac{1}{3}} \log \left (b \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{2}{3}} - \left (-b^{2}\right )^{\frac{1}{3}} b + \left (-b^{2}\right )^{\frac{2}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}}\right ) +{\left (b^{2}\right )}^{\frac{1}{3}} \log \left (b \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{2}{3}} +{\left (b^{2}\right )}^{\frac{1}{3}} b -{\left (b^{2}\right )}^{\frac{2}{3}} \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac{1}{3}} \log \left (b \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}} - \left (-b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b^{2}\right )}^{\frac{1}{3}} \log \left (b \left (\frac{b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac{1}{3}} +{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(3)*(-b^2)^(1/3)*arctan(-1/3*(sqrt(3)*b - 2*sqrt(3)*(-b^2)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(
1/3))/b) + 2*sqrt(3)*(b^2)^(1/3)*arctan(-1/3*(sqrt(3)*b - 2*sqrt(3)*(b^2)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c)
)^(1/3))/b) + (-b^2)^(1/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) - (-b^2)^(1/3)*b + (-b^2)^(2/3)*(b*cosh
(d*x + c)/sinh(d*x + c))^(1/3)) + (b^2)^(1/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) + (b^2)^(1/3)*b - (b
^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 2*(-b^2)^(1/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)
- (-b^2)^(2/3)) - 2*(b^2)^(1/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth{\left (c + d x \right )}\right )^{\frac{2}{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))**(2/3),x)

[Out]

Integral((b*coth(c + d*x))**(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError