3.37 \(\int \frac{1}{(b \coth ^3(c+d x))^{2/3}} \, dx\)

Optimal. Leaf size=50 \[ \frac{x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}-\frac{\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]

[Out]

-(Coth[c + d*x]/(d*(b*Coth[c + d*x]^3)^(2/3))) + (x*Coth[c + d*x]^2)/(b*Coth[c + d*x]^3)^(2/3)

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Rubi [A]  time = 0.0242453, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ \frac{x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}-\frac{\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^3)^(-2/3),x]

[Out]

-(Coth[c + d*x]/(d*(b*Coth[c + d*x]^3)^(2/3))) + (x*Coth[c + d*x]^2)/(b*Coth[c + d*x]^3)^(2/3)

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx &=\frac{\coth ^2(c+d x) \int \tanh ^2(c+d x) \, dx}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ &=-\frac{\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}}+\frac{\coth ^2(c+d x) \int 1 \, dx}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ &=-\frac{\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}}+\frac{x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0599557, size = 40, normalized size = 0.8 \[ \frac{\coth (c+d x) \left (\tanh ^{-1}(\tanh (c+d x)) \coth (c+d x)-1\right )}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x]^3)^(-2/3),x]

[Out]

(Coth[c + d*x]*(-1 + ArcTanh[Tanh[c + d*x]]*Coth[c + d*x]))/(d*(b*Coth[c + d*x]^3)^(2/3))

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Maple [B]  time = 0.093, size = 119, normalized size = 2.4 \begin{align*}{\frac{ \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{2}x}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{2}} \left ({\frac{b \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{3}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{3}}} \right ) ^{-{\frac{2}{3}}}}+2\,{\frac{1+{{\rm e}^{2\,dx+2\,c}}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{2}d} \left ({\frac{b \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{3}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{3}}} \right ) ^{-2/3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*coth(d*x+c)^3)^(2/3),x)

[Out]

1/(b*(1+exp(2*d*x+2*c))^3/(exp(2*d*x+2*c)-1)^3)^(2/3)/(exp(2*d*x+2*c)-1)^2*(1+exp(2*d*x+2*c))^2*x+2/(b*(1+exp(
2*d*x+2*c))^3/(exp(2*d*x+2*c)-1)^3)^(2/3)/(exp(2*d*x+2*c)-1)^2*(1+exp(2*d*x+2*c))/d

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Maxima [A]  time = 1.74955, size = 50, normalized size = 1. \begin{align*} \frac{d x + c}{b^{\frac{2}{3}} d} - \frac{2}{{\left (b^{\frac{2}{3}} e^{\left (-2 \, d x - 2 \, c\right )} + b^{\frac{2}{3}}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^3)^(2/3),x, algorithm="maxima")

[Out]

(d*x + c)/(b^(2/3)*d) - 2/((b^(2/3)*e^(-2*d*x - 2*c) + b^(2/3))*d)

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Fricas [B]  time = 2.23736, size = 724, normalized size = 14.48 \begin{align*} -\frac{{\left (d x \cosh \left (d x + c\right )^{2} -{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x\right )} \sinh \left (d x + c\right )^{2} + d x -{\left (d x \cosh \left (d x + c\right )^{2} + d x + 2\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \,{\left (d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2\right )} \left (\frac{b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac{1}{3}}}{b d \cosh \left (d x + c\right )^{2} +{\left (b d e^{\left (2 \, d x + 2 \, c\right )} + b d\right )} \sinh \left (d x + c\right )^{2} + b d +{\left (b d \cosh \left (d x + c\right )^{2} + b d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (b d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^3)^(2/3),x, algorithm="fricas")

[Out]

-(d*x*cosh(d*x + c)^2 - (d*x*e^(2*d*x + 2*c) - d*x)*sinh(d*x + c)^2 + d*x - (d*x*cosh(d*x + c)^2 + d*x + 2)*e^
(2*d*x + 2*c) - 2*(d*x*cosh(d*x + c)*e^(2*d*x + 2*c) - d*x*cosh(d*x + c))*sinh(d*x + c) + 2)*((b*e^(6*d*x + 6*
c) + 3*b*e^(4*d*x + 4*c) + 3*b*e^(2*d*x + 2*c) + b)/(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) -
1))^(1/3)/(b*d*cosh(d*x + c)^2 + (b*d*e^(2*d*x + 2*c) + b*d)*sinh(d*x + c)^2 + b*d + (b*d*cosh(d*x + c)^2 + b
*d)*e^(2*d*x + 2*c) + 2*(b*d*cosh(d*x + c)*e^(2*d*x + 2*c) + b*d*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)**3)**(2/3),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )^{3}\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^3)^(2/3),x, algorithm="giac")

[Out]

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