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

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

[Out]

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

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Rubi [A]  time = 0.0244887, 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} \[ x \tanh ^2(c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}-\frac{\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^3)^(2/3),x]

[Out]

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

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

Mathematica [C]  time = 0.0275691, size = 41, normalized size = 0.82 \[ -\frac{\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{2/3} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*Coth[c + d*x]^3)^(2/3)*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[c + d*x]^2]*Tanh[c + d*x])/d)

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Maple [B]  time = 0.097, size = 119, normalized size = 2.4 \begin{align*}{\frac{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{2}x}{ \left ( 1+{{\rm e}^{2\,dx+2\,c}} \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{{{\rm e}^{2\,dx+2\,c}}-1}{ \left ( 1+{{\rm e}^{2\,dx+2\,c}} \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((b*coth(d*x+c)^3)^(2/3),x)

[Out]

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

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Maxima [A]  time = 1.61666, size = 46, normalized size = 0.92 \begin{align*} \frac{{\left (d x + c\right )} b^{\frac{2}{3}}}{d} + \frac{2 \, b^{\frac{2}{3}}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.18871, size = 992, normalized size = 19.84 \begin{align*} \frac{{\left (d x \cosh \left (d x + c\right )^{2} +{\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, 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 (4 \, d x + 4 \, c\right )} - 2 \,{\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 (4 \, d x + 4 \, c\right )} - 2 \, 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{2}{3}}}{d \cosh \left (d x + c\right )^{2} +{\left (d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{2} +{\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \,{\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (d \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((b*coth(d*x+c)^3)^(2/3),x, algorithm="giac")

[Out]

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