3.1.0 ∫ 1 ____________ (b coth3(c+dx))2∕3 dx [37]

Integrand size = 14, antiderivative size = 50 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}}+\frac {x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}} \]

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

Rubi [A] (veriﬁed)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 8} \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\frac {x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]

-(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)

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]

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

Rubi steps \begin{align*} \text {integral}& = \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] (veriﬁed)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\frac {\coth (c+d x) (-1+\text {arctanh}(\tanh (c+d x)) \coth (c+d x))}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]

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

Maple [A] (veriﬁed)

Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.78


method	result	size

risch	\(\frac {{\mathrm e}^{4 d x +4 c} d x +2 \,{\mathrm e}^{2 d x +2 c} d x +d x +2 \,{\mathrm e}^{2 d x +2 c}+2}{{\left (\frac {b \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )}^{\frac {2}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d}\)	\(89\)

(exp(4*d*x+4*c)*d*x+2*exp(2*d*x+2*c)*d*x+d*x+2*exp(2*d*x+2*c)+2)/(b*(exp(2*d*x+2*c)+1)^3/(exp(2*d*x+2*c)-1)^3)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (46) = 92\).

Time = 0.26 (sec) , antiderivative size = 287, normalized size of antiderivative = 5.74 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=-\frac {{\left (d x \cosh \left (d x + c\right )^{2} - {\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x\right )} \sinh \left (d x + c\right )^{2} + d x - {\left (d x \cosh \left (d x + c\right )^{2} + d x + 2\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{b d \cosh \left (d x + c\right )^{2} + {\left (b d e^{\left (2 \, d x + 2 \, c\right )} + b d\right )} \sinh \left (d x + c\right )^{2} + b d + {\left (b d \cosh \left (d x + c\right )^{2} + b d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (b d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]

-(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))

Sympy [F]

\[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{\left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]

Maxima [A] (veriﬁcation not implemented)

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\frac {d x + c}{b^{\frac {2}{3}} d} - \frac {2}{{\left (b^{\frac {2}{3}} e^{\left (-2 \, d x - 2 \, c\right )} + b^{\frac {2}{3}}\right )} d} \]

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

Giac [F]

\[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{3}\right )^{\frac {2}{3}}} \,d x } \]

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{2/3}} \,d x \]

\(\int \frac {1}{(b \coth ^3(c+d x))^{2/3}} \, dx\) [37]

Optimal result