\(\int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx\) [124]

Optimal result

Integrand size = 11, antiderivative size = 37 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=-\frac {3 x}{2}+2 \log (\cosh (x))+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))} \]

[Out]

-3/2*x+2*ln(cosh(x))+3/2*tanh(x)-tanh(x)^2+1/2*tanh(x)^2/(1+coth(x))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3633, 3610, 3612, 3556} \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=-\frac {3 x}{2}-\tanh ^2(x)+\frac {3 \tanh (x)}{2}+2 \log (\cosh (x))+\frac {\tanh ^2(x)}{2 (\coth (x)+1)} \]

[In]

Int[Tanh[x]^3/(1 + Coth[x]),x]

[Out]

(-3*x)/2 + 2*Log[Cosh[x]] + (3*Tanh[x])/2 - Tanh[x]^2 + Tanh[x]^2/(2*(1 + Coth[x]))

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3612

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

Rule 3633

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a
)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x]))), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c
+ d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x
] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (-4+3 \coth (x)) \tanh ^3(x) \, dx \\ & = -\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} i \int (-3 i+4 i \coth (x)) \tanh ^2(x) \, dx \\ & = \frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+\frac {1}{2} \int (4-3 \coth (x)) \tanh (x) \, dx \\ & = -\frac {3 x}{2}+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+2 \int \tanh (x) \, dx \\ & = -\frac {3 x}{2}+2 \log (\cosh (x))+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=\frac {1}{2} \left (-3 \text {arctanh}(\tanh (x))+4 \log (\cosh (x))+3 \tanh (x)+\left (-2+\frac {1}{1+\coth (x)}\right ) \tanh ^2(x)\right ) \]

[In]

Integrate[Tanh[x]^3/(1 + Coth[x]),x]

[Out]

(-3*ArcTanh[Tanh[x]] + 4*Log[Cosh[x]] + 3*Tanh[x] + (-2 + (1 + Coth[x])^(-1))*Tanh[x]^2)/2

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {7 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}-\frac {2}{\left (1+{\mathrm e}^{2 x}\right )^{2}}+2 \ln \left (1+{\mathrm e}^{2 x}\right )\)	\(30\)
parallelrisch	\(\frac {\left (-4 \tanh \left (x \right )-4\right ) \ln \left (1-\tanh \left (x \right )\right )-\tanh \left (x \right )^{3}-7 \tanh \left (x \right ) x +\tanh \left (x \right )^{2}-7 x -3}{2+2 \tanh \left (x \right )}\)	\(44\)
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {2 \tanh \left (\frac {x}{2}\right )^{3}-2 \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+2 \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}\)	\(80\)

[In]

int(tanh(x)^3/(1+coth(x)),x,method=_RETURNVERBOSE)

[Out]

-7/2*x-1/4*exp(-2*x)-2/(1+exp(2*x))^2+2*ln(1+exp(2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (31) = 62\).

Time = 0.28 (sec) , antiderivative size = 354, normalized size of antiderivative = 9.57 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=-\frac {14 \, x \cosh \left (x\right )^{6} + 84 \, x \cosh \left (x\right ) \sinh \left (x\right )^{5} + 14 \, x \sinh \left (x\right )^{6} + {\left (28 \, x + 1\right )} \cosh \left (x\right )^{4} + {\left (210 \, x \cosh \left (x\right )^{2} + 28 \, x + 1\right )} \sinh \left (x\right )^{4} + 4 \, {\left (70 \, x \cosh \left (x\right )^{3} + {\left (28 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (7 \, x + 5\right )} \cosh \left (x\right )^{2} + 2 \, {\left (105 \, x \cosh \left (x\right )^{4} + 3 \, {\left (28 \, x + 1\right )} \cosh \left (x\right )^{2} + 7 \, x + 5\right )} \sinh \left (x\right )^{2} - 8 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (21 \, x \cosh \left (x\right )^{5} + {\left (28 \, x + 1\right )} \cosh \left (x\right )^{3} + {\left (7 \, x + 5\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]

[In]

integrate(tanh(x)^3/(1+coth(x)),x, algorithm="fricas")

[Out]

-1/4*(14*x*cosh(x)^6 + 84*x*cosh(x)*sinh(x)^5 + 14*x*sinh(x)^6 + (28*x + 1)*cosh(x)^4 + (210*x*cosh(x)^2 + 28*
x + 1)*sinh(x)^4 + 4*(70*x*cosh(x)^3 + (28*x + 1)*cosh(x))*sinh(x)^3 + 2*(7*x + 5)*cosh(x)^2 + 2*(105*x*cosh(x
)^4 + 3*(28*x + 1)*cosh(x)^2 + 7*x + 5)*sinh(x)^2 - 8*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(
x)^2 + 2)*sinh(x)^4 + 2*cosh(x)^4 + 4*(5*cosh(x)^3 + 2*cosh(x))*sinh(x)^3 + (15*cosh(x)^4 + 12*cosh(x)^2 + 1)*
sinh(x)^2 + cosh(x)^2 + 2*(3*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) +
4*(21*x*cosh(x)^5 + (28*x + 1)*cosh(x)^3 + (7*x + 5)*cosh(x))*sinh(x) + 1)/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 +
sinh(x)^6 + (15*cosh(x)^2 + 2)*sinh(x)^4 + 2*cosh(x)^4 + 4*(5*cosh(x)^3 + 2*cosh(x))*sinh(x)^3 + (15*cosh(x)^4
+ 12*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(3*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \]

[In]

integrate(tanh(x)**3/(1+coth(x)),x)

[Out]

Integral(tanh(x)**3/(coth(x) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=\frac {1}{2} \, x + \frac {2 \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \frac {1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

[In]

integrate(tanh(x)^3/(1+coth(x)),x, algorithm="maxima")

[Out]

1/2*x + 2*(2*e^(-2*x) + 1)/(2*e^(-2*x) + e^(-4*x) + 1) - 1/4*e^(-2*x) + 2*log(e^(-2*x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=-\frac {7}{2} \, x - \frac {{\left (e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} + 2 \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]

[In]

integrate(tanh(x)^3/(1+coth(x)),x, algorithm="giac")

[Out]

-7/2*x - 1/4*(e^(4*x) + 10*e^(2*x) + 1)*e^(-2*x)/(e^(2*x) + 1)^2 + 2*log(e^(2*x) + 1)

Mupad [B] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx=2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )-\frac {7\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {2}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]

[In]

int(tanh(x)^3/(coth(x) + 1),x)

[Out]

2*log(exp(2*x) + 1) - (7*x)/2 - exp(-2*x)/4 - 2/(2*exp(2*x) + exp(4*x) + 1)