\(\int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx\) [112]

Optimal result

Integrand size = 13, antiderivative size = 55 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a} \]

[Out]

x/a-1/15*coth(x)*(15-8*sech(x))/a-1/15*coth(x)^3*(5-4*sech(x))/a-1/5*coth(x)^5*(1-sech(x))/a

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3967, 8} \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a} \]

[In]

Int[Coth[x]^4/(a + a*Sech[x]),x]

[Out]

x/a - (Coth[x]*(15 - 8*Sech[x]))/(15*a) - (Coth[x]^3*(5 - 4*Sech[x]))/(15*a) - (Coth[x]^5*(1 - Sech[x]))/(5*a)

Rule 8

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

Rule 3967

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-(e*Cot[c
+ d*x])^(m + 1))*((a + b*Csc[c + d*x])/(d*e*(m + 1))), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)
*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 3973

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \coth ^6(x) (-a+a \text {sech}(x)) \, dx}{a^2} \\ & = -\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}+\frac {\int \coth ^4(x) (5 a-4 a \text {sech}(x)) \, dx}{5 a^2} \\ & = -\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}-\frac {\int \coth ^2(x) (-15 a+8 a \text {sech}(x)) \, dx}{15 a^2} \\ & = -\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}+\frac {\int 15 a \, dx}{15 a^2} \\ & = \frac {x}{a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.25 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {csch}^3(x) \text {sech}(x) (-25+8 \cosh (x)+16 \cosh (2 x)-16 \cosh (3 x)-23 \cosh (4 x)-90 x \sinh (x)-30 x \sinh (2 x)+30 x \sinh (3 x)+15 x \sinh (4 x))}{120 a (1+\text {sech}(x))} \]

[In]

Integrate[Coth[x]^4/(a + a*Sech[x]),x]

[Out]

(Csch[x]^3*Sech[x]*(-25 + 8*Cosh[x] + 16*Cosh[2*x] - 16*Cosh[3*x] - 23*Cosh[4*x] - 90*x*Sinh[x] - 30*x*Sinh[2*
x] + 30*x*Sinh[3*x] + 15*x*Sinh[4*x]))/(120*a*(1 + Sech[x]))

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}-2 \tanh \left (\frac {x}{2}\right )^{3}-16 \tanh \left (\frac {x}{2}\right )+16 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-16 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {6}{\tanh \left (\frac {x}{2}\right )}}{16 a}\)	\(63\)
risch	\(\frac {x}{a}+\frac {2 \,{\mathrm e}^{7 x}-2 \,{\mathrm e}^{6 x}-\frac {26 \,{\mathrm e}^{5 x}}{3}-\frac {10 \,{\mathrm e}^{4 x}}{3}+\frac {146 \,{\mathrm e}^{3 x}}{15}+\frac {62 \,{\mathrm e}^{2 x}}{15}-\frac {62 \,{\mathrm e}^{x}}{15}-\frac {46}{15}}{a \left ({\mathrm e}^{x}-1\right )^{3} \left ({\mathrm e}^{x}+1\right )^{5}}\)	\(66\)

[In]

int(coth(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

1/16/a*(-1/5*tanh(1/2*x)^5-2*tanh(1/2*x)^3-16*tanh(1/2*x)+16*ln(tanh(1/2*x)+1)-16*ln(tanh(1/2*x)-1)-1/3/tanh(1
/2*x)^3-6/tanh(1/2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (47) = 94\).

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.75 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {23 \, \cosh \left (x\right )^{4} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right ) + 15 \, x + 23\right )} \sinh \left (x\right )^{3} + 23 \, \sinh \left (x\right )^{4} + 16 \, \cosh \left (x\right )^{3} + 2 \, {\left (69 \, \cosh \left (x\right )^{2} + 24 \, \cosh \left (x\right ) - 8\right )} \sinh \left (x\right )^{2} - 16 \, \cosh \left (x\right )^{2} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right )^{3} + 3 \, {\left (15 \, x + 23\right )} \cosh \left (x\right )^{2} - 2 \, {\left (15 \, x + 23\right )} \cosh \left (x\right ) - 45 \, x - 69\right )} \sinh \left (x\right ) - 8 \, \cosh \left (x\right ) + 25}{30 \, {\left ({\left (2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + {\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) - 3 \, a\right )} \sinh \left (x\right )\right )}} \]

[In]

integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-1/30*(23*cosh(x)^4 - 2*(2*(15*x + 23)*cosh(x) + 15*x + 23)*sinh(x)^3 + 23*sinh(x)^4 + 16*cosh(x)^3 + 2*(69*co
sh(x)^2 + 24*cosh(x) - 8)*sinh(x)^2 - 16*cosh(x)^2 - 2*(2*(15*x + 23)*cosh(x)^3 + 3*(15*x + 23)*cosh(x)^2 - 2*
(15*x + 23)*cosh(x) - 45*x - 69)*sinh(x) - 8*cosh(x) + 25)/((2*a*cosh(x) + a)*sinh(x)^3 + (2*a*cosh(x)^3 + 3*a
*cosh(x)^2 - 2*a*cosh(x) - 3*a)*sinh(x))

Sympy [F]

\[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\coth ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(coth(x)**4/(a+a*sech(x)),x)

[Out]

Integral(coth(x)**4/(sech(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (47) = 94\).

Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.91 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {2 \, {\left (31 \, e^{\left (-x\right )} - 31 \, e^{\left (-2 \, x\right )} - 73 \, e^{\left (-3 \, x\right )} + 25 \, e^{\left (-4 \, x\right )} + 65 \, e^{\left (-5 \, x\right )} + 15 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} + 23\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]

[In]

integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="maxima")

[Out]

x/a - 2/15*(31*e^(-x) - 31*e^(-2*x) - 73*e^(-3*x) + 25*e^(-4*x) + 65*e^(-5*x) + 15*e^(-6*x) - 15*e^(-7*x) + 23
)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*x) + a)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {21 \, e^{\left (2 \, x\right )} - 36 \, e^{x} + 19}{24 \, a {\left (e^{x} - 1\right )}^{3}} + \frac {115 \, e^{\left (4 \, x\right )} + 380 \, e^{\left (3 \, x\right )} + 530 \, e^{\left (2 \, x\right )} + 340 \, e^{x} + 91}{40 \, a {\left (e^{x} + 1\right )}^{5}} \]

[In]

integrate(coth(x)^4/(a+a*sech(x)),x, algorithm="giac")

[Out]

x/a - 1/24*(21*e^(2*x) - 36*e^x + 19)/(a*(e^x - 1)^3) + 1/40*(115*e^(4*x) + 380*e^(3*x) + 530*e^(2*x) + 340*e^
x + 91)/(a*(e^x + 1)^5)

Mupad [B] (verification not implemented)

Time = 2.20 (sec) , antiderivative size = 264, normalized size of antiderivative = 4.80 \[ \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\frac {9\,{\mathrm {e}}^{2\,x}}{4\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{2\,a}+\frac {23\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {23}{40\,a}+\frac {3\,{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}+\frac {\frac {9\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {23\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {3}{8\,a}+\frac {9\,{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}+\frac {\frac {23\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {3}{8\,a}+\frac {3\,{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {3}{8\,a}+\frac {23\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {x}{a}-\frac {7}{8\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {23}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]

[In]

int(coth(x)^4/(a + a/cosh(x)),x)

[Out]

((9*exp(2*x))/(4*a) + (3*exp(3*x))/(2*a) + (23*exp(4*x))/(40*a) + 23/(40*a) + (3*exp(x))/(2*a))/(10*exp(2*x) +
10*exp(3*x) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1) + ((9*exp(2*x))/(8*a) + (23*exp(3*x))/(40*a) + 3/(8*a) +
(9*exp(x))/(8*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) + ((23*exp(2*x))/(40*a) + 3/(8*a) + (3*e
xp(x))/(4*a))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) + (3/(8*a) + (23*exp(x))/(40*a))/(exp(2*x) + 2*exp(x) + 1
) + 1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + x/a - 7/(8*a*(exp(x)
- 1)) + 23/(40*a*(exp(x) + 1))