3.7.55 \(\int (\coth (x)+\text {csch}(x))^4 \, dx\) [655]

Optimal. Leaf size=30 \[ x+\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3} \]

[Out]

x+2*sinh(x)/(1-cosh(x))+2/3*sinh(x)^3/(1-cosh(x))^3

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Rubi [A]
time = 0.08, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4477, 2749, 2759, 8} \begin {gather*} x+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac {2 \sinh (x)}{1-\cosh (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Coth[x] + Csch[x])^4,x]

[Out]

x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)

Rule 8

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

Rule 2749

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a/g)^
(2*m), Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 -
b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 4477

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

\begin {align*} \int (\coth (x)+\text {csch}(x))^4 \, dx &=\int (i+i \cosh (x))^4 \text {csch}^4(x) \, dx\\ &=\int \frac {\sinh ^4(x)}{(i-i \cosh (x))^4} \, dx\\ &=\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}-\int \frac {\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\int 1 \, dx\\ &=x+\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} x-\frac {8}{3} \coth \left (\frac {x}{2}\right )-\frac {2}{3} \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^4,x]

[Out]

x - (8*Coth[x/2])/3 - (2*Coth[x/2]*Csch[x/2]^2)/3

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Maple [A]
time = 0.85, size = 23, normalized size = 0.77

method	result	size
risch	\(x -\frac {8 \left (3 \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{x}+2\right )}{3 \left ({\mathrm e}^{x}-1\right )^{3}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x)+csch(x))^4,x,method=_RETURNVERBOSE)

[Out]

x-8/3*(3*exp(2*x)-3*exp(x)+2)/(exp(x)-1)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (24) = 48\).
time = 0.27, size = 183, normalized size = 6.10 \begin {gather*} -2 \, \coth \left (x\right )^{3} + x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {8 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {16 \, e^{\left (-3 \, x\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {8 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {4}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {32}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="maxima")

[Out]

-2*coth(x)^3 + x - 4/3*(3*e^(-2*x) - 3*e^(-4*x) - 2)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 8*e^(-x)/(3*e^
(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 4*e^(-2*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) - 16/3*e^(-3*x)/(3*e
^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 8*e^(-5*x)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) - 4/3/(3*e^(-2*x) -
3*e^(-4*x) + e^(-6*x) - 1) + 32/3/(e^(-x) - e^x)^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
time = 0.36, size = 68, normalized size = 2.27 \begin {gather*} \frac {3 \, x \cosh \left (x\right )^{2} + 3 \, x \sinh \left (x\right )^{2} - 4 \, {\left (3 \, x + 10\right )} \cosh \left (x\right ) + 2 \, {\left (3 \, x \cosh \left (x\right ) - 3 \, x - 4\right )} \sinh \left (x\right ) + 9 \, x + 24}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="fricas")

[Out]

1/3*(3*x*cosh(x)^2 + 3*x*sinh(x)^2 - 4*(3*x + 10)*cosh(x) + 2*(3*x*cosh(x) - 3*x - 4)*sinh(x) + 9*x + 24)/(cos
h(x)^2 + 2*(cosh(x) - 1)*sinh(x) + sinh(x)^2 - 4*cosh(x) + 3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))**4,x)

[Out]

Integral((coth(x) + csch(x))**4, x)

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Giac [A]
time = 0.41, size = 22, normalized size = 0.73 \begin {gather*} x - \frac {8 \, {\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 2\right )}}{3 \, {\left (e^{x} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^4,x, algorithm="giac")

[Out]

x - 8/3*(3*e^(2*x) - 3*e^x + 2)/(e^x - 1)^3

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Mupad [B]
time = 0.05, size = 59, normalized size = 1.97 \begin {gather*} x-\frac {8\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {8}{3}}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}-\frac {8}{3\,\left ({\mathrm {e}}^x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x) + 1/sinh(x))^4,x)

[Out]

x - (8*exp(x))/(3*(exp(2*x) - 2*exp(x) + 1)) + ((8*exp(2*x))/3 + 8/3)/(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1) -
8/(3*(exp(x) - 1))

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