3.196 \(\int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=41 \[ \frac {\coth ^5(x)}{5 a}-\frac {\text {csch}^5(x)}{5 a}-\frac {2 \text {csch}^3(x)}{3 a}-\frac {\text {csch}(x)}{a} \]

[Out]

1/5*coth(x)^5/a-csch(x)/a-2/3*csch(x)^3/a-1/5*csch(x)^5/a

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Rubi [A]  time = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2606, 194} \[ \frac {\coth ^5(x)}{5 a}-\frac {\text {csch}^5(x)}{5 a}-\frac {2 \text {csch}^3(x)}{3 a}-\frac {\text {csch}(x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^4/(a + a*Cosh[x]),x]

[Out]

Coth[x]^5/(5*a) - Csch[x]/a - (2*Csch[x]^3)/(3*a) - Csch[x]^5/(5*a)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\coth ^4(x)}{a+a \cosh (x)} \, dx &=\frac {\int \coth ^5(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth ^4(x) \text {csch}^2(x) \, dx}{a}\\ &=-\frac {i \operatorname {Subst}\left (\int x^4 \, dx,x,i \coth (x)\right )}{a}-\frac {i \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=\frac {\coth ^5(x)}{5 a}-\frac {i \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=\frac {\coth ^5(x)}{5 a}-\frac {\text {csch}(x)}{a}-\frac {2 \text {csch}^3(x)}{3 a}-\frac {\text {csch}^5(x)}{5 a}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 41, normalized size = 1.00 \[ -\frac {(8 \cosh (x)+36 \cosh (2 x)+24 \cosh (3 x)-3 \cosh (4 x)-25) \text {csch}^3(x)}{120 a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/120*((-25 + 8*Cosh[x] + 36*Cosh[2*x] + 24*Cosh[3*x] - 3*Cosh[4*x])*Csch[x]^3)/(a*(1 + Cosh[x]))

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fricas [B]  time = 0.44, size = 224, normalized size = 5.46 \[ -\frac {2 \, {\left (15 \, \cosh \relax (x)^{4} + 6 \, {\left (10 \, \cosh \relax (x) + 3\right )} \sinh \relax (x)^{3} + 15 \, \sinh \relax (x)^{4} + 12 \, \cosh \relax (x)^{3} + 2 \, {\left (45 \, \cosh \relax (x)^{2} + 18 \, \cosh \relax (x) + 2\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 2 \, {\left (30 \, \cosh \relax (x)^{3} + 27 \, \cosh \relax (x)^{2} - 14 \, \cosh \relax (x) - 23\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 13\right )}}{15 \, {\left (a \cosh \relax (x)^{5} + a \sinh \relax (x)^{5} + 2 \, a \cosh \relax (x)^{4} + {\left (5 \, a \cosh \relax (x) + 2 \, a\right )} \sinh \relax (x)^{4} - 3 \, a \cosh \relax (x)^{3} + {\left (10 \, a \cosh \relax (x)^{2} + 8 \, a \cosh \relax (x) - a\right )} \sinh \relax (x)^{3} - 8 \, a \cosh \relax (x)^{2} + {\left (10 \, a \cosh \relax (x)^{3} + 12 \, a \cosh \relax (x)^{2} - 9 \, a \cosh \relax (x) - 8 \, a\right )} \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + {\left (5 \, a \cosh \relax (x)^{4} + 8 \, a \cosh \relax (x)^{3} - 3 \, a \cosh \relax (x)^{2} - 8 \, a \cosh \relax (x) - 2 \, a\right )} \sinh \relax (x) + 6 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*cosh(x)^4 + 6*(10*cosh(x) + 3)*sinh(x)^3 + 15*sinh(x)^4 + 12*cosh(x)^3 + 2*(45*cosh(x)^2 + 18*cosh(x
) + 2)*sinh(x)^2 + 4*cosh(x)^2 + 2*(30*cosh(x)^3 + 27*cosh(x)^2 - 14*cosh(x) - 23)*sinh(x) - 4*cosh(x) + 13)/(
a*cosh(x)^5 + a*sinh(x)^5 + 2*a*cosh(x)^4 + (5*a*cosh(x) + 2*a)*sinh(x)^4 - 3*a*cosh(x)^3 + (10*a*cosh(x)^2 +
8*a*cosh(x) - a)*sinh(x)^3 - 8*a*cosh(x)^2 + (10*a*cosh(x)^3 + 12*a*cosh(x)^2 - 9*a*cosh(x) - 8*a)*sinh(x)^2 +
2*a*cosh(x) + (5*a*cosh(x)^4 + 8*a*cosh(x)^3 - 3*a*cosh(x)^2 - 8*a*cosh(x) - 2*a)*sinh(x) + 6*a)

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giac [A]  time = 0.16, size = 59, normalized size = 1.44 \[ -\frac {15 \, e^{\left (2 \, x\right )} - 24 \, e^{x} + 13}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {165 \, e^{\left (4 \, x\right )} + 480 \, e^{\left (3 \, x\right )} + 650 \, e^{\left (2 \, x\right )} + 400 \, e^{x} + 113}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(15*e^(2*x) - 24*e^x + 13)/(a*(e^x - 1)^3) - 1/120*(165*e^(4*x) + 480*e^(3*x) + 650*e^(2*x) + 400*e^x +
113)/(a*(e^x + 1)^5)

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maple [A]  time = 0.10, size = 45, normalized size = 1.10 \[ \frac {\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}+\frac {4 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+6 \tanh \left (\frac {x}{2}\right )-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {4}{\tanh \left (\frac {x}{2}\right )}}{16 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4/(a+a*cosh(x)),x)

[Out]

1/16/a*(1/5*tanh(1/2*x)^5+4/3*tanh(1/2*x)^3+6*tanh(1/2*x)-1/3/tanh(1/2*x)^3-4/tanh(1/2*x))

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maxima [B]  time = 0.35, size = 469, normalized size = 11.44 \[ -\frac {6 \, e^{\left (-x\right )}}{5 \, {\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 )}} - \frac {14 \, e^{\left (-2 \, x\right )}}{5 \, {\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 )}} - \frac {26 \, e^{\left (-3 \, x\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 )}} + \frac {10 \, e^{\left (-4 \, x\right )}}{3 \, {\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 )}} + \frac {2 \, e^{\left (-5 \, x\right )}}{3 \, {\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 )}} - \frac {2 \, e^{\left (-6 \, x\right )}}{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} - \frac {2 \, e^{\left (-7 \, x\right )}}{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} + \frac {2}{5 \, {\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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*e^(-x)/(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) - 14/5*e^(-2*x)/(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) - 26/15*e^(-3*x)/(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) + 10/3*e^(-4*x)/(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) + 2/3*e^(-5*x)/(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) - 2*e^(-6*x)/(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) - 2*e^(-7*x)/(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) + 2/5/(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)

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mupad [B]  time = 1.03, size = 263, normalized size = 6.41 \[ \frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {11\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{8\,a}+\frac {17\,{\mathrm {e}}^x}{40\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {11\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {17}{120\,a}+\frac {{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{8\,a}+\frac {11\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{20\,a}+\frac {{\mathrm {e}}^{3\,x}}{2\,a}+\frac {11\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {11}{40\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}-\frac {5}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {11}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4/(a + a*cosh(x)),x)

[Out]

1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - ((3*exp(2*x))/(8*a) + (11*exp(3*x))/(40*a) + 1/(8*a) + (17*ex
p(x))/(40*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) - ((11*exp(2*x))/(40*a) + 17/(120*a) + exp(x
)/(4*a))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) - (1/(8*a) + (11*exp(x))/(40*a))/(exp(2*x) + 2*exp(x) + 1) - (
(17*exp(2*x))/(20*a) + exp(3*x)/(2*a) + (11*exp(4*x))/(40*a) + 11/(40*a) + exp(x)/(2*a))/(10*exp(2*x) + 10*exp
(3*x) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) - 5/(8*a*(exp(x) - 1)) - 11/
(40*a*(exp(x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\coth ^{4}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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