3.2.62 \(\int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx\) [162]

Optimal. Leaf size=49 \[ \frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))} \]

[Out]

3/8*arctanh(cosh(x))/a+1/8/(a-a*cosh(x))-1/8*a/(a+a*cosh(x))^2-1/4/(a+a*cosh(x))

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Rubi [A]
time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 212} \begin {gather*} -\frac {a}{8 (a \cosh (x)+a)^2}+\frac {1}{8 (a-a \cosh (x))}-\frac {1}{4 (a \cosh (x)+a)}+\frac {3 \tanh ^{-1}(\cosh (x))}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^3/(a + a*Cosh[x]),x]

[Out]

(3*ArcTanh[Cosh[x]])/(8*a) + 1/(8*(a - a*Cosh[x])) - a/(8*(a + a*Cosh[x])^2) - 1/(4*(a + a*Cosh[x]))

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2746

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx &=a^3 \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \cosh (x)\right )\\ &=a^3 \text {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \cosh (x)\right )\\ &=\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))}+\frac {3}{8} \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right )\\ &=\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 60, normalized size = 1.22 \begin {gather*} -\frac {4+2 \coth ^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )}{16 a (1+\cosh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^3/(a + a*Cosh[x]),x]

[Out]

-1/16*(4 + 2*Coth[x/2]^2 - 12*Cosh[x/2]^2*(Log[Cosh[x/2]] - Log[Sinh[x/2]]) + Sech[x/2]^2)/(a*(1 + Cosh[x]))

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Maple [A]
time = 0.60, size = 38, normalized size = 0.78

method result size
default \(\frac {-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}-3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}\) \(38\)
risch \(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}+3\right )}{4 \left ({\mathrm e}^{x}-1\right )^{2} a \left ({\mathrm e}^{x}+1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}\) \(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^3/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/8/a*(-1/4*tanh(1/2*x)^4+3/2*tanh(1/2*x)^2-1/2/tanh(1/2*x)^2-3*ln(tanh(1/2*x)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (42) = 84\).
time = 0.26, size = 103, normalized size = 2.10 \begin {gather*} -\frac {3 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/4*(3*e^(-x) + 6*e^(-2*x) - 2*e^(-3*x) + 6*e^(-4*x) + 3*e^(-5*x))/(2*a*e^(-x) - a*e^(-2*x) - 4*a*e^(-3*x) -
a*e^(-4*x) + 2*a*e^(-5*x) + a*e^(-6*x) + a) + 3/8*log(e^(-x) + 1)/a - 3/8*log(e^(-x) - 1)/a

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (42) = 84\).
time = 0.37, size = 631, normalized size = 12.88 \begin {gather*} -\frac {6 \, \cosh \left (x\right )^{5} + 6 \, {\left (5 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{4} + 6 \, \sinh \left (x\right )^{5} + 12 \, \cosh \left (x\right )^{4} + 4 \, {\left (15 \, \cosh \left (x\right )^{2} + 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + 12 \, {\left (5 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 12 \, \cosh \left (x\right )^{2} - 3 \, {\left (\cosh \left (x\right )^{6} + 2 \, {\left (3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 2 \, \cosh \left (x\right )^{5} + {\left (15 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 3 \, {\left (\cosh \left (x\right )^{6} + 2 \, {\left (3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 2 \, \cosh \left (x\right )^{5} + {\left (15 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )}{8 \, {\left (a \cosh \left (x\right )^{6} + a \sinh \left (x\right )^{6} + 2 \, a \cosh \left (x\right )^{5} + 2 \, {\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{5} - a \cosh \left (x\right )^{4} + {\left (15 \, a \cosh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{4} - 4 \, a \cosh \left (x\right )^{3} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} - a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{3} - a \cosh \left (x\right )^{2} + {\left (15 \, a \cosh \left (x\right )^{4} + 20 \, a \cosh \left (x\right )^{3} - 6 \, a \cosh \left (x\right )^{2} - 12 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{4} - 2 \, a \cosh \left (x\right )^{3} - 6 \, a \cosh \left (x\right )^{2} - a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

-1/8*(6*cosh(x)^5 + 6*(5*cosh(x) + 2)*sinh(x)^4 + 6*sinh(x)^5 + 12*cosh(x)^4 + 4*(15*cosh(x)^2 + 12*cosh(x) -
1)*sinh(x)^3 - 4*cosh(x)^3 + 12*(5*cosh(x)^3 + 6*cosh(x)^2 - cosh(x) + 1)*sinh(x)^2 + 12*cosh(x)^2 - 3*(cosh(x
)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh
(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6
*cosh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2
- cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) + 1) + 3*(cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^
5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(
x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x)^2 - 12*cosh(x) - 1)*sin
h(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(
x) + 1)*log(cosh(x) + sinh(x) - 1) + 6*(5*cosh(x)^4 + 8*cosh(x)^3 - 2*cosh(x)^2 + 4*cosh(x) + 1)*sinh(x) + 6*c
osh(x))/(a*cosh(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x)^5 - a*cosh(x)^4 + (15*a*cosh(
x)^2 + 10*a*cosh(x) - a)*sinh(x)^4 - 4*a*cosh(x)^3 + 4*(5*a*cosh(x)^3 + 5*a*cosh(x)^2 - a*cosh(x) - a)*sinh(x)
^3 - a*cosh(x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 6*a*cosh(x)^2 - 12*a*cosh(x) - a)*sinh(x)^2 + 2*a*cosh(x
) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x)^4 - 2*a*cosh(x)^3 - 6*a*cosh(x)^2 - a*cosh(x) + a)*sinh(x) + a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**3/(a+a*cosh(x)),x)

[Out]

Integral(csch(x)**3/(cosh(x) + 1), x)/a

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (42) = 84\).
time = 0.41, size = 94, normalized size = 1.92 \begin {gather*} \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 10}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 52 \, e^{\left (-x\right )} + 52 \, e^{x} + 84}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(a+a*cosh(x)),x, algorithm="giac")

[Out]

3/16*log(e^(-x) + e^x + 2)/a - 3/16*log(e^(-x) + e^x - 2)/a + 1/16*(3*e^(-x) + 3*e^x - 10)/(a*(e^(-x) + e^x -
2)) - 1/32*(9*(e^(-x) + e^x)^2 + 52*e^(-x) + 52*e^x + 84)/(a*(e^(-x) + e^x + 2)^2)

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Mupad [B]
time = 0.93, size = 114, normalized size = 2.33 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^3*(a + a*cosh(x))),x)

[Out]

(3*atan((exp(x)*(-a^2)^(1/2))/a))/(4*(-a^2)^(1/2)) - 1/(2*a*(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1
)) - 1/(4*a*(exp(x) - 1)) - 1/(2*a*(exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/(a*(3*exp(2*x) + exp(
3*x) + 3*exp(x) + 1))

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