3.1.77 \(\int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx\) [77]

Optimal. Leaf size=34 \[ \frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x) \]

[Out]

1/8*arctanh(cosh(x))-1/8*coth(x)*csch(x)+1/3*csch(x)^3-1/4*coth(x)*csch(x)^3

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Rubi [A]
time = 0.14, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3599, 3187, 3186, 2686, 30, 2691, 3853, 3855} \begin {gather*} \frac {\text {csch}^3(x)}{3}+\frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{4} \coth (x) \text {csch}^3(x)-\frac {1}{8} \coth (x) \text {csch}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^5/(1 + Tanh[x]),x]

[Out]

ArcTanh[Cosh[x]]/8 - (Coth[x]*Csch[x])/8 + Csch[x]^3/3 - (Coth[x]*Csch[x]^3)/4

Rule 30

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

Rule 2686

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 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3186

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c +
d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]

Rule 3187

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[a^p*b^p, Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin
[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]

Rule 3599

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Sin[e + f*x]^
m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/Cos[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^5(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x) \text {csch}^4(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text {csch}^4(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=-\int \left (\coth (x) \text {csch}^3(x)-\coth ^2(x) \text {csch}^3(x)\right ) \, dx\\ &=-\int \coth (x) \text {csch}^3(x) \, dx+\int \coth ^2(x) \text {csch}^3(x) \, dx\\ &=-\frac {1}{4} \coth (x) \text {csch}^3(x)-i \text {Subst}\left (\int x^2 \, dx,x,-i \text {csch}(x)\right )+\frac {1}{4} \int \text {csch}^3(x) \, dx\\ &=-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x)-\frac {1}{8} \int \text {csch}(x) \, dx\\ &=\frac {1}{8} \tanh ^{-1}(\cosh (x))-\frac {1}{8} \coth (x) \text {csch}(x)+\frac {\text {csch}^3(x)}{3}-\frac {1}{4} \coth (x) \text {csch}^3(x)\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 49, normalized size = 1.44 \begin {gather*} -\frac {1}{192} \text {csch}^4(x) \left (42 \cosh (x)+6 \cosh (3 x)+2 \sinh (x) \left (-32-9 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+3 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^5/(1 + Tanh[x]),x]

[Out]

-1/192*(Csch[x]^4*(42*Cosh[x] + 6*Cosh[3*x] + 2*Sinh[x]*(-32 - 9*Log[Tanh[x/2]]*Sinh[x] + 3*Log[Tanh[x/2]]*Sin
h[3*x])))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(26)=52\).
time = 0.63, size = 55, normalized size = 1.62

method	result	size
risch	\(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}-11 \,{\mathrm e}^{4 x}+53 \,{\mathrm e}^{2 x}+3\right )}{12 \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{x}+1\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{8}\)	\(48\)
default	\(\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^5/(1+tanh(x)),x,method=_RETURNVERBOSE)

[Out]

1/64*tanh(1/2*x)^4-1/24*tanh(1/2*x)^3+1/8*tanh(1/2*x)-1/64/tanh(1/2*x)^4-1/8/tanh(1/2*x)-1/8*ln(tanh(1/2*x))+1
/24/tanh(1/2*x)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
time = 0.27, size = 74, normalized size = 2.18 \begin {gather*} \frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} + 53 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{12 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(1+tanh(x)),x, algorithm="maxima")

[Out]

1/12*(3*e^(-x) - 11*e^(-3*x) + 53*e^(-5*x) + 3*e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1)
+ 1/8*log(e^(-x) + 1) - 1/8*log(e^(-x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (26) = 52\).
time = 0.44, size = 640, normalized size = 18.82 \begin {gather*} -\frac {6 \, \cosh \left (x\right )^{7} + 42 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 6 \, \sinh \left (x\right )^{7} + 2 \, {\left (63 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{5} - 22 \, \cosh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{3} - 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} - 110 \, \cosh \left (x\right )^{2} + 53\right )} \sinh \left (x\right )^{3} + 106 \, \cosh \left (x\right )^{3} + 2 \, {\left (63 \, \cosh \left (x\right )^{5} - 110 \, \cosh \left (x\right )^{3} + 159 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, {\left (21 \, \cosh \left (x\right )^{6} - 55 \, \cosh \left (x\right )^{4} + 159 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )}{24 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(1+tanh(x)),x, algorithm="fricas")

[Out]

-1/24*(6*cosh(x)^7 + 42*cosh(x)*sinh(x)^6 + 6*sinh(x)^7 + 2*(63*cosh(x)^2 - 11)*sinh(x)^5 - 22*cosh(x)^5 + 10*
(21*cosh(x)^3 - 11*cosh(x))*sinh(x)^4 + 2*(105*cosh(x)^4 - 110*cosh(x)^2 + 53)*sinh(x)^3 + 106*cosh(x)^3 + 2*(
63*cosh(x)^5 - 110*cosh(x)^3 + 159*cosh(x))*sinh(x)^2 - 3*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*
cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^
2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*co
sh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(
x) + 1)*log(cosh(x) + sinh(x) + 1) + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh
(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 +
6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)
^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x)
+ sinh(x) - 1) + 2*(21*cosh(x)^6 - 55*cosh(x)^4 + 159*cosh(x)^2 + 3)*sinh(x) + 6*cosh(x))/(cosh(x)^8 + 8*cosh
(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^
5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*s
inh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)
^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{5}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**5/(1+tanh(x)),x)

[Out]

Integral(csch(x)**5/(tanh(x) + 1), x)

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Giac [A]
time = 0.42, size = 49, normalized size = 1.44 \begin {gather*} -\frac {3 \, e^{\left (7 \, x\right )} - 11 \, e^{\left (5 \, x\right )} + 53 \, e^{\left (3 \, x\right )} + 3 \, e^{x}}{12 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + \frac {1}{8} \, \log \left (e^{x} + 1\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(1+tanh(x)),x, algorithm="giac")

[Out]

-1/12*(3*e^(7*x) - 11*e^(5*x) + 53*e^(3*x) + 3*e^x)/(e^(2*x) - 1)^4 + 1/8*log(e^x + 1) - 1/8*log(abs(e^x - 1))

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Mupad [B]
time = 1.09, size = 117, normalized size = 3.44 \begin {gather*} \frac {\ln \left (\frac {{\mathrm {e}}^x}{4}+\frac {1}{4}\right )}{8}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{4}-\frac {1}{4}\right )}{8}-\frac {{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{3\,x}+2\,{\mathrm {e}}^x}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {4\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^5*(tanh(x) + 1)),x)

[Out]

log(exp(x)/4 + 1/4)/8 - log(exp(x)/4 - 1/4)/8 - exp(x)/(4*(exp(2*x) - 1)) - (2*exp(3*x) + 2*exp(x))/(6*exp(4*x
) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - (4*exp(x))/(3*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) + exp(x)
/(6*(exp(4*x) - 2*exp(2*x) + 1))

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