3.79 \(\int \frac{\text{csch}^7(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{\text{csch}^5(x)}{5}-\frac{1}{16} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \coth (x) \text{csch}^5(x)-\frac{1}{24} \coth (x) \text{csch}^3(x)+\frac{1}{16} \coth (x) \text{csch}(x) \]

[Out]

-ArcTanh[Cosh[x]]/16 + (Coth[x]*Csch[x])/16 - (Coth[x]*Csch[x]^3)/24 + Csch[x]^5/5 - (Coth[x]*Csch[x]^5)/6

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Rubi [A]  time = 0.210609, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {3518, 3108, 3107, 2606, 30, 2611, 3768, 3770} \[ \frac{\text{csch}^5(x)}{5}-\frac{1}{16} \tanh ^{-1}(\cosh (x))-\frac{1}{6} \coth (x) \text{csch}^5(x)-\frac{1}{24} \coth (x) \text{csch}^3(x)+\frac{1}{16} \coth (x) \text{csch}(x) \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^7/(1 + Tanh[x]),x]

[Out]

-ArcTanh[Cosh[x]]/16 + (Coth[x]*Csch[x])/16 - (Coth[x]*Csch[x]^3)/24 + Csch[x]^5/5 - (Coth[x]*Csch[x]^5)/6

Rule 3518

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 3108

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 3107

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 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 30

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

Rule 2611

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 3768

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 3770

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}^7(x)}{1+\tanh (x)} \, dx &=\int \frac{\coth (x) \text{csch}^6(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text{csch}^6(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=\int \left (-\coth (x) \text{csch}^5(x)+\coth ^2(x) \text{csch}^5(x)\right ) \, dx\\ &=-\int \coth (x) \text{csch}^5(x) \, dx+\int \coth ^2(x) \text{csch}^5(x) \, dx\\ &=-\frac{1}{6} \coth (x) \text{csch}^5(x)+i \operatorname{Subst}\left (\int x^4 \, dx,x,-i \text{csch}(x)\right )+\frac{1}{6} \int \text{csch}^5(x) \, dx\\ &=-\frac{1}{24} \coth (x) \text{csch}^3(x)+\frac{\text{csch}^5(x)}{5}-\frac{1}{6} \coth (x) \text{csch}^5(x)-\frac{1}{8} \int \text{csch}^3(x) \, dx\\ &=\frac{1}{16} \coth (x) \text{csch}(x)-\frac{1}{24} \coth (x) \text{csch}^3(x)+\frac{\text{csch}^5(x)}{5}-\frac{1}{6} \coth (x) \text{csch}^5(x)+\frac{1}{16} \int \text{csch}(x) \, dx\\ &=-\frac{1}{16} \tanh ^{-1}(\cosh (x))+\frac{1}{16} \coth (x) \text{csch}(x)-\frac{1}{24} \coth (x) \text{csch}^3(x)+\frac{\text{csch}^5(x)}{5}-\frac{1}{6} \coth (x) \text{csch}^5(x)\\ \end{align*}

Mathematica [A]  time = 0.263901, size = 68, normalized size = 1.55 \[ \frac{\text{csch}^6(x) \left (-1140 \cosh (x)-170 \cosh (3 x)+30 \cosh (5 x)+6 \sinh (x) \left (50 \sinh (x) \log \left (\tanh \left (\frac{x}{2}\right )\right )-25 \sinh (3 x) \log \left (\tanh \left (\frac{x}{2}\right )\right )+5 \sinh (5 x) \log \left (\tanh \left (\frac{x}{2}\right )\right )+256\right )\right )}{7680} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^7/(1 + Tanh[x]),x]

[Out]

(Csch[x]^6*(-1140*Cosh[x] - 170*Cosh[3*x] + 30*Cosh[5*x] + 6*Sinh[x]*(256 + 50*Log[Tanh[x/2]]*Sinh[x] - 25*Log
[Tanh[x/2]]*Sinh[3*x] + 5*Log[Tanh[x/2]]*Sinh[5*x])))/7680

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Maple [B]  time = 0.033, size = 103, normalized size = 2.3 \begin{align*}{\frac{1}{384} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{6}}-{\frac{1}{160} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{1}{128} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}+{\frac{1}{32} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{128} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{16}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{128} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{384} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-6}}+{\frac{1}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{32} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{1}{160} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}}+{\frac{1}{16}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{1}{128} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^7/(1+tanh(x)),x)

[Out]

1/384*tanh(1/2*x)^6-1/160*tanh(1/2*x)^5-1/128*tanh(1/2*x)^4+1/32*tanh(1/2*x)^3-1/128*tanh(1/2*x)^2-1/16*tanh(1
/2*x)+1/128/tanh(1/2*x)^2-1/384/tanh(1/2*x)^6+1/16/tanh(1/2*x)-1/32/tanh(1/2*x)^3+1/160/tanh(1/2*x)^5+1/16*ln(
tanh(1/2*x))+1/128/tanh(1/2*x)^4

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Maxima [B]  time = 1.21299, size = 132, normalized size = 3. \begin{align*} -\frac{15 \, e^{\left (-x\right )} - 85 \, e^{\left (-3 \, x\right )} + 198 \, e^{\left (-5 \, x\right )} - 1338 \, e^{\left (-7 \, x\right )} - 85 \, e^{\left (-9 \, x\right )} + 15 \, e^{\left (-11 \, x\right )}}{120 \,{\left (6 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-4 \, x\right )} + 20 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-8 \, x\right )} + 6 \, e^{\left (-10 \, x\right )} - e^{\left (-12 \, x\right )} - 1\right )}} - \frac{1}{16} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{16} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^7/(1+tanh(x)),x, algorithm="maxima")

[Out]

-1/120*(15*e^(-x) - 85*e^(-3*x) + 198*e^(-5*x) - 1338*e^(-7*x) - 85*e^(-9*x) + 15*e^(-11*x))/(6*e^(-2*x) - 15*
e^(-4*x) + 20*e^(-6*x) - 15*e^(-8*x) + 6*e^(-10*x) - e^(-12*x) - 1) - 1/16*log(e^(-x) + 1) + 1/16*log(e^(-x) -
1)

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Fricas [B]  time = 2.31769, size = 4358, normalized size = 99.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^7/(1+tanh(x)),x, algorithm="fricas")

[Out]

1/240*(30*cosh(x)^11 + 330*cosh(x)*sinh(x)^10 + 30*sinh(x)^11 + 10*(165*cosh(x)^2 - 17)*sinh(x)^9 - 170*cosh(x
)^9 + 90*(55*cosh(x)^3 - 17*cosh(x))*sinh(x)^8 + 36*(275*cosh(x)^4 - 170*cosh(x)^2 + 11)*sinh(x)^7 + 396*cosh(
x)^7 + 84*(165*cosh(x)^5 - 170*cosh(x)^3 + 33*cosh(x))*sinh(x)^6 + 12*(1155*cosh(x)^6 - 1785*cosh(x)^4 + 693*c
osh(x)^2 - 223)*sinh(x)^5 - 2676*cosh(x)^5 + 60*(165*cosh(x)^7 - 357*cosh(x)^5 + 231*cosh(x)^3 - 223*cosh(x))*
sinh(x)^4 + 10*(495*cosh(x)^8 - 1428*cosh(x)^6 + 1386*cosh(x)^4 - 2676*cosh(x)^2 - 17)*sinh(x)^3 - 170*cosh(x)
^3 + 6*(275*cosh(x)^9 - 1020*cosh(x)^7 + 1386*cosh(x)^5 - 4460*cosh(x)^3 - 85*cosh(x))*sinh(x)^2 - 15*(cosh(x)
^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 -
3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*
cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x
)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6
+ 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 -
20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh
(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3
- cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) + 1) + 15*(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(
11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*
cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh
(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cos
h(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 +
15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*c
osh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(
x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x)
- 1) + 6*(55*cosh(x)^10 - 255*cosh(x)^8 + 462*cosh(x)^6 - 2230*cosh(x)^4 - 85*cosh(x)^2 + 5)*sinh(x) + 30*cosh
(x))/(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(1
1*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*co
sh(x)^5 - 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 - 315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^
6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cosh(x)^3 - 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 -
84*cosh(x)^6 + 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 4
2*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x
)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 - 5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 +
5*cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{7}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**7/(1+tanh(x)),x)

[Out]

Integral(csch(x)**7/(tanh(x) + 1), x)

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Giac [A]  time = 1.28759, size = 82, normalized size = 1.86 \begin{align*} \frac{15 \, e^{\left (11 \, x\right )} - 85 \, e^{\left (9 \, x\right )} + 198 \, e^{\left (7 \, x\right )} - 1338 \, e^{\left (5 \, x\right )} - 85 \, e^{\left (3 \, x\right )} + 15 \, e^{x}}{120 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{6}} - \frac{1}{16} \, \log \left (e^{x} + 1\right ) + \frac{1}{16} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^7/(1+tanh(x)),x, algorithm="giac")

[Out]

1/120*(15*e^(11*x) - 85*e^(9*x) + 198*e^(7*x) - 1338*e^(5*x) - 85*e^(3*x) + 15*e^x)/(e^(2*x) - 1)^6 - 1/16*log
(e^x + 1) + 1/16*log(abs(e^x - 1))