\(\int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx\) [164]

Optimal result

Integrand size = 13, antiderivative size = 78 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=-\frac {5 \text {arctanh}(\cosh (x))}{16 a}-\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))} \]

[Out]

-5/16*arctanh(cosh(x))/a-1/32*a/(a-a*cosh(x))^2-1/8/(a-a*cosh(x))+1/24*a^2/(a+a*cosh(x))^3+3/32*a/(a+a*cosh(x)
)^2+3/16/(a+a*cosh(x))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 212} \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\frac {a^2}{24 (a \cosh (x)+a)^3}-\frac {5 \text {arctanh}(\cosh (x))}{16 a}-\frac {a}{32 (a-a \cosh (x))^2}+\frac {3 a}{32 (a \cosh (x)+a)^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {3}{16 (a \cosh (x)+a)} \]

[In]

Int[Csch[x]^5/(a + a*Cosh[x]),x]

[Out]

(-5*ArcTanh[Cosh[x]])/(16*a) - a/(32*(a - a*Cosh[x])^2) - 1/(8*(a - a*Cosh[x])) + a^2/(24*(a + a*Cosh[x])^3) +
(3*a)/(32*(a + a*Cosh[x])^2) + 3/(16*(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*} \text {integral}& = -\left (a^5 \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^4} \, dx,x,a \cosh (x)\right )\right ) \\ & = -\left (a^5 \text {Subst}\left (\int \left (\frac {1}{16 a^4 (a-x)^3}+\frac {1}{8 a^5 (a-x)^2}+\frac {1}{8 a^3 (a+x)^4}+\frac {3}{16 a^4 (a+x)^3}+\frac {3}{16 a^5 (a+x)^2}+\frac {5}{16 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \cosh (x)\right )\right ) \\ & = -\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))}-\frac {5}{16} \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right ) \\ & = -\frac {5 \text {arctanh}(\cosh (x))}{16 a}-\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.14 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\frac {\cosh ^2\left (\frac {x}{2}\right ) \left (24 \text {csch}^2\left (\frac {x}{2}\right )-3 \text {csch}^4\left (\frac {x}{2}\right )-120 \log \left (\cosh \left (\frac {x}{2}\right )\right )+120 \log \left (\sinh \left (\frac {x}{2}\right )\right )+36 \text {sech}^2\left (\frac {x}{2}\right )+9 \text {sech}^4\left (\frac {x}{2}\right )+2 \text {sech}^6\left (\frac {x}{2}\right )\right )}{192 (a+a \cosh (x))} \]

[In]

Integrate[Csch[x]^5/(a + a*Cosh[x]),x]

[Out]

(Cosh[x/2]^2*(24*Csch[x/2]^2 - 3*Csch[x/2]^4 - 120*Log[Cosh[x/2]] + 120*Log[Sinh[x/2]] + 36*Sech[x/2]^2 + 9*Se
ch[x/2]^4 + 2*Sech[x/2]^6))/(192*(a + a*Cosh[x]))

Maple [A] (verified)

Time = 5.89 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{6}}{6}+\frac {5 \tanh \left (\frac {x}{2}\right )^{4}}{4}-5 \tanh \left (\frac {x}{2}\right )^{2}+10 \ln \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {1}{4 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {5}{2 \tanh \left (\frac {x}{2}\right )^{2}}}{32 a}\)	\(54\)
risch	\(\frac {{\mathrm e}^{x} \left (15 \,{\mathrm e}^{8 x}+30 \,{\mathrm e}^{7 x}-40 \,{\mathrm e}^{6 x}-110 \,{\mathrm e}^{5 x}+18 \,{\mathrm e}^{4 x}-110 \,{\mathrm e}^{3 x}-40 \,{\mathrm e}^{2 x}+30 \,{\mathrm e}^{x}+15\right )}{24 \left ({\mathrm e}^{x}+1\right )^{6} a \left ({\mathrm e}^{x}-1\right )^{4}}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{16 a}-\frac {5 \ln \left ({\mathrm e}^{x}+1\right )}{16 a}\)	\(89\)

[In]

int(csch(x)^5/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/32/a*(-1/6*tanh(1/2*x)^6+5/4*tanh(1/2*x)^4-5*tanh(1/2*x)^2+10*ln(tanh(1/2*x))-1/4/tanh(1/2*x)^4+5/2/tanh(1/2
*x)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1551 vs. \(2 (68) = 136\).

Time = 0.27 (sec) , antiderivative size = 1551, normalized size of antiderivative = 19.88 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \]

[In]

integrate(csch(x)^5/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/48*(30*cosh(x)^9 + 30*(9*cosh(x) + 2)*sinh(x)^8 + 30*sinh(x)^9 + 60*cosh(x)^8 + 40*(27*cosh(x)^2 + 12*cosh(x
) - 2)*sinh(x)^7 - 80*cosh(x)^7 + 20*(126*cosh(x)^3 + 84*cosh(x)^2 - 28*cosh(x) - 11)*sinh(x)^6 - 220*cosh(x)^
6 + 12*(315*cosh(x)^4 + 280*cosh(x)^3 - 140*cosh(x)^2 - 110*cosh(x) + 3)*sinh(x)^5 + 36*cosh(x)^5 + 20*(189*co
sh(x)^5 + 210*cosh(x)^4 - 140*cosh(x)^3 - 165*cosh(x)^2 + 9*cosh(x) - 11)*sinh(x)^4 - 220*cosh(x)^4 + 40*(63*c
osh(x)^6 + 84*cosh(x)^5 - 70*cosh(x)^4 - 110*cosh(x)^3 + 9*cosh(x)^2 - 22*cosh(x) - 2)*sinh(x)^3 - 80*cosh(x)^
3 + 60*(18*cosh(x)^7 + 28*cosh(x)^6 - 28*cosh(x)^5 - 55*cosh(x)^4 + 6*cosh(x)^3 - 22*cosh(x)^2 - 4*cosh(x) + 1
)*sinh(x)^2 + 60*cosh(x)^2 - 15*(cosh(x)^10 + 2*(5*cosh(x) + 1)*sinh(x)^9 + sinh(x)^10 + 2*cosh(x)^9 + 3*(15*c
osh(x)^2 + 6*cosh(x) - 1)*sinh(x)^8 - 3*cosh(x)^8 + 8*(15*cosh(x)^3 + 9*cosh(x)^2 - 3*cosh(x) - 1)*sinh(x)^7 -
8*cosh(x)^7 + 2*(105*cosh(x)^4 + 84*cosh(x)^3 - 42*cosh(x)^2 - 28*cosh(x) + 1)*sinh(x)^6 + 2*cosh(x)^6 + 12*(
21*cosh(x)^5 + 21*cosh(x)^4 - 14*cosh(x)^3 - 14*cosh(x)^2 + cosh(x) + 1)*sinh(x)^5 + 12*cosh(x)^5 + 2*(105*cos
h(x)^6 + 126*cosh(x)^5 - 105*cosh(x)^4 - 140*cosh(x)^3 + 15*cosh(x)^2 + 30*cosh(x) + 1)*sinh(x)^4 + 2*cosh(x)^
4 + 8*(15*cosh(x)^7 + 21*cosh(x)^6 - 21*cosh(x)^5 - 35*cosh(x)^4 + 5*cosh(x)^3 + 15*cosh(x)^2 + cosh(x) - 1)*s
inh(x)^3 - 8*cosh(x)^3 + 3*(15*cosh(x)^8 + 24*cosh(x)^7 - 28*cosh(x)^6 - 56*cosh(x)^5 + 10*cosh(x)^4 + 40*cosh
(x)^3 + 4*cosh(x)^2 - 8*cosh(x) - 1)*sinh(x)^2 - 3*cosh(x)^2 + 2*(5*cosh(x)^9 + 9*cosh(x)^8 - 12*cosh(x)^7 - 2
8*cosh(x)^6 + 6*cosh(x)^5 + 30*cosh(x)^4 + 4*cosh(x)^3 - 12*cosh(x)^2 - 3*cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1
)*log(cosh(x) + sinh(x) + 1) + 15*(cosh(x)^10 + 2*(5*cosh(x) + 1)*sinh(x)^9 + sinh(x)^10 + 2*cosh(x)^9 + 3*(15
*cosh(x)^2 + 6*cosh(x) - 1)*sinh(x)^8 - 3*cosh(x)^8 + 8*(15*cosh(x)^3 + 9*cosh(x)^2 - 3*cosh(x) - 1)*sinh(x)^7
- 8*cosh(x)^7 + 2*(105*cosh(x)^4 + 84*cosh(x)^3 - 42*cosh(x)^2 - 28*cosh(x) + 1)*sinh(x)^6 + 2*cosh(x)^6 + 12
*(21*cosh(x)^5 + 21*cosh(x)^4 - 14*cosh(x)^3 - 14*cosh(x)^2 + cosh(x) + 1)*sinh(x)^5 + 12*cosh(x)^5 + 2*(105*c
osh(x)^6 + 126*cosh(x)^5 - 105*cosh(x)^4 - 140*cosh(x)^3 + 15*cosh(x)^2 + 30*cosh(x) + 1)*sinh(x)^4 + 2*cosh(x
)^4 + 8*(15*cosh(x)^7 + 21*cosh(x)^6 - 21*cosh(x)^5 - 35*cosh(x)^4 + 5*cosh(x)^3 + 15*cosh(x)^2 + cosh(x) - 1)
*sinh(x)^3 - 8*cosh(x)^3 + 3*(15*cosh(x)^8 + 24*cosh(x)^7 - 28*cosh(x)^6 - 56*cosh(x)^5 + 10*cosh(x)^4 + 40*co
sh(x)^3 + 4*cosh(x)^2 - 8*cosh(x) - 1)*sinh(x)^2 - 3*cosh(x)^2 + 2*(5*cosh(x)^9 + 9*cosh(x)^8 - 12*cosh(x)^7 -
28*cosh(x)^6 + 6*cosh(x)^5 + 30*cosh(x)^4 + 4*cosh(x)^3 - 12*cosh(x)^2 - 3*cosh(x) + 1)*sinh(x) + 2*cosh(x) +
1)*log(cosh(x) + sinh(x) - 1) + 10*(27*cosh(x)^8 + 48*cosh(x)^7 - 56*cosh(x)^6 - 132*cosh(x)^5 + 18*cosh(x)^4
- 88*cosh(x)^3 - 24*cosh(x)^2 + 12*cosh(x) + 3)*sinh(x) + 30*cosh(x))/(a*cosh(x)^10 + a*sinh(x)^10 + 2*a*cosh
(x)^9 + 2*(5*a*cosh(x) + a)*sinh(x)^9 - 3*a*cosh(x)^8 + 3*(15*a*cosh(x)^2 + 6*a*cosh(x) - a)*sinh(x)^8 - 8*a*c
osh(x)^7 + 8*(15*a*cosh(x)^3 + 9*a*cosh(x)^2 - 3*a*cosh(x) - a)*sinh(x)^7 + 2*a*cosh(x)^6 + 2*(105*a*cosh(x)^4
+ 84*a*cosh(x)^3 - 42*a*cosh(x)^2 - 28*a*cosh(x) + a)*sinh(x)^6 + 12*a*cosh(x)^5 + 12*(21*a*cosh(x)^5 + 21*a*
cosh(x)^4 - 14*a*cosh(x)^3 - 14*a*cosh(x)^2 + a*cosh(x) + a)*sinh(x)^5 + 2*a*cosh(x)^4 + 2*(105*a*cosh(x)^6 +
126*a*cosh(x)^5 - 105*a*cosh(x)^4 - 140*a*cosh(x)^3 + 15*a*cosh(x)^2 + 30*a*cosh(x) + a)*sinh(x)^4 - 8*a*cosh(
x)^3 + 8*(15*a*cosh(x)^7 + 21*a*cosh(x)^6 - 21*a*cosh(x)^5 - 35*a*cosh(x)^4 + 5*a*cosh(x)^3 + 15*a*cosh(x)^2 +
a*cosh(x) - a)*sinh(x)^3 - 3*a*cosh(x)^2 + 3*(15*a*cosh(x)^8 + 24*a*cosh(x)^7 - 28*a*cosh(x)^6 - 56*a*cosh(x)
^5 + 10*a*cosh(x)^4 + 40*a*cosh(x)^3 + 4*a*cosh(x)^2 - 8*a*cosh(x) - a)*sinh(x)^2 + 2*a*cosh(x) + 2*(5*a*cosh(
x)^9 + 9*a*cosh(x)^8 - 12*a*cosh(x)^7 - 28*a*cosh(x)^6 + 6*a*cosh(x)^5 + 30*a*cosh(x)^4 + 4*a*cosh(x)^3 - 12*a
*cosh(x)^2 - 3*a*cosh(x) + a)*sinh(x) + a)

Sympy [F]

\[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\operatorname {csch}^{5}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(csch(x)**5/(a+a*cosh(x)),x)

[Out]

Integral(csch(x)**5/(cosh(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (68) = 136\).

Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.99 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\frac {15 \, e^{\left (-x\right )} + 30 \, e^{\left (-2 \, x\right )} - 40 \, e^{\left (-3 \, x\right )} - 110 \, e^{\left (-4 \, x\right )} + 18 \, e^{\left (-5 \, x\right )} - 110 \, e^{\left (-6 \, x\right )} - 40 \, e^{\left (-7 \, x\right )} + 30 \, e^{\left (-8 \, x\right )} + 15 \, e^{\left (-9 \, x\right )}}{24 \, {\left (2 \, a e^{\left (-x\right )} - 3 \, a e^{\left (-2 \, x\right )} - 8 \, a e^{\left (-3 \, x\right )} + 2 \, a e^{\left (-4 \, x\right )} + 12 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 8 \, a e^{\left (-7 \, x\right )} - 3 \, a e^{\left (-8 \, x\right )} + 2 \, a e^{\left (-9 \, x\right )} + a e^{\left (-10 \, x\right )} + a\right )}} - \frac {5 \, \log \left (e^{\left (-x\right )} + 1\right )}{16 \, a} + \frac {5 \, \log \left (e^{\left (-x\right )} - 1\right )}{16 \, a} \]

[In]

integrate(csch(x)^5/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

1/24*(15*e^(-x) + 30*e^(-2*x) - 40*e^(-3*x) - 110*e^(-4*x) + 18*e^(-5*x) - 110*e^(-6*x) - 40*e^(-7*x) + 30*e^(
-8*x) + 15*e^(-9*x))/(2*a*e^(-x) - 3*a*e^(-2*x) - 8*a*e^(-3*x) + 2*a*e^(-4*x) + 12*a*e^(-5*x) + 2*a*e^(-6*x) -
8*a*e^(-7*x) - 3*a*e^(-8*x) + 2*a*e^(-9*x) + a*e^(-10*x) + a) - 5/16*log(e^(-x) + 1)/a + 5/16*log(e^(-x) - 1)
/a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=-\frac {5 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{32 \, a} + \frac {5 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{32 \, a} - \frac {15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 76 \, e^{\left (-x\right )} - 76 \, e^{x} + 100}{64 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}^{2}} + \frac {55 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 402 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 1020 \, e^{\left (-x\right )} + 1020 \, e^{x} + 936}{192 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{3}} \]

[In]

integrate(csch(x)^5/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-5/32*log(e^(-x) + e^x + 2)/a + 5/32*log(e^(-x) + e^x - 2)/a - 1/64*(15*(e^(-x) + e^x)^2 - 76*e^(-x) - 76*e^x
+ 100)/(a*(e^(-x) + e^x - 2)^2) + 1/192*(55*(e^(-x) + e^x)^3 + 402*(e^(-x) + e^x)^2 + 1020*e^(-x) + 1020*e^x +
936)/(a*(e^(-x) + e^x + 2)^3)

Mupad [B] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.13 \[ \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx=\frac {1}{a\,\left (10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1\right )}+\frac {1}{4\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}+\frac {1}{8\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}-\frac {1}{8\,a\,\left (6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1\right )}-\frac {5}{8\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}+\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {3}{8\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{8\,\sqrt {-a^2}}-\frac {1}{3\,a\,\left (15\,{\mathrm {e}}^{2\,x}+20\,{\mathrm {e}}^{3\,x}+15\,{\mathrm {e}}^{4\,x}+6\,{\mathrm {e}}^{5\,x}+{\mathrm {e}}^{6\,x}+6\,{\mathrm {e}}^x+1\right )}-\frac {5}{12\,a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \]

[In]

int(1/(sinh(x)^5*(a + a*cosh(x))),x)

[Out]

1/(a*(10*exp(2*x) + 10*exp(3*x) + 5*exp(4*x) + exp(5*x) + 5*exp(x) + 1)) + 1/(4*a*(3*exp(2*x) - exp(3*x) - 3*e
xp(x) + 1)) + 1/(8*a*(exp(2*x) - 2*exp(x) + 1)) - 1/(8*a*(6*exp(2*x) - 4*exp(3*x) + exp(4*x) - 4*exp(x) + 1))
- 5/(8*a*(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1)) + 1/(4*a*(exp(x) - 1)) + 3/(8*a*(exp(x) + 1)) -
(5*atan((exp(x)*(-a^2)^(1/2))/a))/(8*(-a^2)^(1/2)) - 1/(3*a*(15*exp(2*x) + 20*exp(3*x) + 15*exp(4*x) + 6*exp(5
*x) + exp(6*x) + 6*exp(x) + 1)) - 5/(12*a*(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1))