3.2.87 \(\int \frac {\tanh ^6(x)}{a+a \cosh (x)} \, dx\) [187]

Optimal. Leaf size=46 \[ \frac {3 \text {ArcTan}(\sinh (x))}{8 a}-\frac {3 \text {sech}(x) \tanh (x)}{8 a}-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}-\frac {\tanh ^5(x)}{5 a} \]

[Out]

3/8*arctan(sinh(x))/a-3/8*sech(x)*tanh(x)/a-1/4*sech(x)*tanh(x)^3/a-1/5*tanh(x)^5/a

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Rubi [A]
time = 0.07, antiderivative size = 46, 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 = {2785, 2687, 30, 2691, 3855} \begin {gather*} \frac {3 \text {ArcTan}(\sinh (x))}{8 a}-\frac {\tanh ^5(x)}{5 a}-\frac {\tanh ^3(x) \text {sech}(x)}{4 a}-\frac {3 \tanh (x) \text {sech}(x)}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^6/(a + a*Cosh[x]),x]

[Out]

(3*ArcTan[Sinh[x]])/(8*a) - (3*Sech[x]*Tanh[x])/(8*a) - (Sech[x]*Tanh[x]^3)/(4*a) - Tanh[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 2687

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 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 2785

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]

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 {\tanh ^6(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \tanh ^4(x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh ^4(x) \, dx}{a}\\ &=-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}+\frac {i \text {Subst}\left (\int x^4 \, dx,x,i \tanh (x)\right )}{a}+\frac {3 \int \text {sech}(x) \tanh ^2(x) \, dx}{4 a}\\ &=-\frac {3 \text {sech}(x) \tanh (x)}{8 a}-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}-\frac {\tanh ^5(x)}{5 a}+\frac {3 \int \text {sech}(x) \, dx}{8 a}\\ &=\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {3 \text {sech}(x) \tanh (x)}{8 a}-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}-\frac {\tanh ^5(x)}{5 a}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 58, normalized size = 1.26 \begin {gather*} \frac {\cosh ^2\left (\frac {x}{2}\right ) \left (30 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\left (-8-25 \text {sech}(x)+16 \text {sech}^2(x)+10 \text {sech}^3(x)-8 \text {sech}^4(x)\right ) \tanh (x)\right )}{20 a (1+\cosh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^6/(a + a*Cosh[x]),x]

[Out]

(Cosh[x/2]^2*(30*ArcTan[Tanh[x/2]] + (-8 - 25*Sech[x] + 16*Sech[x]^2 + 10*Sech[x]^3 - 8*Sech[x]^4)*Tanh[x]))/(
20*a*(1 + Cosh[x]))

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Maple [A]
time = 0.60, size = 64, normalized size = 1.39

method	result	size
default	\(\frac {\frac {64 \left (\frac {3 \left (\tanh ^{9}\left (\frac {x}{2}\right )\right )}{256}+\frac {7 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{128}-\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{10}-\frac {7 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{128}-\frac {3 \tanh \left (\frac {x}{2}\right )}{256}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{a}\)	\(64\)
risch	\(-\frac {25 \,{\mathrm e}^{9 x}-40 \,{\mathrm e}^{8 x}+10 \,{\mathrm e}^{7 x}-80 \,{\mathrm e}^{4 x}-10 \,{\mathrm e}^{3 x}-25 \,{\mathrm e}^{x}-8}{20 \left (1+{\mathrm e}^{2 x}\right )^{5} a}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8 a}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8 a}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

64/a*((3/256*tanh(1/2*x)^9+7/128*tanh(1/2*x)^7-1/10*tanh(1/2*x)^5-7/128*tanh(1/2*x)^3-3/256*tanh(1/2*x))/(tanh
(1/2*x)^2+1)^5+3/256*arctan(tanh(1/2*x)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (38) = 76\).
time = 0.49, size = 89, normalized size = 1.93 \begin {gather*} -\frac {25 \, e^{\left (-x\right )} + 10 \, e^{\left (-3 \, x\right )} + 80 \, e^{\left (-4 \, x\right )} - 10 \, e^{\left (-7 \, x\right )} + 40 \, e^{\left (-8 \, x\right )} - 25 \, e^{\left (-9 \, x\right )} + 8}{20 \, {\left (5 \, a e^{\left (-2 \, x\right )} + 10 \, a e^{\left (-4 \, x\right )} + 10 \, a e^{\left (-6 \, x\right )} + 5 \, a e^{\left (-8 \, x\right )} + a e^{\left (-10 \, x\right )} + a\right )}} - \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/20*(25*e^(-x) + 10*e^(-3*x) + 80*e^(-4*x) - 10*e^(-7*x) + 40*e^(-8*x) - 25*e^(-9*x) + 8)/(5*a*e^(-2*x) + 10
*a*e^(-4*x) + 10*a*e^(-6*x) + 5*a*e^(-8*x) + a*e^(-10*x) + a) - 3/4*arctan(e^(-x))/a

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (38) = 76\).
time = 0.52, size = 750, normalized size = 16.30 \begin {gather*} -\frac {25 \, \cosh \left (x\right )^{9} + 5 \, {\left (45 \, \cosh \left (x\right ) - 8\right )} \sinh \left (x\right )^{8} + 25 \, \sinh \left (x\right )^{9} - 40 \, \cosh \left (x\right )^{8} + 10 \, {\left (90 \, \cosh \left (x\right )^{2} - 32 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{7} + 10 \, \cosh \left (x\right )^{7} + 70 \, {\left (30 \, \cosh \left (x\right )^{3} - 16 \, \cosh \left (x\right )^{2} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{6} + 70 \, {\left (45 \, \cosh \left (x\right )^{4} - 32 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{5} + 10 \, {\left (315 \, \cosh \left (x\right )^{5} - 280 \, \cosh \left (x\right )^{4} + 35 \, \cosh \left (x\right )^{3} - 8\right )} \sinh \left (x\right )^{4} - 80 \, \cosh \left (x\right )^{4} + 10 \, {\left (210 \, \cosh \left (x\right )^{6} - 224 \, \cosh \left (x\right )^{5} + 35 \, \cosh \left (x\right )^{4} - 32 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 10 \, \cosh \left (x\right )^{3} + 10 \, {\left (90 \, \cosh \left (x\right )^{7} - 112 \, \cosh \left (x\right )^{6} + 21 \, \cosh \left (x\right )^{5} - 48 \, \cosh \left (x\right )^{2} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 15 \, {\left (\cosh \left (x\right )^{10} + 10 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + \sinh \left (x\right )^{10} + 5 \, {\left (9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{8} + 5 \, \cosh \left (x\right )^{8} + 40 \, {\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 10 \, {\left (21 \, \cosh \left (x\right )^{4} + 14 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{6} + 4 \, {\left (63 \, \cosh \left (x\right )^{5} + 70 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{6} + 35 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 10 \, \cosh \left (x\right )^{4} + 40 \, {\left (3 \, \cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, {\left (9 \, \cosh \left (x\right )^{8} + 28 \, \cosh \left (x\right )^{6} + 30 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 5 \, \cosh \left (x\right )^{2} + 10 \, {\left (\cosh \left (x\right )^{9} + 4 \, \cosh \left (x\right )^{7} + 6 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 5 \, {\left (45 \, \cosh \left (x\right )^{8} - 64 \, \cosh \left (x\right )^{7} + 14 \, \cosh \left (x\right )^{6} - 64 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right ) - 25 \, \cosh \left (x\right ) - 8}{20 \, {\left (a \cosh \left (x\right )^{10} + 10 \, a \cosh \left (x\right ) \sinh \left (x\right )^{9} + a \sinh \left (x\right )^{10} + 5 \, a \cosh \left (x\right )^{8} + 5 \, {\left (9 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{8} + 40 \, {\left (3 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 10 \, a \cosh \left (x\right )^{6} + 10 \, {\left (21 \, a \cosh \left (x\right )^{4} + 14 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (63 \, a \cosh \left (x\right )^{5} + 70 \, a \cosh \left (x\right )^{3} + 15 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, a \cosh \left (x\right )^{4} + 10 \, {\left (21 \, a \cosh \left (x\right )^{6} + 35 \, a \cosh \left (x\right )^{4} + 15 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 40 \, {\left (3 \, a \cosh \left (x\right )^{7} + 7 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} + 5 \, {\left (9 \, a \cosh \left (x\right )^{8} + 28 \, a \cosh \left (x\right )^{6} + 30 \, a \cosh \left (x\right )^{4} + 12 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 10 \, {\left (a \cosh \left (x\right )^{9} + 4 \, a \cosh \left (x\right )^{7} + 6 \, a \cosh \left (x\right )^{5} + 4 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

-1/20*(25*cosh(x)^9 + 5*(45*cosh(x) - 8)*sinh(x)^8 + 25*sinh(x)^9 - 40*cosh(x)^8 + 10*(90*cosh(x)^2 - 32*cosh(
x) + 1)*sinh(x)^7 + 10*cosh(x)^7 + 70*(30*cosh(x)^3 - 16*cosh(x)^2 + cosh(x))*sinh(x)^6 + 70*(45*cosh(x)^4 - 3
2*cosh(x)^3 + 3*cosh(x)^2)*sinh(x)^5 + 10*(315*cosh(x)^5 - 280*cosh(x)^4 + 35*cosh(x)^3 - 8)*sinh(x)^4 - 80*co
sh(x)^4 + 10*(210*cosh(x)^6 - 224*cosh(x)^5 + 35*cosh(x)^4 - 32*cosh(x) - 1)*sinh(x)^3 - 10*cosh(x)^3 + 10*(90
*cosh(x)^7 - 112*cosh(x)^6 + 21*cosh(x)^5 - 48*cosh(x)^2 - 3*cosh(x))*sinh(x)^2 - 15*(cosh(x)^10 + 10*cosh(x)*
sinh(x)^9 + sinh(x)^10 + 5*(9*cosh(x)^2 + 1)*sinh(x)^8 + 5*cosh(x)^8 + 40*(3*cosh(x)^3 + cosh(x))*sinh(x)^7 +
10*(21*cosh(x)^4 + 14*cosh(x)^2 + 1)*sinh(x)^6 + 10*cosh(x)^6 + 4*(63*cosh(x)^5 + 70*cosh(x)^3 + 15*cosh(x))*s
inh(x)^5 + 10*(21*cosh(x)^6 + 35*cosh(x)^4 + 15*cosh(x)^2 + 1)*sinh(x)^4 + 10*cosh(x)^4 + 40*(3*cosh(x)^7 + 7*
cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*sinh(x)^3 + 5*(9*cosh(x)^8 + 28*cosh(x)^6 + 30*cosh(x)^4 + 12*cosh(x)^2 + 1
)*sinh(x)^2 + 5*cosh(x)^2 + 10*(cosh(x)^9 + 4*cosh(x)^7 + 6*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*sinh(x) + 1)*ar
ctan(cosh(x) + sinh(x)) + 5*(45*cosh(x)^8 - 64*cosh(x)^7 + 14*cosh(x)^6 - 64*cosh(x)^3 - 6*cosh(x)^2 - 5)*sinh
(x) - 25*cosh(x) - 8)/(a*cosh(x)^10 + 10*a*cosh(x)*sinh(x)^9 + a*sinh(x)^10 + 5*a*cosh(x)^8 + 5*(9*a*cosh(x)^2
+ a)*sinh(x)^8 + 40*(3*a*cosh(x)^3 + a*cosh(x))*sinh(x)^7 + 10*a*cosh(x)^6 + 10*(21*a*cosh(x)^4 + 14*a*cosh(x
)^2 + a)*sinh(x)^6 + 4*(63*a*cosh(x)^5 + 70*a*cosh(x)^3 + 15*a*cosh(x))*sinh(x)^5 + 10*a*cosh(x)^4 + 10*(21*a*
cosh(x)^6 + 35*a*cosh(x)^4 + 15*a*cosh(x)^2 + a)*sinh(x)^4 + 40*(3*a*cosh(x)^7 + 7*a*cosh(x)^5 + 5*a*cosh(x)^3
+ a*cosh(x))*sinh(x)^3 + 5*a*cosh(x)^2 + 5*(9*a*cosh(x)^8 + 28*a*cosh(x)^6 + 30*a*cosh(x)^4 + 12*a*cosh(x)^2
+ a)*sinh(x)^2 + 10*(a*cosh(x)^9 + 4*a*cosh(x)^7 + 6*a*cosh(x)^5 + 4*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tanh ^{6}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**6/(a+a*cosh(x)),x)

[Out]

Integral(tanh(x)**6/(cosh(x) + 1), x)/a

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Giac [A]
time = 0.42, size = 58, normalized size = 1.26 \begin {gather*} \frac {3 \, \arctan \left (e^{x}\right )}{4 \, a} - \frac {25 \, e^{\left (9 \, x\right )} - 40 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (7 \, x\right )} - 80 \, e^{\left (4 \, x\right )} - 10 \, e^{\left (3 \, x\right )} - 25 \, e^{x} - 8}{20 \, a {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+a*cosh(x)),x, algorithm="giac")

[Out]

3/4*arctan(e^x)/a - 1/20*(25*e^(9*x) - 40*e^(8*x) + 10*e^(7*x) - 80*e^(4*x) - 10*e^(3*x) - 25*e^x - 8)/(a*(e^(
2*x) + 1)^5)

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Mupad [B]
time = 1.06, size = 183, normalized size = 3.98 \begin {gather*} \frac {\frac {16}{a}-\frac {6\,{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {8}{a}-\frac {9\,{\mathrm {e}}^x}{2\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {32}{5\,a\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {\frac {16}{a}-\frac {4\,{\mathrm {e}}^x}{a}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {2}{a}-\frac {5\,{\mathrm {e}}^x}{4\,a}}{{\mathrm {e}}^{2\,x}+1}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{4\,\sqrt {a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a + a*cosh(x)),x)

[Out]

(16/a - (6*exp(x))/a)/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - (8/a - (9*exp(x))/(2*a))/(2*exp(2*x) + exp(4*
x) + 1) + 32/(5*a*(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*x) + exp(10*x) + 1)) - (16/a - (4*exp(x))/
a)/(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1) + (2/a - (5*exp(x))/(4*a))/(exp(2*x) + 1) + (3*atan((
exp(x)*(a^2)^(1/2))/a))/(4*(a^2)^(1/2))

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