3.2.3 \(\int \frac {\tanh ^6(x)}{a+a \text {sech}(x)} \, dx\) [103]

Optimal. Leaf size=48 \[ \frac {x}{a}-\frac {3 \text {ArcTan}(\sinh (x))}{8 a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a} \]

[Out]

x/a-3/8*arctan(sinh(x))/a-1/8*(8-3*sech(x))*tanh(x)/a-1/12*(4-3*sech(x))*tanh(x)^3/a

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3966, 3855} \begin {gather*} -\frac {3 \text {ArcTan}(\sinh (x))}{8 a}+\frac {x}{a}-\frac {\tanh ^3(x) (4-3 \text {sech}(x))}{12 a}-\frac {\tanh (x) (8-3 \text {sech}(x))}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^6/(a + a*Sech[x]),x]

[Out]

x/a - (3*ArcTan[Sinh[x]])/(8*a) - ((8 - 3*Sech[x])*Tanh[x])/(8*a) - ((4 - 3*Sech[x])*Tanh[x]^3)/(12*a)

Rule 3855

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

Rule 3966

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-e)*(e*Cot
[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc[c + d*x])/(d*m*(m - 1))), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m -
2)*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3973

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\tanh ^6(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\int (-a+a \text {sech}(x)) \tanh ^4(x) \, dx}{a^2}\\ &=-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {\int (-4 a+3 a \text {sech}(x)) \tanh ^2(x) \, dx}{4 a^2}\\ &=-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {\int (-8 a+3 a \text {sech}(x)) \, dx}{8 a^2}\\ &=\frac {x}{a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {3 \int \text {sech}(x) \, dx}{8 a}\\ &=\frac {x}{a}-\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 60, normalized size = 1.25 \begin {gather*} \frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (6 \left (4 x-3 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )\right )+\left (-32+15 \text {sech}(x)+8 \text {sech}^2(x)-6 \text {sech}^3(x)\right ) \tanh (x)\right )}{12 a (1+\text {sech}(x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[x/2]^2*Sech[x]*(6*(4*x - 3*ArcTan[Tanh[x/2]]) + (-32 + 15*Sech[x] + 8*Sech[x]^2 - 6*Sech[x]^3)*Tanh[x]))
/(12*a*(1 + Sech[x]))

________________________________________________________________________________________

Maple [A]
time = 0.69, size = 75, normalized size = 1.56

method	result	size
default	\(\frac {\frac {2 \left (-\frac {11 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{8}-\frac {137 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{24}-\frac {71 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {5 \tanh \left (\frac {x}{2}\right )}{8}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\)	\(75\)
risch	\(\frac {x}{a}+\frac {15 \,{\mathrm e}^{7 x}+48 \,{\mathrm e}^{6 x}-9 \,{\mathrm e}^{5 x}+96 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{3 x}+80 \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{x}+32}{12 \left (1+{\mathrm e}^{2 x}\right )^{4} a}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8 a}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8 a}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

64/a*(1/32*(-11/8*tanh(1/2*x)^7-137/24*tanh(1/2*x)^5-71/24*tanh(1/2*x)^3-5/8*tanh(1/2*x))/(tanh(1/2*x)^2+1)^4-
3/256*arctan(tanh(1/2*x))+1/64*ln(tanh(1/2*x)+1)-1/64*ln(tanh(1/2*x)-1))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (42) = 84\).
time = 0.49, size = 93, normalized size = 1.94 \begin {gather*} \frac {x}{a} + \frac {15 \, e^{\left (-x\right )} - 80 \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 96 \, e^{\left (-4 \, x\right )} + 9 \, e^{\left (-5 \, x\right )} - 48 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} - 32}{12 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, 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*sech(x)),x, algorithm="maxima")

[Out]

x/a + 1/12*(15*e^(-x) - 80*e^(-2*x) - 9*e^(-3*x) - 96*e^(-4*x) + 9*e^(-5*x) - 48*e^(-6*x) - 15*e^(-7*x) - 32)/
(4*a*e^(-2*x) + 6*a*e^(-4*x) + 4*a*e^(-6*x) + a*e^(-8*x) + a) + 3/4*arctan(e^(-x))/a

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (42) = 84\).
time = 0.39, size = 686, normalized size = 14.29 \begin {gather*} \frac {12 \, x \cosh \left (x\right )^{8} + 12 \, x \sinh \left (x\right )^{8} + 3 \, {\left (32 \, x \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{7} + 48 \, {\left (x + 1\right )} \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{7} + 3 \, {\left (112 \, x \cosh \left (x\right )^{2} + 16 \, x + 35 \, \cosh \left (x\right ) + 16\right )} \sinh \left (x\right )^{6} + 3 \, {\left (224 \, x \cosh \left (x\right )^{3} + 96 \, {\left (x + 1\right )} \cosh \left (x\right ) + 105 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{5} + 24 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{5} + 3 \, {\left (280 \, x \cosh \left (x\right )^{4} + 240 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + 175 \, \cosh \left (x\right )^{3} + 24 \, x - 15 \, \cosh \left (x\right ) + 32\right )} \sinh \left (x\right )^{4} + 3 \, {\left (224 \, x \cosh \left (x\right )^{5} + 320 \, {\left (x + 1\right )} \cosh \left (x\right )^{3} + 175 \, \cosh \left (x\right )^{4} + 32 \, {\left (3 \, x + 4\right )} \cosh \left (x\right ) - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 16 \, {\left (3 \, x + 5\right )} \cosh \left (x\right )^{2} + 9 \, \cosh \left (x\right )^{3} + {\left (336 \, x \cosh \left (x\right )^{6} + 720 \, {\left (x + 1\right )} \cosh \left (x\right )^{4} + 315 \, \cosh \left (x\right )^{5} + 144 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{2} - 90 \, \cosh \left (x\right )^{3} + 48 \, x + 27 \, \cosh \left (x\right ) + 80\right )} \sinh \left (x\right )^{2} - 9 \, {\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 )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (96 \, x \cosh \left (x\right )^{7} + 288 \, {\left (x + 1\right )} \cosh \left (x\right )^{5} + 105 \, \cosh \left (x\right )^{6} + 96 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{3} - 45 \, \cosh \left (x\right )^{4} + 32 \, {\left (3 \, x + 5\right )} \cosh \left (x\right ) + 27 \, \cosh \left (x\right )^{2} - 15\right )} \sinh \left (x\right ) + 12 \, x - 15 \, \cosh \left (x\right ) + 32}{12 \, {\left (a \cosh \left (x\right )^{8} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + a \sinh \left (x\right )^{8} + 4 \, a \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{6} + 8 \, {\left (7 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 6 \, a \cosh \left (x\right )^{4} + 2 \, {\left (35 \, a \cosh \left (x\right )^{4} + 30 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (7 \, a \cosh \left (x\right )^{5} + 10 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + 4 \, {\left (7 \, a \cosh \left (x\right )^{6} + 15 \, a \cosh \left (x\right )^{4} + 9 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 8 \, {\left (a \cosh \left (x\right )^{7} + 3 \, a \cosh \left (x\right )^{5} + 3 \, 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*sech(x)),x, algorithm="fricas")

[Out]

1/12*(12*x*cosh(x)^8 + 12*x*sinh(x)^8 + 3*(32*x*cosh(x) + 5)*sinh(x)^7 + 48*(x + 1)*cosh(x)^6 + 15*cosh(x)^7 +
3*(112*x*cosh(x)^2 + 16*x + 35*cosh(x) + 16)*sinh(x)^6 + 3*(224*x*cosh(x)^3 + 96*(x + 1)*cosh(x) + 105*cosh(x
)^2 - 3)*sinh(x)^5 + 24*(3*x + 4)*cosh(x)^4 - 9*cosh(x)^5 + 3*(280*x*cosh(x)^4 + 240*(x + 1)*cosh(x)^2 + 175*c
osh(x)^3 + 24*x - 15*cosh(x) + 32)*sinh(x)^4 + 3*(224*x*cosh(x)^5 + 320*(x + 1)*cosh(x)^3 + 175*cosh(x)^4 + 32
*(3*x + 4)*cosh(x) - 30*cosh(x)^2 + 3)*sinh(x)^3 + 16*(3*x + 5)*cosh(x)^2 + 9*cosh(x)^3 + (336*x*cosh(x)^6 + 7
20*(x + 1)*cosh(x)^4 + 315*cosh(x)^5 + 144*(3*x + 4)*cosh(x)^2 - 90*cosh(x)^3 + 48*x + 27*cosh(x) + 80)*sinh(x
)^2 - 9*(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)*arctan(cosh(x) + sinh(x)) + (96*x*cosh(x)^
7 + 288*(x + 1)*cosh(x)^5 + 105*cosh(x)^6 + 96*(3*x + 4)*cosh(x)^3 - 45*cosh(x)^4 + 32*(3*x + 5)*cosh(x) + 27*
cosh(x)^2 - 15)*sinh(x) + 12*x - 15*cosh(x) + 32)/(a*cosh(x)^8 + 8*a*cosh(x)*sinh(x)^7 + a*sinh(x)^8 + 4*a*cos
h(x)^6 + 4*(7*a*cosh(x)^2 + a)*sinh(x)^6 + 8*(7*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^5 + 6*a*cosh(x)^4 + 2*(35*a
*cosh(x)^4 + 30*a*cosh(x)^2 + 3*a)*sinh(x)^4 + 8*(7*a*cosh(x)^5 + 10*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 4*
a*cosh(x)^2 + 4*(7*a*cosh(x)^6 + 15*a*cosh(x)^4 + 9*a*cosh(x)^2 + a)*sinh(x)^2 + 8*(a*cosh(x)^7 + 3*a*cosh(x)^
5 + 3*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tanh ^{6}{\left (x \right )}}{\operatorname {sech}{\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*sech(x)),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.39, size = 69, normalized size = 1.44 \begin {gather*} \frac {x}{a} - \frac {3 \, \arctan \left (e^{x}\right )}{4 \, a} + \frac {15 \, e^{\left (7 \, x\right )} + 48 \, e^{\left (6 \, x\right )} - 9 \, e^{\left (5 \, x\right )} + 96 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (3 \, x\right )} + 80 \, e^{\left (2 \, x\right )} - 15 \, e^{x} + 32}{12 \, a {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a - 3/4*arctan(e^x)/a + 1/12*(15*e^(7*x) + 48*e^(6*x) - 9*e^(5*x) + 96*e^(4*x) + 9*e^(3*x) + 80*e^(2*x) - 15
*e^x + 32)/(a*(e^(2*x) + 1)^4)

________________________________________________________________________________________

Mupad [B]
time = 1.46, size = 143, normalized size = 2.98 \begin {gather*} \frac {\frac {8}{3\,a}+\frac {6\,{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4}{a}+\frac {9\,{\mathrm {e}}^x}{2\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {x}{a}+\frac {\frac {4}{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}}-\frac {4\,{\mathrm {e}}^x}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a + a/cosh(x)),x)

[Out]

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

________________________________________________________________________________________