\(\int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx\) [105]

Optimal result

Integrand size = 13, antiderivative size = 31 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}-\frac {\arctan (\sinh (x))}{2 a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a} \]

[Out]

x/a-1/2*arctan(sinh(x))/a-1/2*(2-sech(x))*tanh(x)/a

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 31, 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 = {3973, 3966, 3855} \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {\arctan (\sinh (x))}{2 a}+\frac {x}{a}-\frac {\tanh (x) (2-\text {sech}(x))}{2 a} \]

[In]

Int[Tanh[x]^4/(a + a*Sech[x]),x]

[Out]

x/a - ArcTan[Sinh[x]]/(2*a) - ((2 - Sech[x])*Tanh[x])/(2*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*} \text {integral}& = -\frac {\int (-a+a \text {sech}(x)) \tanh ^2(x) \, dx}{a^2} \\ & = -\frac {(2-\text {sech}(x)) \tanh (x)}{2 a}-\frac {\int (-2 a+a \text {sech}(x)) \, dx}{2 a^2} \\ & = \frac {x}{a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a}-\frac {\int \text {sech}(x) \, dx}{2 a} \\ & = \frac {x}{a}-\frac {\arctan (\sinh (x))}{2 a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (2 \left (x-\arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )+(-2+\text {sech}(x)) \tanh (x)\right )}{a (1+\text {sech}(x))} \]

[In]

Integrate[Tanh[x]^4/(a + a*Sech[x]),x]

[Out]

(Cosh[x/2]^2*Sech[x]*(2*(x - ArcTan[Tanh[x/2]]) + (-2 + Sech[x])*Tanh[x]))/(a*(1 + Sech[x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(27)=54\).

Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90

method	result	size
default	\(\frac {\frac {2 \left (-\frac {3 \tanh \left (\frac {x}{2}\right )^{3}}{2}-\frac {\tanh \left (\frac {x}{2}\right )}{2}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}-\arctan \left (\tanh \left (\frac {x}{2}\right )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\)	\(59\)
risch	\(\frac {x}{a}+\frac {{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-{\mathrm e}^{x}+2}{\left (1+{\mathrm e}^{2 x}\right )^{2} a}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2 a}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2 a}\)	\(59\)

[In]

int(tanh(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

16/a*(1/8*(-3/2*tanh(1/2*x)^3-1/2*tanh(1/2*x))/(1+tanh(1/2*x)^2)^2-1/16*arctan(tanh(1/2*x))+1/16*ln(tanh(1/2*x
)+1)-1/16*ln(tanh(1/2*x)-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (25) = 50\).

Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 6.77 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} + {\left (4 \, x \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 2 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + \cosh \left (x\right )^{3} + {\left (6 \, x \cosh \left (x\right )^{2} + 2 \, x + 3 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (4 \, x \cosh \left (x\right )^{3} + 4 \, {\left (x + 1\right )} \cosh \left (x\right ) + 3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) + x - \cosh \left (x\right ) + 2}{a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \]

[In]

integrate(tanh(x)^4/(a+a*sech(x)),x, algorithm="fricas")

[Out]

(x*cosh(x)^4 + x*sinh(x)^4 + (4*x*cosh(x) + 1)*sinh(x)^3 + 2*(x + 1)*cosh(x)^2 + cosh(x)^3 + (6*x*cosh(x)^2 +
2*x + 3*cosh(x) + 2)*sinh(x)^2 - (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2
+ 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + (4*x*cosh(x)^3 + 4*(x + 1)*co
sh(x) + 3*cosh(x)^2 - 1)*sinh(x) + x - cosh(x) + 2)/(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*a*c
osh(x)^2 + 2*(3*a*cosh(x)^2 + a)*sinh(x)^2 + 4*(a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)

Sympy [F]

\[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\tanh ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(tanh(x)**4/(a+a*sech(x)),x)

[Out]

Integral(tanh(x)**4/(sech(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} + \frac {e^{\left (-x\right )} - 2 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - 2}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} + \frac {\arctan \left (e^{\left (-x\right )}\right )}{a} \]

[In]

integrate(tanh(x)^4/(a+a*sech(x)),x, algorithm="maxima")

[Out]

x/a + (e^(-x) - 2*e^(-2*x) - e^(-3*x) - 2)/(2*a*e^(-2*x) + a*e^(-4*x) + a) + arctan(e^(-x))/a

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} - \frac {\arctan \left (e^{x}\right )}{a} + \frac {e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - e^{x} + 2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]

[In]

integrate(tanh(x)^4/(a+a*sech(x)),x, algorithm="giac")

[Out]

x/a - arctan(e^x)/a + (e^(3*x) + 2*e^(2*x) - e^x + 2)/(a*(e^(2*x) + 1)^2)

Mupad [B] (verification not implemented)

Time = 2.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.16 \[ \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a}+\frac {\frac {2}{a}+\frac {{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}}-\frac {2\,{\mathrm {e}}^x}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]

[In]

int(tanh(x)^4/(a + a/cosh(x)),x)

[Out]

x/a + (2/a + exp(x)/a)/(exp(2*x) + 1) - atan((exp(x)*(a^2)^(1/2))/a)/(a^2)^(1/2) - (2*exp(x))/(a*(2*exp(2*x) +
exp(4*x) + 1))