\(\int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx\) [104]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=-\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]

[Out]

-(a^2-b^2)*ln(a+b*tanh(x))/b^3+a*tanh(x)/b^2-1/2*tanh(x)^2/b

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3587, 711} \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=-\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]

[In]

Int[Sech[x]^4/(a + b*Tanh[x]),x]

[Out]

-(((a^2 - b^2)*Log[a + b*Tanh[x]])/b^3) + (a*Tanh[x])/b^2 - Tanh[x]^2/(2*b)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 3587

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-\frac {x^2}{b^2}}{a+x} \, dx,x,b \tanh (x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a}{b^2}-\frac {x}{b^2}+\frac {-a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \tanh (x)\right )}{b} \\ & = -\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=\frac {2 \left (-a^2+b^2\right ) \log (a+b \tanh (x))+2 a b \tanh (x)-b^2 \tanh ^2(x)}{2 b^3} \]

[In]

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

[Out]

(2*(-a^2 + b^2)*Log[a + b*Tanh[x]] + 2*a*b*Tanh[x] - b^2*Tanh[x]^2)/(2*b^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(38)=76\).

Time = 12.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.48

method	result	size
default	\(\frac {\frac {2 \left (a b \tanh \left (\frac {x}{2}\right )^{3}-\tanh \left (\frac {x}{2}\right )^{2} b^{2}+a b \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\left (a^{2}-b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{b^{3}}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{b^{3}}\)	\(99\)
risch	\(-\frac {2 \left (a \,{\mathrm e}^{2 x}-b \,{\mathrm e}^{2 x}+a \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} b^{2}}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right ) a^{2}}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{b}\)	\(102\)

[In]

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

[Out]

2/b^3*((a*b*tanh(1/2*x)^3-tanh(1/2*x)^2*b^2+a*b*tanh(1/2*x))/(1+tanh(1/2*x)^2)^2+1/2*(a^2-b^2)*ln(1+tanh(1/2*x
)^2))-(a^2-b^2)/b^3*ln(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (38) = 76\).

Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 10.75 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=-\frac {2 \, {\left (a b - b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (a b - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, {\left (a b - b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a b + {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b^{3} \cosh \left (x\right )^{4} + 4 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{3} \sinh \left (x\right )^{4} + 2 \, b^{3} \cosh \left (x\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (x\right )^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{3} \cosh \left (x\right )^{3} + b^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]

[In]

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

[Out]

-(2*(a*b - b^2)*cosh(x)^2 + 4*(a*b - b^2)*cosh(x)*sinh(x) + 2*(a*b - b^2)*sinh(x)^2 + 2*a*b + ((a^2 - b^2)*cos
h(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*
cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))*log(2*
(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) - ((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*cosh(x)*sinh(x)^3 + (a^2
- b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 - b^2 +
4*((a^2 - b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))))/(b^3*cosh(x)^4 +
4*b^3*cosh(x)*sinh(x)^3 + b^3*sinh(x)^4 + 2*b^3*cosh(x)^2 + b^3 + 2*(3*b^3*cosh(x)^2 + b^3)*sinh(x)^2 + 4*(b^3
*cosh(x)^3 + b^3*cosh(x))*sinh(x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (38) = 76\).

Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.22 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=\frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} + a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} + b^{2} e^{\left (-4 \, x\right )} + b^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{3}} \]

[In]

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

[Out]

2*((a + b)*e^(-2*x) + a)/(2*b^2*e^(-2*x) + b^2*e^(-4*x) + b^2) - (a^2 - b^2)*log(-(a - b)*e^(-2*x) - a - b)/b^
3 + (a^2 - b^2)*log(e^(-2*x) + 1)/b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (38) = 76\).

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.60 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=-\frac {{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{3} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{3}} - \frac {2 \, {\left (a b + {\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3} {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]

[In]

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

[Out]

-(a^3 + a^2*b - a*b^2 - b^3)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a*b^3 + b^4) + (a^2 - b^2)*log(e^(2*x) +
1)/b^3 - 2*(a*b + (a*b - b^2)*e^(2*x))/(b^3*(e^(2*x) + 1)^2)

Mupad [B] (verification not implemented)

Time = 1.89 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.20 \[ \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}-\frac {2\,\left (a-b\right )}{b^2\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}-\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]

[In]

int(1/(cosh(x)^4*(a + b*tanh(x))),x)

[Out]

(log(exp(2*x) + 1)*(a + b)*(a - b))/b^3 - (2*(a - b))/(b^2*(exp(2*x) + 1)) - (log(a - b + a*exp(2*x) + b*exp(2
*x))*(a + b)*(a - b))/b^3 - 2/(b*(2*exp(2*x) + exp(4*x) + 1))