\(\int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx\) [73]

Optimal result

Integrand size = 13, antiderivative size = 20 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a+a \text {sech}(x)} \]

[Out]

arctan(sinh(x))/a-tanh(x)/(a+a*sech(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3874, 3855, 3879} \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a \text {sech}(x)+a} \]

[In]

Int[Sech[x]^2/(a + a*Sech[x]),x]

[Out]

ArcTan[Sinh[x]]/a - Tanh[x]/(a + a*Sech[x])

Rule 3855

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

Rule 3874

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3879

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \text {sech}(x) \, dx}{a}-\int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx \\ & = \frac {\arctan (\sinh (x))}{a}-\frac {\tanh (x)}{a+a \text {sech}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\arctan (\sinh (x))+\arctan (\sinh (x)) \text {sech}(x)-\tanh (x)}{a+a \text {sech}(x)} \]

[In]

Integrate[Sech[x]^2/(a + a*Sech[x]),x]

[Out]

(ArcTan[Sinh[x]] + ArcTan[Sinh[x]]*Sech[x] - Tanh[x])/(a + a*Sech[x])

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {-\tanh \left (\frac {x}{2}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\)	\(19\)
parallelrisch	\(\frac {-i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )+i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )-\tanh \left (\frac {x}{2}\right )}{a}\)	\(34\)
risch	\(\frac {2}{\left ({\mathrm e}^{x}+1\right ) a}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{a}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{a}\)	\(37\)

[In]

int(sech(x)^2/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*(-tanh(1/2*x)+2*arctan(tanh(1/2*x)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {2 \, {\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 1\right )}}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \]

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="fricas")

[Out]

2*((cosh(x) + sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 1)/(a*cosh(x) + a*sinh(x) + a)

Sympy [F]

\[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(sech(x)**2/(a+a*sech(x)),x)

[Out]

Integral(sech(x)**2/(sech(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=-\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="giac")

[Out]

2*arctan(e^x)/a + 2/(a*(e^x + 1))

Mupad [B] (verification not implemented)

Time = 2.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx=\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]

[In]

int(1/(cosh(x)^2*(a + a/cosh(x))),x)

[Out]

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