\(\int \frac {1}{a+a \text {sech}(c+d x)} \, dx\) [76]

Optimal result

Integrand size = 12, antiderivative size = 29 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))} \]

[Out]

x/a-tanh(d*x+c)/d/(a+a*sech(d*x+c))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3862, 8} \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {\tanh (c+d x)}{d (a \text {sech}(c+d x)+a)} \]

[In]

Int[(a + a*Sech[c + d*x])^(-1),x]

[Out]

x/a - Tanh[c + d*x]/(d*(a + a*Sech[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3862

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-Cot[c + d*x])*((a + b*Csc[c + d*x])^n/(d*
(2*n + 1))), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*
x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))}+\frac {\int a \, dx}{a^2} \\ & = \frac {x}{a}-\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (d x \cosh \left (\frac {d x}{2}\right )+d x \cosh \left (c+\frac {d x}{2}\right )-2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]

[In]

Integrate[(a + a*Sech[c + d*x])^(-1),x]

[Out]

(Sech[c/2]*Sech[(c + d*x)/2]*(d*x*Cosh[(d*x)/2] + d*x*Cosh[c + (d*x)/2] - 2*Sinh[(d*x)/2]))/(2*a*d)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {d x -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}\)	\(23\)
risch	\(\frac {x}{a}+\frac {2}{d a \left ({\mathrm e}^{d x +c}+1\right )}\)	\(25\)
derivativedivides	\(\frac {-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(46\)
default	\(\frac {-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(46\)

[In]

int(1/(a+sech(d*x+c)*a),x,method=_RETURNVERBOSE)

[Out]

(d*x-tanh(1/2*d*x+1/2*c))/d/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {d x \cosh \left (d x + c\right ) + d x \sinh \left (d x + c\right ) + d x + 2}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) + a d} \]

[In]

integrate(1/(a+a*sech(d*x+c)),x, algorithm="fricas")

[Out]

(d*x*cosh(d*x + c) + d*x*sinh(d*x + c) + d*x + 2)/(a*d*cosh(d*x + c) + a*d*sinh(d*x + c) + a*d)

Sympy [F]

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

[In]

integrate(1/(a+a*sech(d*x+c)),x)

[Out]

Integral(1/(sech(c + d*x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {d x + c}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + a\right )} d} \]

[In]

integrate(1/(a+a*sech(d*x+c)),x, algorithm="maxima")

[Out]

(d*x + c)/(a*d) - 2/((a*e^(-d*x - c) + a)*d)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {\frac {d x + c}{a} + \frac {2}{a {\left (e^{\left (d x + c\right )} + 1\right )}}}{d} \]

[In]

integrate(1/(a+a*sech(d*x+c)),x, algorithm="giac")

[Out]

((d*x + c)/a + 2/(a*(e^(d*x + c) + 1)))/d

Mupad [B] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a+a \text {sech}(c+d x)} \, dx=\frac {x}{a}+\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}+1\right )} \]

[In]

int(1/(a + a/cosh(c + d*x)),x)

[Out]

x/a + 2/(a*d*(exp(c + d*x) + 1))