\(\int \frac {1}{a-a \text {sech}(c+d x)} \, dx\) [77]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {\tanh (c+d x)}{d (a-a \text {sech}(c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3862, 8} \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {\tanh (c+d x)}{d (a-a \text {sech}(c+d x))} \]

[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 = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right ) \left (-d x \cosh \left (\frac {d x}{2}\right )+d x \cosh \left (c+\frac {d x}{2}\right )+2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]

[In]

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

[Out]

(Csch[c/2]*Csch[(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 = 25, normalized size of antiderivative = 0.83

method	result	size
risch	\(\frac {x}{a}-\frac {2}{d a \left ({\mathrm e}^{d x +c}-1\right )}\)	\(25\)
parallelrisch	\(\frac {-1+x \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) d}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a d}\)	\(33\)
derivativedivides	\(\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}\)	\(48\)
default	\(\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {d x \cosh \left (d x + c\right ) + d x \sinh \left (d x + c\right ) - d x - 2}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) - a d} \]

[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 = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {d x + c}{a d} + \frac {2}{{\left (a e^{\left (-d x - c\right )} - a\right )} d} \]

[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 = 0.97 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {\frac {d x + c}{a} - \frac {2}{a {\left (e^{\left (d x + c\right )} - 1\right )}}}{d} \]

[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 = 1.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1}{a-a \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-1\right )} \]

[In]

int(1/(a - a/cosh(c + d*x)),x)

[Out]

x/a - 2/(a*d*(exp(c + d*x) - 1))