3.470 \(\int \text {csch}(a+b x) \text {sech}(a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac {\log (\tanh (a+b x))}{b} \]

[Out]

ln(tanh(b*x+a))/b

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2620, 29} \[ \frac {\log (\tanh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]*Sech[a + b*x],x]

[Out]

Log[Tanh[a + b*x]]/b

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rubi steps

\begin {align*} \int \text {csch}(a+b x) \text {sech}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\log (\tanh (a+b x))}{b}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 31, normalized size = 2.82 \[ 2 \left (\frac {\log (\sinh (a+b x))}{2 b}-\frac {\log (\cosh (a+b x))}{2 b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]*Sech[a + b*x],x]

[Out]

2*(-1/2*Log[Cosh[a + b*x]]/b + Log[Sinh[a + b*x]]/(2*b))

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fricas [B]  time = 0.41, size = 60, normalized size = 5.45 \[ -\frac {\log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a),x, algorithm="fricas")

[Out]

-(log(2*cosh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))) - log(2*sinh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))))
/b

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giac [B]  time = 0.11, size = 41, normalized size = 3.73 \[ -\frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 12, normalized size = 1.09 \[ \frac {\ln \left (\tanh \left (b x +a \right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)*sech(b*x+a),x)

[Out]

ln(tanh(b*x+a))/b

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maxima [B]  time = 0.41, size = 50, normalized size = 4.55 \[ \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.45, size = 30, normalized size = 2.73 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)*sinh(a + b*x)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)*sech(b*x+a),x)

[Out]

Integral(csch(a + b*x)*sech(a + b*x), x)

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