3.112 \(\int \coth (a+b x) \text {csch}(a+b x) \, dx\)

Optimal. Leaf size=11 \[ -\frac {\text {csch}(a+b x)}{b} \]

[Out]

-csch(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 = {2606, 8} \[ -\frac {\text {csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]*Csch[a + b*x],x]

[Out]

-(Csch[a + b*x]/b)

Rule 8

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

Rule 2606

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

Rubi steps

\begin {align*} \int \coth (a+b x) \text {csch}(a+b x) \, dx &=-\frac {i \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(a+b x))}{b}\\ &=-\frac {\text {csch}(a+b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ -\frac {\text {csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csch[a + b*x]/b)

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fricas [B]  time = 0.38, size = 56, normalized size = 5.09 \[ -\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.13, size = 24, normalized size = 2.18 \[ -\frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 12, normalized size = 1.09 \[ -\frac {\mathrm {csch}\left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-csch(b*x+a)/b

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maxima [B]  time = 0.40, size = 25, normalized size = 2.27 \[ -\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(b*(e^(b*x + a) - e^(-b*x - a)))

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mupad [B]  time = 0.06, size = 24, normalized size = 2.18 \[ -\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x)/sinh(a + b*x),x)

[Out]

-(2*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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