3.2.12 \(\int \coth (a+b x) \text {csch}(a+b x) \, dx\) [112]

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 = {2686, 8} \begin {gather*} -\frac {\text {csch}(a+b x)}{b} \end {gather*}

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 2686

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 \text {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 \begin {gather*} -\frac {\text {csch}(a+b x)}{b} \end {gather*}

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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Maple [A]
time = 0.73, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(-\frac {\mathrm {csch}\left (b x +a \right )}{b}\)	\(12\)
default	\(-\frac {\mathrm {csch}\left (b x +a \right )}{b}\)	\(12\)
risch	\(-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a)*csch(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-csch(b*x+a)/b

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.27, size = 25, normalized size = 2.27 \begin {gather*} -\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (11) = 22\).
time = 0.37, size = 56, normalized size = 5.09 \begin {gather*} -\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} \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.40, size = 25, normalized size = 2.27 \begin {gather*} -\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \end {gather*}

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/(b*(e^(b*x + a) - e^(-b*x - a)))

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

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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