3.2 \(\int \text {csch}^2(a+b x) \, dx\)

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

[Out]

-coth(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 = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3767, 8} \[ -\frac {\coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^2,x]

[Out]

-(Coth[a + b*x]/b)

Rule 8

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \text {csch}^2(a+b x) \, dx &=-\frac {i \operatorname {Subst}(\int 1 \, dx,x,-i \coth (a+b x))}{b}\\ &=-\frac {\coth (a+b x)}{b}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^2,x]

[Out]

-(Coth[a + b*x]/b)

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fricas [B]  time = 0.64, size = 43, normalized size = 3.91 \[ -\frac {2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2,x, algorithm="fricas")

[Out]

-2/(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 [A]  time = 0.13, size = 18, normalized size = 1.64 \[ -\frac {2}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2,x, algorithm="giac")

[Out]

-2/(b*(e^(2*b*x + 2*a) - 1))

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maple [A]  time = 0.31, size = 12, normalized size = 1.09 \[ -\frac {\coth \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^2,x)

[Out]

-coth(b*x+a)/b

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maxima [A]  time = 0.36, size = 18, normalized size = 1.64 \[ \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2,x, algorithm="maxima")

[Out]

2/(b*(e^(-2*b*x - 2*a) - 1))

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mupad [B]  time = 0.07, size = 18, normalized size = 1.64 \[ -\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(a + b*x)^2,x)

[Out]

-2/(b*(exp(2*a + 2*b*x) - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**2,x)

[Out]

Integral(csch(a + b*x)**2, x)

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