3.435 \(\int \coth ^2(a+b x) \, dx\)

Optimal. Leaf size=13 \[ x-\frac {\coth (a+b x)}{b} \]

[Out]

x-coth(b*x+a)/b

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Rubi [A]  time = 0.01, antiderivative size = 13, 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 = {3473, 8} \[ x-\frac {\coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^2,x]

[Out]

x - Coth[a + b*x]/b

Rule 8

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 27, normalized size = 2.08 \[ -\frac {\coth (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x]^2,x]

[Out]

-((Coth[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[a + b*x]^2])/b)

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fricas [B]  time = 0.40, size = 33, normalized size = 2.54 \[ \frac {{\left (b x + 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \sinh \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + 1)*sinh(b*x + a) - cosh(b*x + a))/(b*sinh(b*x + a))

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giac [A]  time = 0.14, size = 24, normalized size = 1.85 \[ \frac {b x + a - \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a - 2/(e^(2*b*x + 2*a) - 1))/b

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maple [A]  time = 0.12, size = 18, normalized size = 1.38 \[ \frac {b x +a -\coth \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(b*x+a-coth(b*x+a))

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maxima [A]  time = 0.31, size = 25, normalized size = 1.92 \[ x + \frac {a}{b} + \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 20, normalized size = 1.54 \[ x-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2/(b*(exp(2*a + 2*b*x) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x)**2*csch(a + b*x)**2, x)

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