3.1.8 \(\int \coth ^2(a+b x) \, dx\) [8]

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 = {3554, 8} \begin {gather*} x-\frac {\coth (a+b x)}{b} \end {gather*}

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 3554

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] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.01, size = 27, normalized size = 2.08 \begin {gather*} -\frac {\coth (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(a+b x)\right )}{b} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(13)=26\).
time = 0.25, size = 36, normalized size = 2.77

method	result	size
risch	\(x -\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\)	\(21\)
derivativedivides	\(\frac {-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\)	\(36\)
default	\(\frac {-\coth \left (b x +a \right )-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/b*(-coth(b*x+a)-1/2*ln(coth(b*x+a)-1)+1/2*ln(coth(b*x+a)+1))

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Maxima [A]
time = 0.27, size = 25, normalized size = 1.92 \begin {gather*} x + \frac {a}{b} + \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
time = 0.53, size = 33, normalized size = 2.54 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (8) = 16\).
time = 0.63, size = 87, normalized size = 6.69 \begin {gather*} \begin {cases} - \frac {\log {\left (- e^{- b x} \right )} \coth ^{2}{\left (b x + \log {\left (- e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\log {\left (e^{- b x} \right )} \coth ^{2}{\left (b x + \log {\left (e^{- b x} \right )} \right )}}{b} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x \coth ^{2}{\left (a \right )} & \text {for}\: b = 0 \\x - \frac {1}{b \tanh {\left (a + b x \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(-exp(-b*x))*coth(b*x + log(-exp(-b*x)))**2/b, Eq(a, log(-exp(-b*x)))), (-log(exp(-b*x))*coth(b
*x + log(exp(-b*x)))**2/b, Eq(a, log(exp(-b*x)))), (x*coth(a)**2, Eq(b, 0)), (x - 1/(b*tanh(a + b*x)), True))

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Giac [A]
time = 0.40, size = 24, normalized size = 1.85 \begin {gather*} \frac {b x + a - \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a - 2/(e^(2*b*x + 2*a) - 1))/b

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Mupad [B]
time = 1.02, size = 13, normalized size = 1.00 \begin {gather*} x-\frac {\mathrm {coth}\left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x)^2,x)

[Out]

x - coth(a + b*x)/b

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