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

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

[Out]

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

________________________________________________________________________________________

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*}

________________________________________________________________________________________

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)

________________________________________________________________________________________

fricas [B]  time = 0.64, 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(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))

________________________________________________________________________________________

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(coth(b*x+a)^2,x, algorithm="giac")

[Out]

(b*x + a - 2/(e^(2*b*x + 2*a) - 1))/b

________________________________________________________________________________________

maple [B]  time = 0.01, size = 41, normalized size = 3.15 \[ -\frac {\coth \left (b x +a \right )}{b}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2 b}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.18, 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(coth(b*x+a)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

sympy [A]  time = 1.26, size = 36, normalized size = 2.77 \[ \begin {cases} x \coth ^{2}{\relax (a )} & \text {for}\: b = 0 \\\tilde {\infty } x & \text {for}\: a = \log {\left (- e^{- b x} \right )} \vee a = \log {\left (e^{- b x} \right )} \\x - \frac {1}{b \tanh {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________