3.10 \(\int \coth ^4(a+b x) \, dx\)

Optimal. Leaf size=28 \[ -\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b}+x \]

[Out]

x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0161894, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^4,x]

[Out]

x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b)

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]

Rule 8

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

Rubi steps

\begin{align*} \int \coth ^4(a+b x) \, dx &=-\frac{\coth ^3(a+b x)}{3 b}+\int \coth ^2(a+b x) \, dx\\ &=-\frac{\coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x-\frac{\coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [C]  time = 0.0098578, size = 31, normalized size = 1.11 \[ -\frac{\coth ^3(a+b x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\tanh ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x]^4,x]

[Out]

-(Coth[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[a + b*x]^2])/(3*b)

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Maple [B]  time = 0.002, size = 54, normalized size = 1.9 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{3}}{3\,b}}-{\frac{{\rm coth} \left (bx+a\right )}{b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}+{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a)^4,x)

[Out]

-1/3*coth(b*x+a)^3/b-coth(b*x+a)/b-1/2/b*ln(coth(b*x+a)-1)+1/2/b*ln(coth(b*x+a)+1)

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Maxima [B]  time = 1.04297, size = 96, normalized size = 3.43 \begin{align*} x + \frac{a}{b} - \frac{4 \,{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 2\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^4,x, algorithm="maxima")

[Out]

x + a/b - 4/3*(3*e^(-2*b*x - 2*a) - 3*e^(-4*b*x - 4*a) - 2)/(b*(3*e^(-2*b*x - 2*a) - 3*e^(-4*b*x - 4*a) + e^(-
6*b*x - 6*a) - 1))

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Fricas [B]  time = 2.11061, size = 286, normalized size = 10.21 \begin{align*} \frac{{\left (3 \, b x + 4\right )} \sinh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )^{3} - 12 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \,{\left ({\left (3 \, b x + 4\right )} \cosh \left (b x + a\right )^{2} - 3 \, b x - 4\right )} \sinh \left (b x + a\right )}{3 \,{\left (b \sinh \left (b x + a\right )^{3} + 3 \,{\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*((3*b*x + 4)*sinh(b*x + a)^3 - 4*cosh(b*x + a)^3 - 12*cosh(b*x + a)*sinh(b*x + a)^2 + 3*((3*b*x + 4)*cosh(
b*x + a)^2 - 3*b*x - 4)*sinh(b*x + a))/(b*sinh(b*x + a)^3 + 3*(b*cosh(b*x + a)^2 - b)*sinh(b*x + a))

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Sympy [A]  time = 9.25608, size = 49, normalized size = 1.75 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\x \coth ^{4}{\left (a \right )} & \text{for}\: b = 0 \\x - \frac{1}{b \tanh{\left (a + b x \right )}} - \frac{1}{3 b \tanh ^{3}{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)**4,x)

[Out]

Piecewise((zoo*x, Eq(a, log(exp(-b*x))) | Eq(a, log(-exp(-b*x)))), (x*coth(a)**4, Eq(b, 0)), (x - 1/(b*tanh(a
+ b*x)) - 1/(3*b*tanh(a + b*x)**3), True))

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Giac [A]  time = 1.21794, size = 70, normalized size = 2.5 \begin{align*} \frac{b x + a}{b} - \frac{4 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 2\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^4,x, algorithm="giac")

[Out]

(b*x + a)/b - 4/3*(3*e^(4*b*x + 4*a) - 3*e^(2*b*x + 2*a) + 2)/(b*(e^(2*b*x + 2*a) - 1)^3)