3.12 \(\int \coth ^6(a+b x) \, dx\)

Optimal. Leaf size=43 \[ -\frac{\coth ^5(a+b x)}{5 b}-\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b}+x \]

[Out]

x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b) - Coth[a + b*x]^5/(5*b)

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Rubi [A]  time = 0.0241105, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\coth ^5(a+b x)}{5 b}-\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^6,x]

[Out]

x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b) - Coth[a + b*x]^5/(5*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 ^6(a+b x) \, dx &=-\frac{\coth ^5(a+b x)}{5 b}+\int \coth ^4(a+b x) \, dx\\ &=-\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth ^5(a+b x)}{5 b}+\int \coth ^2(a+b x) \, dx\\ &=-\frac{\coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth ^5(a+b x)}{5 b}+\int 1 \, dx\\ &=x-\frac{\coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}-\frac{\coth ^5(a+b x)}{5 b}\\ \end{align*}

Mathematica [C]  time = 0.0089742, size = 31, normalized size = 0.72 \[ -\frac{\coth ^5(a+b x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\tanh ^2(a+b x)\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x]^6,x]

[Out]

-(Coth[a + b*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, Tanh[a + b*x]^2])/(5*b)

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Maple [A]  time = 0.002, size = 67, normalized size = 1.6 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{5}}{5\,b}}-{\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)^6,x)

[Out]

-1/5*coth(b*x+a)^5/b-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 = 1.03757, size = 155, normalized size = 3.6 \begin{align*} x + \frac{a}{b} - \frac{2 \,{\left (70 \, e^{\left (-2 \, b x - 2 \, a\right )} - 140 \, e^{\left (-4 \, b x - 4 \, a\right )} + 90 \, e^{\left (-6 \, b x - 6 \, a\right )} - 45 \, e^{\left (-8 \, b x - 8 \, a\right )} - 23\right )}}{15 \, b{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^6,x, algorithm="maxima")

[Out]

x + a/b - 2/15*(70*e^(-2*b*x - 2*a) - 140*e^(-4*b*x - 4*a) + 90*e^(-6*b*x - 6*a) - 45*e^(-8*b*x - 8*a) - 23)/(
b*(5*e^(-2*b*x - 2*a) - 10*e^(-4*b*x - 4*a) + 10*e^(-6*b*x - 6*a) - 5*e^(-8*b*x - 8*a) + e^(-10*b*x - 10*a) -
1))

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Fricas [B]  time = 2.28793, size = 666, normalized size = 15.49 \begin{align*} \frac{{\left (15 \, b x + 23\right )} \sinh \left (b x + a\right )^{5} - 23 \, \cosh \left (b x + a\right )^{5} - 115 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 5 \,{\left (2 \,{\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{2} - 15 \, b x - 23\right )} \sinh \left (b x + a\right )^{3} + 25 \, \cosh \left (b x + a\right )^{3} - 5 \,{\left (46 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 5 \,{\left ({\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{4} - 3 \,{\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{2} + 30 \, b x + 46\right )} \sinh \left (b x + a\right ) - 50 \, \cosh \left (b x + a\right )}{15 \,{\left (b \sinh \left (b x + a\right )^{5} + 5 \,{\left (2 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{3} + 5 \,{\left (b \cosh \left (b x + a\right )^{4} - 3 \, b \cosh \left (b x + a\right )^{2} + 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)^6,x, algorithm="fricas")

[Out]

1/15*((15*b*x + 23)*sinh(b*x + a)^5 - 23*cosh(b*x + a)^5 - 115*cosh(b*x + a)*sinh(b*x + a)^4 + 5*(2*(15*b*x +
23)*cosh(b*x + a)^2 - 15*b*x - 23)*sinh(b*x + a)^3 + 25*cosh(b*x + a)^3 - 5*(46*cosh(b*x + a)^3 - 15*cosh(b*x
+ a))*sinh(b*x + a)^2 + 5*((15*b*x + 23)*cosh(b*x + a)^4 - 3*(15*b*x + 23)*cosh(b*x + a)^2 + 30*b*x + 46)*sinh
(b*x + a) - 50*cosh(b*x + a))/(b*sinh(b*x + a)^5 + 5*(2*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^3 + 5*(b*cosh(b*x
+ a)^4 - 3*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a))

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Sympy [A]  time = 44.8374, size = 63, normalized size = 1.47 \begin{align*} \begin{cases} x \coth ^{6}{\left (a \right )} & \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 )}} - \frac{1}{3 b \tanh ^{3}{\left (a + b x \right )}} - \frac{1}{5 b \tanh ^{5}{\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)**6,x)

[Out]

Piecewise((x*coth(a)**6, Eq(b, 0)), (zoo*x, Eq(a, log(exp(-b*x))) | Eq(a, log(-exp(-b*x)))), (x - 1/(b*tanh(a
+ b*x)) - 1/(3*b*tanh(a + b*x)**3) - 1/(5*b*tanh(a + b*x)**5), True))

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Giac [A]  time = 1.16831, size = 100, normalized size = 2.33 \begin{align*} \frac{b x + a}{b} - \frac{2 \,{\left (45 \, e^{\left (8 \, b x + 8 \, a\right )} - 90 \, e^{\left (6 \, b x + 6 \, a\right )} + 140 \, e^{\left (4 \, b x + 4 \, a\right )} - 70 \, e^{\left (2 \, b x + 2 \, a\right )} + 23\right )}}{15 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)^6,x, algorithm="giac")

[Out]

(b*x + a)/b - 2/15*(45*e^(8*b*x + 8*a) - 90*e^(6*b*x + 6*a) + 140*e^(4*b*x + 4*a) - 70*e^(2*b*x + 2*a) + 23)/(
b*(e^(2*b*x + 2*a) - 1)^5)