3.6 \(\int \text{csch}^6(a+b x) \, dx\)

Optimal. Leaf size=42 \[ -\frac{\coth ^5(a+b x)}{5 b}+\frac{2 \coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0153031, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ -\frac{\coth ^5(a+b x)}{5 b}+\frac{2 \coth ^3(a+b x)}{3 b}-\frac{\coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^6,x]

[Out]

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \text{csch}^6(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (a+b x)\right )}{b}\\ &=-\frac{\coth (a+b x)}{b}+\frac{2 \coth ^3(a+b x)}{3 b}-\frac{\coth ^5(a+b x)}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0161284, size = 56, normalized size = 1.33 \[ -\frac{8 \coth (a+b x)}{15 b}-\frac{\coth (a+b x) \text{csch}^4(a+b x)}{5 b}+\frac{4 \coth (a+b x) \text{csch}^2(a+b x)}{15 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^6,x]

[Out]

(-8*Coth[a + b*x])/(15*b) + (4*Coth[a + b*x]*Csch[a + b*x]^2)/(15*b) - (Coth[a + b*x]*Csch[a + b*x]^4)/(5*b)

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Maple [A]  time = 0.012, size = 33, normalized size = 0.8 \begin{align*}{\frac{{\rm coth} \left (bx+a\right )}{b} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}}{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^6,x)

[Out]

1/b*(-8/15-1/5*csch(b*x+a)^4+4/15*csch(b*x+a)^2)*coth(b*x+a)

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Maxima [B]  time = 1.01119, size = 277, normalized size = 6.6 \begin{align*} -\frac{16 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, 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 )}} + \frac{32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{3 \, 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 )}} + \frac{16}{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(csch(b*x+a)^6,x, algorithm="maxima")

[Out]

-16/3*e^(-2*b*x - 2*a)/(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)) + 32/3*e^(-4*b*x - 4*a)/(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)) + 16/15/(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 = 1.49635, size = 954, normalized size = 22.71 \begin{align*} -\frac{16 \,{\left (11 \, \cosh \left (b x + a\right )^{2} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 11 \, \sinh \left (b x + a\right )^{2} - 5\right )}}{15 \,{\left (b \cosh \left (b x + a\right )^{8} + 8 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + b \sinh \left (b x + a\right )^{8} - 5 \, b \cosh \left (b x + a\right )^{6} +{\left (28 \, b \cosh \left (b x + a\right )^{2} - 5 \, b\right )} \sinh \left (b x + a\right )^{6} + 2 \,{\left (28 \, b \cosh \left (b x + a\right )^{3} - 15 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \, b \cosh \left (b x + a\right )^{4} + 5 \,{\left (14 \, b \cosh \left (b x + a\right )^{4} - 15 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \,{\left (14 \, b \cosh \left (b x + a\right )^{5} - 25 \, b \cosh \left (b x + a\right )^{3} + 10 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 11 \, b \cosh \left (b x + a\right )^{2} +{\left (28 \, b \cosh \left (b x + a\right )^{6} - 75 \, b \cosh \left (b x + a\right )^{4} + 60 \, b \cosh \left (b x + a\right )^{2} - 11 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \,{\left (4 \, b \cosh \left (b x + a\right )^{7} - 15 \, b \cosh \left (b x + a\right )^{5} + 20 \, b \cosh \left (b x + a\right )^{3} - 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 5 \, b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^6,x, algorithm="fricas")

[Out]

-16/15*(11*cosh(b*x + a)^2 + 18*cosh(b*x + a)*sinh(b*x + a) + 11*sinh(b*x + a)^2 - 5)/(b*cosh(b*x + a)^8 + 8*b
*cosh(b*x + a)*sinh(b*x + a)^7 + b*sinh(b*x + a)^8 - 5*b*cosh(b*x + a)^6 + (28*b*cosh(b*x + a)^2 - 5*b)*sinh(b
*x + a)^6 + 2*(28*b*cosh(b*x + a)^3 - 15*b*cosh(b*x + a))*sinh(b*x + a)^5 + 10*b*cosh(b*x + a)^4 + 5*(14*b*cos
h(b*x + a)^4 - 15*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^4 + 4*(14*b*cosh(b*x + a)^5 - 25*b*cosh(b*x + a)^3 +
10*b*cosh(b*x + a))*sinh(b*x + a)^3 - 11*b*cosh(b*x + a)^2 + (28*b*cosh(b*x + a)^6 - 75*b*cosh(b*x + a)^4 + 60
*b*cosh(b*x + a)^2 - 11*b)*sinh(b*x + a)^2 + 2*(4*b*cosh(b*x + a)^7 - 15*b*cosh(b*x + a)^5 + 20*b*cosh(b*x + a
)^3 - 9*b*cosh(b*x + a))*sinh(b*x + a) + 5*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}^{6}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**6,x)

[Out]

Integral(csch(a + b*x)**6, x)

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Giac [A]  time = 1.17068, size = 57, normalized size = 1.36 \begin{align*} -\frac{16 \,{\left (10 \, e^{\left (4 \, b x + 4 \, a\right )} - 5 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\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(csch(b*x+a)^6,x, algorithm="giac")

[Out]

-16/15*(10*e^(4*b*x + 4*a) - 5*e^(2*b*x + 2*a) + 1)/(b*(e^(2*b*x + 2*a) - 1)^5)