3.5 \(\int \text{csch}^5(a+b x) \, dx\)

Optimal. Leaf size=55 \[ -\frac{3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}+\frac{3 \coth (a+b x) \text{csch}(a+b x)}{8 b} \]

[Out]

(-3*ArcTanh[Cosh[a + b*x]])/(8*b) + (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]*Csch[a + b*x]^3)/(4
*b)

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Rubi [A]  time = 0.0430128, antiderivative size = 55, 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 = {3768, 3770} \[ -\frac{3 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}+\frac{3 \coth (a+b x) \text{csch}(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^5,x]

[Out]

(-3*ArcTanh[Cosh[a + b*x]])/(8*b) + (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]*Csch[a + b*x]^3)/(4
*b)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \text{csch}^5(a+b x) \, dx &=-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}-\frac{3}{4} \int \text{csch}^3(a+b x) \, dx\\ &=\frac{3 \coth (a+b x) \text{csch}(a+b x)}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}+\frac{3}{8} \int \text{csch}(a+b x) \, dx\\ &=-\frac{3 \tanh ^{-1}(\cosh (a+b x))}{8 b}+\frac{3 \coth (a+b x) \text{csch}(a+b x)}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0160215, size = 95, normalized size = 1.73 \[ -\frac{\text{csch}^4\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{3 \text{csch}^2\left (\frac{1}{2} (a+b x)\right )}{32 b}+\frac{\text{sech}^4\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{3 \text{sech}^2\left (\frac{1}{2} (a+b x)\right )}{32 b}+\frac{3 \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^5,x]

[Out]

(3*Csch[(a + b*x)/2]^2)/(32*b) - Csch[(a + b*x)/2]^4/(64*b) + (3*Log[Tanh[(a + b*x)/2]])/(8*b) + (3*Sech[(a +
b*x)/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b)

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Maple [A]  time = 0.011, size = 41, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( \left ( -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (bx+a\right )}{8}} \right ){\rm coth} \left (bx+a\right )-{\frac{3\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^5,x)

[Out]

1/b*((-1/4*csch(b*x+a)^3+3/8*csch(b*x+a))*coth(b*x+a)-3/4*arctanh(exp(b*x+a)))

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Maxima [B]  time = 1.03042, size = 180, normalized size = 3.27 \begin{align*} -\frac{3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac{3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} - \frac{3 \, e^{\left (-b x - a\right )} - 11 \, e^{\left (-3 \, b x - 3 \, a\right )} - 11 \, e^{\left (-5 \, b x - 5 \, a\right )} + 3 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^5,x, algorithm="maxima")

[Out]

-3/8*log(e^(-b*x - a) + 1)/b + 3/8*log(e^(-b*x - a) - 1)/b - 1/4*(3*e^(-b*x - a) - 11*e^(-3*b*x - 3*a) - 11*e^
(-5*b*x - 5*a) + 3*e^(-7*b*x - 7*a))/(b*(4*e^(-2*b*x - 2*a) - 6*e^(-4*b*x - 4*a) + 4*e^(-6*b*x - 6*a) - e^(-8*
b*x - 8*a) - 1))

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Fricas [B]  time = 1.72173, size = 3087, normalized size = 56.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^5,x, algorithm="fricas")

[Out]

1/8*(6*cosh(b*x + a)^7 + 42*cosh(b*x + a)*sinh(b*x + a)^6 + 6*sinh(b*x + a)^7 + 2*(63*cosh(b*x + a)^2 - 11)*si
nh(b*x + a)^5 - 22*cosh(b*x + a)^5 + 10*(21*cosh(b*x + a)^3 - 11*cosh(b*x + a))*sinh(b*x + a)^4 + 2*(105*cosh(
b*x + a)^4 - 110*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^3 - 22*cosh(b*x + a)^3 + 2*(63*cosh(b*x + a)^5 - 110*cosh
(b*x + a)^3 - 33*cosh(b*x + a))*sinh(b*x + a)^2 - 3*(cosh(b*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x + a)^7 + sinh(
b*x + a)^8 + 4*(7*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^6 - 4*cosh(b*x + a)^6 + 8*(7*cosh(b*x + a)^3 - 3*cosh(b*x
+ a))*sinh(b*x + a)^5 + 2*(35*cosh(b*x + a)^4 - 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^4 + 6*cosh(b*x + a)^4 +
8*(7*cosh(b*x + a)^5 - 10*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*cosh(b*x + a)^6 - 15*cosh
(b*x + a)^4 + 9*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 4*cosh(b*x + a)^2 + 8*(cosh(b*x + a)^7 - 3*cosh(b*x + a
)^5 + 3*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 3*(cosh(b
*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x + a)^7 + sinh(b*x + a)^8 + 4*(7*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^6 - 4*
cosh(b*x + a)^6 + 8*(7*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^5 + 2*(35*cosh(b*x + a)^4 - 30*cosh(b*
x + a)^2 + 3)*sinh(b*x + a)^4 + 6*cosh(b*x + a)^4 + 8*(7*cosh(b*x + a)^5 - 10*cosh(b*x + a)^3 + 3*cosh(b*x + a
))*sinh(b*x + a)^3 + 4*(7*cosh(b*x + a)^6 - 15*cosh(b*x + a)^4 + 9*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 4*co
sh(b*x + a)^2 + 8*(cosh(b*x + a)^7 - 3*cosh(b*x + a)^5 + 3*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)
*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*(21*cosh(b*x + a)^6 - 55*cosh(b*x + a)^4 - 33*cosh(b*x + a)^2 + 3)
*sinh(b*x + a) + 6*cosh(b*x + a))/(b*cosh(b*x + a)^8 + 8*b*cosh(b*x + a)*sinh(b*x + a)^7 + b*sinh(b*x + a)^8 -
4*b*cosh(b*x + a)^6 + 4*(7*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^6 + 8*(7*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a
))*sinh(b*x + a)^5 + 6*b*cosh(b*x + a)^4 + 2*(35*b*cosh(b*x + a)^4 - 30*b*cosh(b*x + a)^2 + 3*b)*sinh(b*x + a)
^4 + 8*(7*b*cosh(b*x + a)^5 - 10*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^3 - 4*b*cosh(b*x + a)^2
+ 4*(7*b*cosh(b*x + a)^6 - 15*b*cosh(b*x + a)^4 + 9*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^2 + 8*(b*cosh(b*x + a
)^7 - 3*b*cosh(b*x + a)^5 + 3*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a) + b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}^{5}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**5,x)

[Out]

Integral(csch(a + b*x)**5, x)

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Giac [B]  time = 1.15139, size = 154, normalized size = 2.8 \begin{align*} -\frac{3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right )}{16 \, b} + \frac{3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{16 \, b} + \frac{3 \,{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 20 \, e^{\left (b x + a\right )} - 20 \, e^{\left (-b x - a\right )}}{4 \,{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^5,x, algorithm="giac")

[Out]

-3/16*log(e^(b*x + a) + e^(-b*x - a) + 2)/b + 3/16*log(e^(b*x + a) + e^(-b*x - a) - 2)/b + 1/4*(3*(e^(b*x + a)
+ e^(-b*x - a))^3 - 20*e^(b*x + a) - 20*e^(-b*x - a))/(((e^(b*x + a) + e^(-b*x - a))^2 - 4)^2*b)