3.167 \(\int \frac{\text{csch}^3(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

ArcTanh[Cosh[a + b*Log[c*x^n]]]/(2*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(2*b*n)

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Rubi [A]  time = 0.0448704, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^3/x,x]

[Out]

ArcTanh[Cosh[a + b*Log[c*x^n]]]/(2*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(2*b*n)

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 \frac{\text{csch}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\operatorname{Subst}\left (\int \text{csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}\left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0591374, size = 81, normalized size = 1.47 \[ -\frac{\log \left (\tanh \left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac{\text{sech}^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac{\text{csch}^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^3/x,x]

[Out]

-Csch[(a + b*Log[c*x^n])/2]^2/(8*b*n) - Log[Tanh[(a + b*Log[c*x^n])/2]]/(2*b*n) - Sech[(a + b*Log[c*x^n])/2]^2
/(8*b*n)

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Maple [A]  time = 0.019, size = 51, normalized size = 0.9 \begin{align*} -{\frac{{\rm csch} \left (a+b\ln \left ( c{x}^{n} \right ) \right ){\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}{2\,bn}}+{\frac{{\it Artanh} \left ({{\rm e}^{a+b\ln \left ( c{x}^{n} \right ) }} \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(a+b*ln(c*x^n))^3/x,x)

[Out]

-1/2*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))/b/n+1/b/n*arctanh(exp(a+b*ln(c*x^n)))

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Maxima [B]  time = 1.15472, size = 203, normalized size = 3.69 \begin{align*} -\frac{c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + \frac{\log \left (\frac{{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} - \frac{\log \left (\frac{{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

-(c^(3*b)*e^(3*b*log(x^n) + 3*a) + c^b*e^(b*log(x^n) + a))/(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 2*b*c^(2*b)*n
*e^(2*b*log(x^n) + 2*a) + b*n) + 1/2*log((c^b*e^(b*log(x^n) + a) + 1)*e^(-a)/c^b)/(b*n) - 1/2*log((c^b*e^(b*lo
g(x^n) + a) - 1)*e^(-a)/c^b)/(b*n)

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Fricas [B]  time = 1.84464, size = 2109, normalized size = 38.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

-1/2*(2*cosh(b*n*log(x) + b*log(c) + a)^3 + 6*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^
2 + 2*sinh(b*n*log(x) + b*log(c) + a)^3 - (cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) +
a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a
)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(
c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(cosh(b*n*log(x) + b*log(
c) + a) + sinh(b*n*log(x) + b*log(c) + a) + 1) + (cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*lo
g(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log
(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) +
b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(cosh(b*n*log(x) +
b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a) - 1) + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*l
og(x) + b*log(c) + a) + 2*cosh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh
(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*sinh(b*n*log(x) + b*log(c) + a)^4 - 2*b*n*
cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 - b*n)*sinh(b*n*log(x) + b*log(
c) + a)^2 + b*n + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log
(x) + b*log(c) + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*ln(c*x**n))**3/x,x)

[Out]

Integral(csch(a + b*log(c*x**n))**3/x, x)

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Giac [B]  time = 1.20888, size = 278, normalized size = 5.05 \begin{align*} \frac{1}{4} \, c^{3 \, b}{\left (\frac{c^{b} e^{\left (-3 \, a\right )} \log \left (2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{4 \, b} n} - \frac{c^{b} e^{\left (-3 \, a\right )} \log \left (-2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{4 \, b} n} - \frac{4 \,{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

1/4*c^(3*b)*(c^b*e^(-3*a)*log(2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*pi*b)*e^a + x^(2*b*n)*abs(c)^(2*b)
*e^(2*a) + 1)/(b*c^(4*b)*n) - c^b*e^(-3*a)*log(-2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*pi*b)*e^a + x^(2
*b*n)*abs(c)^(2*b)*e^(2*a) + 1)/(b*c^(4*b)*n) - 4*(c^(2*b)*x^(3*b*n)*e^(2*a) + x^(b*n))*e^(-2*a)/((c^(2*b)*x^(
2*b*n)*e^(2*a) - 1)^2*b*c^(2*b)*n))*e^(3*a)