3.191 \(\int \frac{\coth ^3(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.0402723, antiderivative size = 43, 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 = {3473, 3475} \[ \frac{\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[c*x^n]]^3/x,x]

[Out]

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

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 3475

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

Rubi steps

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

Mathematica [A]  time = 0.206839, size = 52, normalized size = 1.21 \[ -\frac{-2 \log \left (\tanh \left (a+b \log \left (c x^n\right )\right )\right )-2 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )+\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 67, normalized size = 1.6 \begin{align*} -{\frac{ \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{2\,bn}}-{\frac{\ln \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )-1 \right ) }{2\,bn}}-{\frac{\ln \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )+1 \right ) }{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.37249, size = 446, normalized size = 10.37 \begin{align*} -\frac{4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \,{\left (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\right )}} - \frac{3 \,{\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{4 \,{\left (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\right )}} + \frac{2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \,{\left (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\right )}} - \frac{3}{4 \,{\left (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\right )}} + \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 )}{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 )}{b n} - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.67292, size = 1841, normalized size = 42.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*n*cosh(b*n*log(x) + b*log(c) + a)^4*log(x) + 4*b*n*cosh(b*n*log(x) + b*log(c) + a)*log(x)*sinh(b*n*log(x)
+ b*log(c) + a)^3 + b*n*log(x)*sinh(b*n*log(x) + b*log(c) + a)^4 - 2*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(
c) + a)^2 + b*n*log(x) + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) + a)^2*log(x) - b*n*log(x) + 1)*sinh(b*n*log(x) +
b*log(c) + a)^2 - (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*lo
g(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(2*sinh(b*n*log(x) + b*log(c) + a)/(cosh(b*n*log
(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a))) + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3*log(x) - (b
*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a))*sinh(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)*si
nh(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.42911, size = 170, normalized size = 3.95 \begin{align*} \frac{\log \left (-2 \, x^{2 \, b n}{\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm{sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n}{\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1\right )}{2 \, b n} - \frac{3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \,{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b n} - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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