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]

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

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.22, 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]

-1/2*(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]]])/(b*n)

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fricas [B]  time = 0.63, size = 572, normalized size = 13.30 \[ -\frac {b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} \log \relax (x) + 4 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \log \relax (x) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + b n \log \relax (x) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 2 \, {\left (b n \log \relax (x) - 1\right )} \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n \log \relax (x) + 2 \, {\left (3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} \log \relax (x) - b n \log \relax (x) + 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}\right ) + 4 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} \log \relax (x) - {\left (b n \log \relax (x) - 1\right )} \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 4 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + b n \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 2 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - b n\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - b n \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )} \]

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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giac [B]  time = 0.30, size = 127, normalized size = 2.95 \[ \frac {\log \left (\sqrt {-2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\relax (c) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{2} b n} - \log \relax (x) \]

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]

log(sqrt(-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)

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maple [A]  time = 0.02, size = 67, normalized size = 1.56 \[ -\frac {\coth ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2 n b}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2 n b} \]

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 = 0.43, size = 330, normalized size = 7.67 \[ -\frac {4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3 \, {\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{b n} + \frac {\log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{b n} - \log \relax (x) \]

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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mupad [B]  time = 1.23, size = 95, normalized size = 2.21 \[ \frac {2}{b\,n-b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\ln \relax (x)-\frac {2}{b\,n-2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*log(c*x^n))^3/x,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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