3.193 \(\int \frac{\coth ^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=66 \[ \frac{\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

-Coth[a + b*Log[c*x^n]]^2/(2*b*n) - Coth[a + b*Log[c*x^n]]^4/(4*b*n) + Log[Sinh[a + b*Log[c*x^n]]]/(b*n)

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Rubi [A]  time = 0.0573933, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3473, 3475} \[ \frac{\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[c*x^n]]^5/x,x]

[Out]

-Coth[a + b*Log[c*x^n]]^2/(2*b*n) - Coth[a + b*Log[c*x^n]]^4/(4*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 ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\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{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 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{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.279886, size = 67, normalized size = 1.02 \[ -\frac{-4 \log \left (\tanh \left (a+b \log \left (c x^n\right )\right )\right )-4 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )+\coth ^4\left (a+b \log \left (c x^n\right )\right )+2 \coth ^2\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*Log[c*x^n]]^5/x,x]

[Out]

-(2*Coth[a + b*Log[c*x^n]]^2 + Coth[a + b*Log[c*x^n]]^4 - 4*Log[Cosh[a + b*Log[c*x^n]]] - 4*Log[Tanh[a + b*Log
[c*x^n]]])/(4*b*n)

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Maple [A]  time = 0.003, size = 88, normalized size = 1.3 \begin{align*} -{\frac{ \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{4\,bn}}-{\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))^5/x,x)

[Out]

-1/4*coth(a+b*ln(c*x^n))^4/b/n-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 = 1.63914, size = 1154, normalized size = 17.48 \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))^5/x,x, algorithm="maxima")

[Out]

-1/24*(48*c^(6*b)*e^(6*b*log(x^n) + 6*a) - 108*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 88*c^(2*b)*e^(2*b*log(x^n) + 2
*a) - 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*lo
g(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) + 1/24*(12*c^(6*b)*e^(6*b*log(x^n) + 6*a) - 42*c^(
4*b)*e^(4*b*log(x^n) + 4*a) + 52*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*
b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*
a) + b*n) - 5/8*(4*c^(6*b)*e^(6*b*log(x^n) + 6*a) - 6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c^(2*b)*e^(2*b*log(x^
n) + 2*a) - 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4
*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 5/12*(6*c^(4*b)*e^(4*b*log(x^n) + 4*a) - 4*
c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a)
+ 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 5/12*(4*c^(2*b)*e^(2*b
*log(x^n) + 2*a) - 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)
*n*e^(4*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 5/8/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8
*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) - 4*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 = 2.68697, size = 5060, normalized size = 76.67 \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))^5/x,x, algorithm="fricas")

[Out]

-(b*n*cosh(b*n*log(x) + b*log(c) + a)^8*log(x) + 8*b*n*cosh(b*n*log(x) + b*log(c) + a)*log(x)*sinh(b*n*log(x)
+ b*log(c) + a)^7 + b*n*log(x)*sinh(b*n*log(x) + b*log(c) + a)^8 - 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(
c) + a)^6 + 4*(7*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
)^6 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3*log(x) - 3*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a))*
sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a)^4 + 2*(35*b*n*cosh(b*
n*log(x) + b*log(c) + a)^4*log(x) - 30*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n*log(x) - 2)*
sinh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5*log(x) - 10*(b*n*log(x) - 1)*co
sh(b*n*log(x) + b*log(c) + a)^3 + (3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(
c) + a)^3 - 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n*log(x) + 4*(7*b*n*cosh(b*n*log(x) + b*l
og(c) + a)^6*log(x) - 15*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^4 + 3*(3*b*n*log(x) - 2)*cosh(b*n*lo
g(x) + b*log(c) + a)^2 - b*n*log(x) + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b*n*log(x) + b*log(c) + a)^
8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 +
4*(7*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^6 - 4*cosh(b*n*log(x) + b*log(c) +
a)^6 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^3 - 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c)
+ a)^5 + 2*(35*cosh(b*n*log(x) + b*log(c) + a)^4 - 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x)
+ b*log(c) + a)^4 + 6*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*cosh(b*n
*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b
*n*log(x) + b*log(c) + a)^6 - 15*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*
sinh(b*n*log(x) + b*log(c) + a)^2 - 4*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7
- 3*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*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(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))) + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7*log(x) - 3*(b*n*log(x) - 1)*cos
h(b*n*log(x) + b*log(c) + a)^5 + (3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a)^3 - (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)^8 + 8*b*n*co
sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sinh(b*n*log(x) + b*log(c) + a)^8 - 4*b*
n*cosh(b*n*log(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 - b*n)*sinh(b*n*log(x) + b*lo
g(c) + a)^6 + 6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - 3*b*n*cos
h(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4
- 30*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^4 - 4*b*n*cosh(b*n*log(x)
+ b*log(c) + a)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*
b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) +
a)^6 - 15*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*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 + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 - 3*b*n*cosh(b*n*log(x) + b*log(c) + a)
^5 + 3*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))**5/x,x)

[Out]

Timed out

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Giac [B]  time = 1.34413, size = 216, normalized size = 3.27 \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{25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} - 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \,{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} 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))^5/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)
- 1/12*(25*c^(8*b)*x^(8*b*n)*e^(8*a) - 52*c^(6*b)*x^(6*b*n)*e^(6*a) + 102*c^(4*b)*x^(4*b*n)*e^(4*a) - 52*c^(2*
b)*x^(2*b*n)*e^(2*a) + 25)/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^4*b*n) - log(x)