3.192 \(\int \frac{\coth ^4(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n)) - Coth[a + b*Log[c*x^n]]^3/(3*b*n) + Log[x]

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Rubi [A]  time = 0.0374456, antiderivative size = 45, 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, 8} \[ -\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[c*x^n]]^4/x,x]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n)) - Coth[a + b*Log[c*x^n]]^3/(3*b*n) + Log[x]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\coth ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\operatorname{Subst}\left (\int \coth ^2(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 )}{b n}-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\log (x)\\ \end{align*}

Mathematica [C]  time = 0.114257, size = 44, normalized size = 0.98 \[ -\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\tanh ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*Log[c*x^n]]^4/x,x]

[Out]

-(Coth[a + b*Log[c*x^n]]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[a + b*Log[c*x^n]]^2])/(3*b*n)

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Maple [A]  time = 0.003, size = 86, normalized size = 1.9 \begin{align*} -{\frac{ \left ({\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{3\,bn}}-{\frac{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}{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))^4/x,x)

[Out]

-1/3*coth(a+b*ln(c*x^n))^3/b/n-coth(a+b*ln(c*x^n))/b/n-1/2/n/b*ln(coth(a+b*ln(c*x^n))-1)+1/2/n/b*ln(coth(a+b*l
n(c*x^n))+1)

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Maxima [B]  time = 1.51077, size = 674, normalized size = 14.98 \begin{align*} -\frac{18 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 27 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 11}{12 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac{6 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 15 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 11}{12 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac{2 \,{\left (3 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{3 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac{3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1}{2 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac{2}{3 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} + \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))^4/x,x, algorithm="maxima")

[Out]

-1/12*(18*c^(4*b)*e^(4*b*log(x^n) + 4*a) - 27*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 11)/(b*c^(6*b)*n*e^(6*b*log(x^n
) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 1/12*(6*c^(4*b
)*e^(4*b*log(x^n) + 4*a) - 15*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 11)/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c
^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 2/3*(3*c^(4*b)*e^(4*b*log(x^n)
+ 4*a) - 3*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log
(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 1/2*(3*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 1)/(b*c^(
6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) -
b*n) - 2/3/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*l
og(x^n) + 2*a) - b*n) + log(x)

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Fricas [B]  time = 2.50828, size = 541, normalized size = 12.02 \begin{align*} \frac{{\left (3 \, b n \log \left (x\right ) + 4\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 12 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 3 \,{\left ({\left (3 \, b n \log \left (x\right ) + 4\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, b n \log \left (x\right ) - 4\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \,{\left (b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/3*((3*b*n*log(x) + 4)*sinh(b*n*log(x) + b*log(c) + a)^3 - 4*cosh(b*n*log(x) + b*log(c) + a)^3 - 12*cosh(b*n*
log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + 3*((3*b*n*log(x) + 4)*cosh(b*n*log(x) + b*log(c) +
a)^2 - 3*b*n*log(x) - 4)*sinh(b*n*log(x) + b*log(c) + a))/(b*n*sinh(b*n*log(x) + b*log(c) + a)^3 + 3*(b*n*cosh
(b*n*log(x) + b*log(c) + a)^2 - b*n)*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))**4/x,x)

[Out]

Timed out

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Giac [A]  time = 1.43402, size = 90, normalized size = 2. \begin{align*} -\frac{4 \,{\left (3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 2\right )}}{3 \,{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{3} 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))^4/x,x, algorithm="giac")

[Out]

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