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

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

[Out]

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

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Rubi [A]  time = 0.0394355, 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} \[ -\frac{\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\tanh \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[a + b*Log[c*x^n]]^4/x,x]

[Out]

Log[x] - Tanh[a + b*Log[c*x^n]]/(b*n) - Tanh[a + b*Log[c*x^n]]^3/(3*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 8

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

Rubi steps

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

Mathematica [A]  time = 0.102834, size = 62, normalized size = 1.38 \[ -\frac{\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\tanh ^{-1}\left (\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\tanh \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 86, normalized size = 1.9 \begin{align*} -{\frac{ \left ( \tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{3\,bn}}-{\frac{\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}}-{\frac{\ln \left ( \tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) -1 \right ) }{2\,bn}}+{\frac{\ln \left ( \tanh \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(tanh(a+b*ln(c*x^n))^4/x,x)

[Out]

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

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Maxima [B]  time = 1.53108, size = 667, normalized size = 14.82 \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(tanh(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*lo
g(x^n) + 2*a) + b*n) + log(x)

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Fricas [B]  time = 2.00437, size = 630, normalized size = 14. \begin{align*} \frac{{\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 )^{3} + 3 \,{\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 ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 12 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 4 \, \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\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 )}{3 \,{\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, b n \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 \, b n \cosh \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(tanh(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 28.8841, size = 71, normalized size = 1.58 \begin{align*} \begin{cases} \log{\left (x \right )} \tanh ^{4}{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\\log{\left (x \right )} \tanh ^{4}{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\log{\left (x \right )} \tanh ^{4}{\left (a \right )} & \text{for}\: b = 0 \\\log{\left (x \right )} - \frac{\tanh ^{3}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{3 b n} - \frac{\tanh{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+b*ln(c*x**n))**4/x,x)

[Out]

Piecewise((log(x)*tanh(a)**4, Eq(b, 0) & Eq(n, 0)), (log(x)*tanh(a + b*log(c))**4, Eq(n, 0)), (log(x)*tanh(a)*
*4, Eq(b, 0)), (log(x) - tanh(a + b*n*log(x) + b*log(c))**3/(3*b*n) - tanh(a + b*n*log(x) + b*log(c))/(b*n), T
rue))

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Giac [A]  time = 1.3631, 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(tanh(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)