3.194 \(\int \frac{\text{sech}^4(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tanh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

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 = 0.0337831, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3767} \[ \frac{\tanh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]^4/x,x]

[Out]

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0475731, size = 42, normalized size = 1. \[ \frac{\tanh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 = 0.017, size = 36, normalized size = 0.9 \begin{align*}{\frac{\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))^4/x,x)

[Out]

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

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Maxima [B]  time = 1.20158, size = 123, normalized size = 2.93 \begin{align*} -\frac{4 \,{\left (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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(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)

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Fricas [B]  time = 3.05937, size = 869, normalized size = 20.69 \begin{align*} -\frac{8 \,{\left (2 \, \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 )\right )}}{3 \,{\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 5 \, 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 )^{4} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} +{\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 3 \, b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 4 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 9 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (5 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 9 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 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(sech(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

-8/3*(2*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
)^5 + 5*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^4 + b*n*sinh(b*n*log(x) + b*log(c)
+ a)^5 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + (10*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*
n*log(x) + b*log(c) + a)^3 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a) + (10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3
+ 9*b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^2 + (5*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 + 2*b*n)*sinh(b*n*log(x) + b*log(c) + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{4}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(a + b*log(c*x**n))**4/x, x)

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Giac [A]  time = 1.17025, size = 63, normalized size = 1.5 \begin{align*} -\frac{4 \,{\left (3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}}{3 \,{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{3} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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