3.193 \(\int \frac{\text{sech}^3(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b*n) + (Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(2*b*n)

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Rubi [A]  time = 0.0396532, antiderivative size = 55, normalized size of antiderivative = 1., 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 = {3768, 3770} \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]^3/x,x]

[Out]

ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b*n) + (Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(2*b*n)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0552964, size = 55, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*Log[c*x^n]]^3/x,x]

[Out]

ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b*n) + (Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(2*b*n)

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Maple [A]  time = 0.02, size = 51, normalized size = 0.9 \begin{align*}{\frac{{\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right )\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\frac{\arctan \left ({{\rm e}^{a+b\ln \left ( c{x}^{n} \right ) }} \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))^3/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 8 \, c^{b} \int \frac{e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \,{\left (c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + x\right )}}\,{d x} + \frac{c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

8*c^b*integrate(1/8*e^(b*log(x^n) + a)/(c^(2*b)*x*e^(2*b*log(x^n) + 2*a) + x), x) + (c^(3*b)*e^(3*b*log(x^n) +
3*a) - c^b*e^(b*log(x^n) + a))/(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)

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Fricas [B]  time = 3.1923, size = 1486, normalized size = 27.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

(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)^2 + sin
h(b*n*log(x) + b*log(c) + a)^3 + (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*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*arctan(cosh(b*n*log(x) + b*log(c) + a
) + sinh(b*n*log(x) + b*log(c) + a)) + (3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) +
a) - cosh(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*l
og(c) + a)^2 + 2*(3*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 + 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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{3}{\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))**3/x,x)

[Out]

Integral(sech(a + b*log(c*x**n))**3/x, x)

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Giac [B]  time = 1.17208, size = 155, normalized size = 2.82 \begin{align*} c^{3 \, b}{\left (\frac{\arctan \left (\frac{c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac{{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

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