3.260 \(\int \frac {1}{\cosh ^{\frac {3}{2}}(a+\frac {2 \log (c x^n)}{n})} \, dx\)

Optimal. Leaf size=42 \[ -\frac {x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]

[Out]

-1/2*x*(1+1/exp(2*a)/((c*x^n)^(4/n)))/cosh(a+2*ln(c*x^n)/n)^(3/2)

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Rubi [A]  time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5526, 5534, 264} \[ -\frac {x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + (2*Log[c*x^n])/n]^(-3/2),x]

[Out]

-(x*(1 + 1/(E^(2*a)*(c*x^n)^(4/n))))/(2*Cosh[a + (2*Log[c*x^n])/n]^(3/2))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 5526

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5534

Int[Cosh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[Cosh[d*(a + b*Log[x])]^p/(x
^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[(e*x)^m*x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; Fr
eeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 61, normalized size = 1.45 \[ \frac {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}-2 \log (x)\right )-\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}-2 \log (x)\right )}{x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + (2*Log[c*x^n])/n]^(-3/2),x]

[Out]

(-Cosh[a - 2*Log[x] + (2*Log[c*x^n])/n] + Sinh[a - 2*Log[x] + (2*Log[c*x^n])/n])/(x*Sqrt[Cosh[a + (2*Log[c*x^n
])/n]])

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fricas [A]  time = 0.49, size = 68, normalized size = 1.62 \[ -\frac {2 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \relax (c)}{2 \, n}\right )}}{x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(a+2*ln(c*x^n)/n)^(3/2),x)

[Out]

int(1/cosh(a+2*ln(c*x^n)/n)^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="maxima")

[Out]

integrate(cosh(a + 2*log(c*x^n)/n)^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(a + (2*log(c*x^n))/n)^(3/2),x)

[Out]

int(1/cosh(a + (2*log(c*x^n))/n)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cosh ^{\frac {3}{2}}{\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cosh(a+2*ln(c*x**n)/n)**(3/2),x)

[Out]

Integral(cosh(a + 2*log(c*x**n)/n)**(-3/2), x)

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