3.188 \(\int \text {sech}^3(a+2 \log (\frac {c}{\sqrt {x}})) \, dx\)

Optimal. Leaf size=25 \[ \frac {2 e^{-3 a} c^2}{\left (e^{-2 a}+\frac {c^4}{x^2}\right )^2} \]

[Out]

2*c^2/exp(3*a)/(exp(-2*a)+c^4/x^2)^2

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Rubi [A]  time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5545, 5547, 263, 261} \[ \frac {2 e^{-3 a} c^2}{\left (e^{-2 a}+\frac {c^4}{x^2}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + 2*Log[c/Sqrt[x]]]^3,x]

[Out]

(2*c^2)/(E^(3*a)*(E^(-2*a) + c^4/x^2)^2)

Rule 261

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 5545

Int[Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Sech[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 5547

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

Rubi steps

\begin {align*} \int \text {sech}^3\left (a+2 \log \left (\frac {c}{\sqrt {x}}\right )\right ) \, dx &=-\left (\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {\text {sech}^3(a+2 \log (x))}{x^3} \, dx,x,\frac {c}{\sqrt {x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {e^{-2 a}}{x^4}\right )^3 x^9} \, dx,x,\frac {c}{\sqrt {x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (e^{-2 a}+x^4\right )^3} \, dx,x,\frac {c}{\sqrt {x}}\right )\right )\\ &=\frac {2 c^2 e^{-3 a}}{\left (e^{-2 a}+\frac {c^4}{x^2}\right )^2}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 64, normalized size = 2.56 \[ -\frac {2 c^6 (\sinh (2 a)+\cosh (2 a)) \left (\sinh (a) \left (c^4-2 x^2\right )+\cosh (a) \left (c^4+2 x^2\right )\right )}{\left (\sinh (a) \left (c^4-x^2\right )+\cosh (a) \left (c^4+x^2\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + 2*Log[c/Sqrt[x]]]^3,x]

[Out]

(-2*c^6*((c^4 + 2*x^2)*Cosh[a] + (c^4 - 2*x^2)*Sinh[a])*(Cosh[2*a] + Sinh[2*a]))/((c^4 + x^2)*Cosh[a] + (c^4 -
x^2)*Sinh[a])^2

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fricas [B]  time = 0.40, size = 49, normalized size = 1.96 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c^10*e^(5*a) + 2*c^6*x^2*e^(3*a))/(c^8*e^(4*a) + 2*c^4*x^2*e^(2*a) + x^4)

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giac [A]  time = 0.13, size = 37, normalized size = 1.48 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{{\left (c^{4} e^{\left (2 \, a\right )} + x^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c^10*e^(5*a) + 2*c^6*x^2*e^(3*a))/(c^4*e^(2*a) + x^2)^2

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maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a +2 \ln \left (\frac {c}{\sqrt {x}}\right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.34, size = 49, normalized size = 1.96 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c^10*e^(5*a) + 2*c^6*x^2*e^(3*a))/(c^8*e^(4*a) + 2*c^4*x^2*e^(2*a) + x^4)

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mupad [B]  time = 1.45, size = 36, normalized size = 1.44 \[ \frac {2\,c^2\,x^4\,{\mathrm {e}}^a}{{\mathrm {e}}^{4\,a}\,c^8+2\,{\mathrm {e}}^{2\,a}\,c^4\,x^2+x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(a + 2*log(c/x^(1/2)))^3,x)

[Out]

(2*c^2*x^4*exp(a))/(c^8*exp(4*a) + x^4 + 2*c^4*x^2*exp(2*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(a + 2*log(c/sqrt(x)))**3, x)

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