3.159 \(\int \frac{x^4}{\sqrt{\text{sech}(2 \log (c x))}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^5 \left (c^4+\frac{1}{x^4}\right )}{6 c^4 \sqrt{\text{sech}(2 \log (c x))}} \]

[Out]

((c^4 + x^(-4))*x^5)/(6*c^4*Sqrt[Sech[2*Log[c*x]]])

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Rubi [A]  time = 0.0423691, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5551, 5549, 264} \[ \frac{x^5 \left (c^4+\frac{1}{x^4}\right )}{6 c^4 \sqrt{\text{sech}(2 \log (c x))}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[Sech[2*Log[c*x]]],x]

[Out]

((c^4 + x^(-4))*x^5)/(6*c^4*Sqrt[Sech[2*Log[c*x]]])

Rule 5551

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

Rule 5549

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

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]

Rubi steps

\begin{align*} \int \frac{x^4}{\sqrt{\text{sech}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\sqrt{\text{sech}(2 \log (x))}} \, dx,x,c x\right )}{c^5}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1+\frac{1}{x^4}} x^5 \, dx,x,c x\right )}{c^6 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{\left (c^4+\frac{1}{x^4}\right ) x^5}{6 c^4 \sqrt{\text{sech}(2 \log (c x))}}\\ \end{align*}

Mathematica [A]  time = 0.0472009, size = 44, normalized size = 1.57 \[ \frac{\left (c^4 x^4+1\right )^2 \sqrt{\frac{c^2 x^2}{2 c^4 x^4+2}}}{6 c^6 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[Sech[2*Log[c*x]]],x]

[Out]

((1 + c^4*x^4)^2*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)])/(6*c^6*x)

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Maple [A]  time = 0.035, size = 39, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}x \left ({c}^{4}{x}^{4}+1 \right ) }{12\,{c}^{4}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/sech(2*ln(c*x))^(1/2),x)

[Out]

1/12*2^(1/2)*x/(c^2*x^2/(c^4*x^4+1))^(1/2)*(c^4*x^4+1)/c^4

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Maxima [A]  time = 1.81053, size = 41, normalized size = 1.46 \begin{align*} \frac{{\left (\sqrt{2} c^{4} x^{4} + \sqrt{2}\right )} \sqrt{c^{4} x^{4} + 1}}{12 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/sech(2*log(c*x))^(1/2),x, algorithm="maxima")

[Out]

1/12*(sqrt(2)*c^4*x^4 + sqrt(2))*sqrt(c^4*x^4 + 1)/c^5

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Fricas [B]  time = 3.02418, size = 103, normalized size = 3.68 \begin{align*} \frac{\sqrt{2}{\left (c^{8} x^{8} + 2 \, c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{12 \, c^{6} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/sech(2*log(c*x))^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(2)*(c^8*x^8 + 2*c^4*x^4 + 1)*sqrt(c^2*x^2/(c^4*x^4 + 1))/(c^6*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{\operatorname{sech}{\left (2 \log{\left (c x \right )} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/sech(2*ln(c*x))**(1/2),x)

[Out]

Integral(x**4/sqrt(sech(2*log(c*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/sech(2*log(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(sech(2*log(c*x))), x)