3.171 \(\int \frac{x^6}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^7 \left (c^4+\frac{1}{x^4}\right )}{10 c^4 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

((c^4 + x^(-4))*x^7)/(10*c^4*Sech[2*Log[c*x]]^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0434737, 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} \[ \frac{x^7 \left (c^4+\frac{1}{x^4}\right )}{10 c^4 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[x^6/Sech[2*Log[c*x]]^(3/2),x]

[Out]

((c^4 + x^(-4))*x^7)/(10*c^4*Sech[2*Log[c*x]]^(3/2))

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^6}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^9 \, dx,x,c x\right )}{c^{10} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{\left (c^4+\frac{1}{x^4}\right ) x^7}{10 c^4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.0536219, size = 44, normalized size = 1.57 \[ \frac{\left (c^4 x^4+1\right )^3 \sqrt{\frac{c^2 x^2}{2 c^4 x^4+2}}}{20 c^8 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sech[2*Log[c*x]]^(3/2),x]

[Out]

((1 + c^4*x^4)^3*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)])/(20*c^8*x)

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 47, normalized size = 1.7 \begin{align*}{\frac{\sqrt{2}x \left ({c}^{8}{x}^{8}+2\,{c}^{4}{x}^{4}+1 \right ) }{40\,{c}^{6}}{\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^6/sech(2*ln(c*x))^(3/2),x)

[Out]

1/40*2^(1/2)/c^6*x/(c^2*x^2/(c^4*x^4+1))^(1/2)*(c^8*x^8+2*c^4*x^4+1)

________________________________________________________________________________________

Maxima [A]  time = 1.90706, size = 41, normalized size = 1.46 \begin{align*} \frac{{\left (\sqrt{2} c^{4} x^{4} + \sqrt{2}\right )}{\left (c^{4} x^{4} + 1\right )}^{\frac{3}{2}}}{40 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/sech(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

1/40*(sqrt(2)*c^4*x^4 + sqrt(2))*(c^4*x^4 + 1)^(3/2)/c^7

________________________________________________________________________________________

Fricas [B]  time = 2.92552, size = 122, normalized size = 4.36 \begin{align*} \frac{\sqrt{2}{\left (c^{12} x^{12} + 3 \, c^{8} x^{8} + 3 \, c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{40 \, c^{8} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/sech(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

1/40*sqrt(2)*(c^12*x^12 + 3*c^8*x^8 + 3*c^4*x^4 + 1)*sqrt(c^2*x^2/(c^4*x^4 + 1))/(c^8*x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/sech(2*ln(c*x))**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/sech(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(x^6/sech(2*log(c*x))^(3/2), x)