3.2.71 \(\int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx\) [171]

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

[Out]

1/10*(c^4+1/x^4)*x^7/c^4/sech(2*ln(c*x))^(3/2)

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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5670, 5668, 270} \begin {gather*} \frac {x^7 \left (c^4+\frac {1}{x^4}\right )}{10 c^4 \text {sech}^{\frac {3}{2}}(2 \log (c x))} \end {gather*}

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 270

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 5668

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*(1/(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 5670

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])

Rubi steps

\begin {align*} \int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\text {sech}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7}\\ &=\frac {\text {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*}

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Mathematica [A]
time = 0.04, size = 44, normalized size = 1.57 \begin {gather*} \frac {\left (1+c^4 x^4\right )^3 \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}}{20 c^8 x} \end {gather*}

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)

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Maple [A]
time = 1.18, size = 47, normalized size = 1.68

method	result	size
risch	\(\frac {\sqrt {2}\, x \left (c^{8} x^{8}+2 c^{4} x^{4}+1\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/sech(2*ln(c*x))^(3/2),x,method=_RETURNVERBOSE)

[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)

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Maxima [A]
time = 0.51, size = 30, normalized size = 1.07 \begin {gather*} \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 {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
time = 0.38, size = 56, normalized size = 2.00 \begin {gather*} \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 {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [B]
time = 1.45, size = 42, normalized size = 1.50 \begin {gather*} \frac {{\left (c^4\,x^4+1\right )}^3\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{40\,c^8\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(1/cosh(2*log(c*x)))^(3/2),x)

[Out]

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

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