3.2.58 \(\int \frac {x^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx\) [158]

Optimal. Leaf size=108 \[ \frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{21 c^5 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \]

[Out]

2/21*x^2/c^4/sech(2*ln(c*x))^(1/2)+1/7*x^6/sech(2*ln(c*x))^(1/2)+1/21*(c^2+1/x^2)*(cos(2*arccot(c*x))^2)^(1/2)
/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),1/2*2^(1/2))*((c^4+1/x^4)/(c^2+1/x^2)^2)^(1/2)/c^5/(c^4+1/x^4
)/x/sech(2*ln(c*x))^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5670, 5668, 342, 283, 331, 226} \begin {gather*} \frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{21 c^5 x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[Sech[2*Log[c*x]]],x]

[Out]

(2*x^2)/(21*c^4*Sqrt[Sech[2*Log[c*x]]]) + x^6/(7*Sqrt[Sech[2*Log[c*x]]]) + (Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))
^2]*(c^2 + x^(-2))*EllipticF[2*ArcCot[c*x], 1/2])/(21*c^5*(c^4 + x^(-4))*x*Sqrt[Sech[2*Log[c*x]]])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 283

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

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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^5}{\sqrt {\text {sech}(2 \log (c x))}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^5}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^6}\\ &=\frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^6 \, dx,x,c x\right )}{c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^8} \, dx,x,\frac {1}{c x}\right )}{c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{7 c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{21 c^7 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {2 x^2}{21 c^4 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^6}{7 \sqrt {\text {sech}(2 \log (c x))}}+\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{21 c^5 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.12, size = 77, normalized size = 0.71 \begin {gather*} \frac {\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}} \left (\left (1+c^4 x^4\right )^{3/2}-\, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-c^4 x^4\right )\right )}{7 c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[Sech[2*Log[c*x]]],x]

[Out]

(Sqrt[1 + c^4*x^4]*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)]*((1 + c^4*x^4)^(3/2) - Hypergeometric2F1[-1/2, 1/4, 5/4, -(
c^4*x^4)]))/(7*c^6)

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Maple [C] Result contains complex when optimal does not.
time = 1.43, size = 130, normalized size = 1.20

method	result	size
risch	\(\frac {x^{2} \left (3 c^{4} x^{4}+2\right ) \sqrt {2}}{42 c^{4} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}-\frac {\sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \EllipticF \left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, x}{21 c^{4} \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/sech(2*ln(c*x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/42*x^2*(3*c^4*x^4+2)/c^4*2^(1/2)/(c^2*x^2/(c^4*x^4+1))^(1/2)-1/21/c^4/(I*c^2)^(1/2)*(1-I*c^2*x^2)^(1/2)*(1+I
*c^2*x^2)^(1/2)/(c^4*x^4+1)*EllipticF(x*(I*c^2)^(1/2),I)*2^(1/2)*x/(c^2*x^2/(c^4*x^4+1))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/sech(2*log(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^5/sqrt(sech(2*log(c*x))), x)

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Fricas [A]
time = 0.11, size = 80, normalized size = 0.74 \begin {gather*} -\frac {2 \, \sqrt {2} \sqrt {c^{4}} c \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}, -1\right ) - \sqrt {2} {\left (3 \, c^{8} x^{8} + 5 \, c^{4} x^{4} + 2\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{42 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/sech(2*log(c*x))^(1/2),x, algorithm="fricas")

[Out]

-1/42*(2*sqrt(2)*sqrt(c^4)*c*(-1/c^4)^(3/4)*ellipticF((-1/c^4)^(1/4)/x, -1) - sqrt(2)*(3*c^8*x^8 + 5*c^4*x^4 +
2)*sqrt(c^2*x^2/(c^4*x^4 + 1)))/c^6

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/sech(2*ln(c*x))**(1/2),x)

[Out]

Integral(x**5/sqrt(sech(2*log(c*x))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/sech(2*log(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate(x^5/sqrt(sech(2*log(c*x))), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(1/cosh(2*log(c*x)))^(1/2),x)

[Out]

int(x^5/(1/cosh(2*log(c*x)))^(1/2), x)

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