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

Optimal. Leaf size=214 \[ -\frac{6 c \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )}{5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{6}{5 x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{12}{5 x^4 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 c \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

-12/(5*(c^4 + x^(-4))*(c^2 + x^(-2))*x^4*Sech[2*Log[c*x]]^(3/2)) + 6/(5*(c^4 + x^(-4))*x^2*Sech[2*Log[c*x]]^(3
/2)) + x^2/(5*Sech[2*Log[c*x]]^(3/2)) + (12*c*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^2]*(c^2 + x^(-2))*EllipticE[2
*ArcCot[c*x], 1/2])/(5*(c^4 + x^(-4))^2*x^3*Sech[2*Log[c*x]]^(3/2)) - (6*c*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^
2]*(c^2 + x^(-2))*EllipticF[2*ArcCot[c*x], 1/2])/(5*(c^4 + x^(-4))^2*x^3*Sech[2*Log[c*x]]^(3/2))

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Rubi [A]  time = 0.116287, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {5551, 5549, 335, 277, 305, 220, 1196} \[ \frac{6}{5 x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{12}{5 x^4 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{6 c \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 c \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-12/(5*(c^4 + x^(-4))*(c^2 + x^(-2))*x^4*Sech[2*Log[c*x]]^(3/2)) + 6/(5*(c^4 + x^(-4))*x^2*Sech[2*Log[c*x]]^(3
/2)) + x^2/(5*Sech[2*Log[c*x]]^(3/2)) + (12*c*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^2]*(c^2 + x^(-2))*EllipticE[2
*ArcCot[c*x], 1/2])/(5*(c^4 + x^(-4))^2*x^3*Sech[2*Log[c*x]]^(3/2)) - (6*c*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^
2]*(c^2 + x^(-2))*EllipticF[2*ArcCot[c*x], 1/2])/(5*(c^4 + x^(-4))^2*x^3*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 335

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 277

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 305

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

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{x}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^4 \, dx,x,c x\right )}{c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^4\right )^{3/2}}{x^6} \, dx,x,\frac{1}{c x}\right )}{c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{6 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^4}}{x^2} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{6}{5 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{6}{5 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 \operatorname{Subst}\left (\int \frac{1-x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{12}{5 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) x^4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{6}{5 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 c \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 ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 \left (c^4+\frac{1}{x^4}\right )^2 x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{6 c \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 )}{5 \left (c^4+\frac{1}{x^4}\right )^2 x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}

Mathematica [C]  time = 0.114773, size = 65, normalized size = 0.3 \[ -\frac{\, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-c^4 x^4\right )}{2 \sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \sqrt{c^4 x^4+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.039, size = 159, normalized size = 0.7 \begin{align*}{\frac{ \left ({c}^{8}{x}^{8}-4\,{c}^{4}{x}^{4}-5 \right ) \sqrt{2}}{ \left ( 20\,{c}^{4}{x}^{4}+20 \right ){c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}+{\frac{{\frac{3\,i}{5}}\sqrt{2}x}{{c}^{4}{x}^{4}+1}\sqrt{1-i{c}^{2}{x}^{2}}\sqrt{1+i{c}^{2}{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{i{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{i{c}^{2}},i \right ) \right ){\frac{1}{\sqrt{i{c}^{2}}}}{\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/sech(2*ln(c*x))^(3/2),x)

[Out]

1/20*(c^8*x^8-4*c^4*x^4-5)/(c^4*x^4+1)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4+1))^(1/2)+3/5*I/(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)-EllipticE(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., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/sech(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/sech(2*log(c*x))^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/sech(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

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