\(\int (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}) \, dx\) [16]

Optimal result

Integrand size = 24, antiderivative size = 66 \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}-\frac {16}{27} i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {\text {sech}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}} \]

[Out]

-8/9*x/sech(x)^(3/2)+16/27*sinh(x)/sech(x)^(1/2)+2/3*x^2*sinh(x)/sech(x)^(1/2)-16/27*I*(cosh(1/2*x)^2)^(1/2)/c
osh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))*cosh(x)^(1/2)*sech(x)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274, 3854, 3856, 2720} \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}-\frac {16}{27} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \]

[In]

Int[x^2/Sech[x]^(3/2) - (x^2*Sqrt[Sech[x]])/3,x]

[Out]

(-8*x)/(9*Sech[x]^(3/2)) - ((16*I)/27)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x]] + (16*Sinh[x])/(27*Sq
rt[Sech[x]]) + (2*x^2*Sinh[x])/(3*Sqrt[Sech[x]])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4273

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Csc[e + f*x])^n/(f^2*n^2)), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x], x] + Simp[(c + d*x)^m*Cos[e + f*x
]*((b*Csc[e + f*x])^(n + 1)/(b*f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 1]

Rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \int x^2 \sqrt {\text {sech}(x)} \, dx\right )+\int \frac {x^2}{\text {sech}^{\frac {3}{2}}(x)} \, dx \\ & = -\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \int x^2 \sqrt {\text {sech}(x)} \, dx+\frac {8}{9} \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)} \, dx-\frac {1}{3} \left (\sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int \frac {x^2}{\sqrt {\cosh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {8}{27} \int \sqrt {\text {sech}(x)} \, dx \\ & = -\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{27} \left (8 \sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx \\ & = -\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}-\frac {16}{27} i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {\text {sech}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\frac {2}{27} \sqrt {\text {sech}(x)} \left (-8 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\cosh (x) \left (-12 x \cosh (x)+\left (8+9 x^2\right ) \sinh (x)\right )\right ) \]

[In]

Integrate[x^2/Sech[x]^(3/2) - (x^2*Sqrt[Sech[x]])/3,x]

[Out]

(2*Sqrt[Sech[x]]*((-8*I)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2] + Cosh[x]*(-12*x*Cosh[x] + (8 + 9*x^2)*Sinh[x])))
/27

Maple [F]

\[\int \left (\frac {x^{2}}{\operatorname {sech}\left (x \right )^{\frac {3}{2}}}-\frac {x^{2} \sqrt {\operatorname {sech}\left (x \right )}}{3}\right )d x\]

[In]

int(x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2),x)

[Out]

int(x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=- \frac {\int \left (- \frac {3 x^{2}}{\operatorname {sech}^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x^{2} \sqrt {\operatorname {sech}{\left (x \right )}}\, dx}{3} \]

[In]

integrate(x**2/sech(x)**(3/2)-1/3*x**2*sech(x)**(1/2),x)

[Out]

-(Integral(-3*x**2/sech(x)**(3/2), x) + Integral(x**2*sqrt(sech(x)), x))/3

Maxima [F]

\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\left (x\right )} + \frac {x^{2}}{\operatorname {sech}\left (x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-1/3*x^2*sqrt(sech(x)) + x^2/sech(x)^(3/2), x)

Giac [F]

\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\left (x\right )} + \frac {x^{2}}{\operatorname {sech}\left (x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^2/sech(x)^(3/2)-1/3*x^2*sech(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/3*x^2*sqrt(sech(x)) + x^2/sech(x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=-\int \frac {x^2\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{3}-\frac {x^2}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{3/2}} \,d x \]

[In]

int(x^2/(1/cosh(x))^(3/2) - (x^2*(1/cosh(x))^(1/2))/3,x)

[Out]

-int((x^2*(1/cosh(x))^(1/2))/3 - x^2/(1/cosh(x))^(3/2), x)