3.97 \(\int (\frac{x}{\text{sech}^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\text{sech}(x)}) \, dx\)

Optimal. Leaf size=47 \[ -\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}} \]

[Out]

-4/(49*Sech[x]^(7/2)) - 20/(63*Sech[x]^(3/2)) + (2*x*Sinh[x])/(7*Sech[x]^(5/2)) + (10*x*Sinh[x])/(21*Sqrt[Sech
[x]])

________________________________________________________________________________________

Rubi [A]  time = 0.104468, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ -\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sech[x]^(7/2) - (5*x*Sqrt[Sech[x]])/21,x]

[Out]

-4/(49*Sech[x]^(7/2)) - 20/(63*Sech[x]^(3/2)) + (2*x*Sinh[x])/(7*Sech[x]^(5/2)) + (10*x*Sinh[x])/(21*Sqrt[Sech
[x]])

Rule 4187

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(d*(b*Csc[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x], x] + Simp[((c + d*x)*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]

Rule 4189

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*} \int \left (\frac{x}{\text{sech}^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\text{sech}(x)}\right ) \, dx &=-\left (\frac{5}{21} \int x \sqrt{\text{sech}(x)} \, dx\right )+\int \frac{x}{\text{sech}^{\frac{7}{2}}(x)} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{5}{7} \int \frac{x}{\text{sech}^{\frac{3}{2}}(x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}-\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}}+\frac{5}{21} \int x \sqrt{\text{sech}(x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{49 \text{sech}^{\frac{7}{2}}(x)}-\frac{20}{63 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{7 \text{sech}^{\frac{5}{2}}(x)}+\frac{10 x \sinh (x)}{21 \sqrt{\text{sech}(x)}}\\ \end{align*}

Mathematica [A]  time = 0.103575, size = 45, normalized size = 0.96 \[ \sqrt{\text{sech}(x)} \left (\frac{13}{42} x \sinh (2 x)+\frac{1}{28} x \sinh (4 x)-\frac{88}{441} \cosh (2 x)-\frac{1}{98} \cosh (4 x)-\frac{167}{882}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sech[x]^(7/2) - (5*x*Sqrt[Sech[x]])/21,x]

[Out]

Sqrt[Sech[x]]*(-167/882 - (88*Cosh[2*x])/441 - Cosh[4*x]/98 + (13*x*Sinh[2*x])/42 + (x*Sinh[4*x])/28)

________________________________________________________________________________________

Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x \left ({\rm sech} \left (x\right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,x}{21}\sqrt{{\rm sech} \left (x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2),x)

[Out]

int(x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-5/21*x*sqrt(sech(x)) + x/sech(x)^(7/2), x)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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/sech(x)**(7/2)-5/21*x*sech(x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sech(x)^(7/2)-5/21*x*sech(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-5/21*x*sqrt(sech(x)) + x/sech(x)^(7/2), x)