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

Optimal. Leaf size=24 \[ \frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)} \]

[Out]

-4/(9*Sech[x]^(3/2)) + (2*x*Sinh[x])/(3*Sqrt[Sech[x]])

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Rubi [A]  time = 0.0868486, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ \frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(9*Sech[x]^(3/2)) + (2*x*Sinh[x])/(3*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{3}{2}}(x)}-\frac{1}{3} x \sqrt{\text{sech}(x)}\right ) \, dx &=-\left (\frac{1}{3} \int x \sqrt{\text{sech}(x)} \, dx\right )+\int \frac{x}{\text{sech}^{\frac{3}{2}}(x)} \, dx\\ &=-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}+\frac{1}{3} \int x \sqrt{\text{sech}(x)} \, dx-\frac{1}{3} \left (\sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0880857, size = 17, normalized size = 0.71 \[ \frac{2 (3 x \tanh (x)-2)}{9 \text{sech}^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 + 3*x*Tanh[x]))/(9*Sech[x]^(3/2))

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Maple [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{x \left ({\rm sech} \left (x\right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{3}\sqrt{{\rm sech} \left (x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{3} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{3} \, x \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\operatorname{sech}\left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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