3.1.13 \(\int (\frac {x}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x \sqrt {\text {sech}(x)}) \, dx\) [13]

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

[Out]

-4/9/sech(x)^(3/2)+2/3*x*sinh(x)/sech(x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \begin {gather*} \frac {2 x \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {4}{9 \text {sech}^{\frac {3}{2}}(x)} \end {gather*}

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 4272

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 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*} \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*}

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Mathematica [A]
time = 0.05, size = 17, normalized size = 0.71 \begin {gather*} \frac {2 (-2+3 x \tanh (x))}{9 \text {sech}^{\frac {3}{2}}(x)} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x}{\mathrm {sech}\left (x \right )^{\frac {3}{2}}}-\frac {x \sqrt {\mathrm {sech}\left (x \right )}}{3}\, dx\]

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 = 0.00, 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/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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \left (- \frac {3 x}{\operatorname {sech}^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x \sqrt {\operatorname {sech}{\left (x \right )}}\, dx}{3} \end {gather*}

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 = 0.00, 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/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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {x\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{3}-\frac {x}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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