\(\int (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}) \, dx\) [96]

Optimal result

Integrand size = 20, antiderivative size = 24 \[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=-\frac {4}{25 \text {sech}^{\frac {5}{2}}(x)}+\frac {2 x \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)} \]

[Out]

-4/25/sech(x)^(5/2)+2/5*x*sinh(x)/sech(x)^(3/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=\frac {2 x \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}-\frac {4}{25 \text {sech}^{\frac {5}{2}}(x)} \]

[In]

Int[x/Sech[x]^(5/2) - (3*x)/(5*Sqrt[Sech[x]]),x]

[Out]

-4/(25*Sech[x]^(5/2)) + (2*x*Sinh[x])/(5*Sech[x]^(3/2))

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*} \text {integral}& = -\left (\frac {3}{5} \int \frac {x}{\sqrt {\text {sech}(x)}} \, dx\right )+\int \frac {x}{\text {sech}^{\frac {5}{2}}(x)} \, dx \\ & = -\frac {4}{25 \text {sech}^{\frac {5}{2}}(x)}+\frac {2 x \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}+\frac {3}{5} \int \frac {x}{\sqrt {\text {sech}(x)}} \, dx-\frac {1}{5} \left (3 \sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int x \sqrt {\cosh (x)} \, dx \\ & = -\frac {4}{25 \text {sech}^{\frac {5}{2}}(x)}+\frac {2 x \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=\frac {2 (-2+5 x \tanh (x))}{25 \text {sech}^{\frac {5}{2}}(x)} \]

[In]

Integrate[x/Sech[x]^(5/2) - (3*x)/(5*Sqrt[Sech[x]]),x]

[Out]

(2*(-2 + 5*x*Tanh[x]))/(25*Sech[x]^(5/2))

Maple [F]

\[\int \left (\frac {x}{\operatorname {sech}\left (x \right )^{\frac {5}{2}}}-\frac {3 x}{5 \sqrt {\operatorname {sech}\left (x \right )}}\right )d x\]

[In]

int(x/sech(x)^(5/2)-3/5*x/sech(x)^(1/2),x)

[Out]

int(x/sech(x)^(5/2)-3/5*x/sech(x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/sech(x)^(5/2)-3/5*x/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]

\[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=- \frac {\int \left (- \frac {5 x}{\operatorname {sech}^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {3 x}{\sqrt {\operatorname {sech}{\left (x \right )}}}\, dx}{5} \]

[In]

integrate(x/sech(x)**(5/2)-3/5*x/sech(x)**(1/2),x)

[Out]

-(Integral(-5*x/sech(x)**(5/2), x) + Integral(3*x/sqrt(sech(x)), x))/5

Maxima [F]

\[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\operatorname {sech}\left (x\right )}} + \frac {x}{\operatorname {sech}\left (x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/sech(x)^(5/2)-3/5*x/sech(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-3/5*x/sqrt(sech(x)) + x/sech(x)^(5/2), x)

Giac [F]

\[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\operatorname {sech}\left (x\right )}} + \frac {x}{\operatorname {sech}\left (x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/sech(x)^(5/2)-3/5*x/sech(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-3/5*x/sqrt(sech(x)) + x/sech(x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x}{\text {sech}^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\text {sech}(x)}}\right ) \, dx=-\int \frac {3\,x}{5\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}-\frac {x}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{5/2}} \,d x \]

[In]

int(x/(1/cosh(x))^(5/2) - (3*x)/(5*(1/cosh(x))^(1/2)),x)

[Out]

-int((3*x)/(5*(1/cosh(x))^(1/2)) - x/(1/cosh(x))^(5/2), x)