\(\int (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}) \, dx\) [97]

Optimal result

Integrand size = 20, antiderivative size = 47 \[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=-\frac {4}{49 \text {sech}^{\frac {7}{2}}(x)}-\frac {20}{63 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {10 x \sinh (x)}{21 \sqrt {\text {sech}(x)}} \]

[Out]

-4/49/sech(x)^(7/2)-20/63/sech(x)^(3/2)+2/7*x*sinh(x)/sech(x)^(5/2)+10/21*x*sinh(x)/sech(x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=-\frac {20}{63 \text {sech}^{\frac {3}{2}}(x)}-\frac {4}{49 \text {sech}^{\frac {7}{2}}(x)}+\frac {2 x \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {10 x \sinh (x)}{21 \sqrt {\text {sech}(x)}} \]

[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 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 {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] (verified)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=\sqrt {\text {sech}(x)} \left (-\frac {167}{882}-\frac {88}{441} \cosh (2 x)-\frac {1}{98} \cosh (4 x)+\frac {13}{42} x \sinh (2 x)+\frac {1}{28} x \sinh (4 x)\right ) \]

[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]

\[\int \left (\frac {x}{\operatorname {sech}\left (x \right )^{\frac {7}{2}}}-\frac {5 x \sqrt {\operatorname {sech}\left (x \right )}}{21}\right )d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/sech(x)^(7/2)-5/21*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]

\[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=- \frac {\int \left (- \frac {21 x}{\operatorname {sech}^{\frac {7}{2}}{\left (x \right )}}\right )\, dx + \int 5 x \sqrt {\operatorname {sech}{\left (x \right )}}\, dx}{21} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {5}{21} \, x \sqrt {\operatorname {sech}\left (x\right )} + \frac {x}{\operatorname {sech}\left (x\right )^{\frac {7}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {5}{21} \, x \sqrt {\operatorname {sech}\left (x\right )} + \frac {x}{\operatorname {sech}\left (x\right )^{\frac {7}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x}{\text {sech}^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\text {sech}(x)}\right ) \, dx=-\int \frac {5\,x\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{21}-\frac {x}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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