\(\int (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}) \, dx\) [72]

Optimal result

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

[Out]

2/3*x*sinh(x)/cosh(x)^(3/2)+4/3/cosh(x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \[ \int \left (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}\right ) \, dx=\frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)} \]

[In]

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

[Out]

4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))

Rule 3396

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)*Cos[e + f*x]*((b*Si
n[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[d*((b*Sin[e + f*x])^(n + 2)/(b^2*f^2*(n + 1)*(n + 2))), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \left (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}\right ) \, dx=\frac {2 (2+x \tanh (x))}{3 \sqrt {\cosh (x)}} \]

[In]

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

[Out]

(2*(2 + x*Tanh[x]))/(3*Sqrt[Cosh[x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (16) = 32\).

Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.54 \[ \int \left (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}\right ) \, dx=\frac {4 \, {\left ({\left (x + 2\right )} \cosh \left (x\right )^{3} + 3 \, {\left (x + 2\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (x + 2\right )} \sinh \left (x\right )^{3} - {\left (x - 2\right )} \cosh \left (x\right ) + {\left (3 \, {\left (x + 2\right )} \cosh \left (x\right )^{2} - x + 2\right )} \sinh \left (x\right )\right )} \sqrt {\cosh \left (x\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]

[In]

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

[Out]

4/3*((x + 2)*cosh(x)^3 + 3*(x + 2)*cosh(x)*sinh(x)^2 + (x + 2)*sinh(x)^3 - (x - 2)*cosh(x) + (3*(x + 2)*cosh(x
)^2 - x + 2)*sinh(x))*sqrt(cosh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)
^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(-1/3*x/sqrt(cosh(x)) + x/cosh(x)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(-1/3*x/sqrt(cosh(x)) + x/cosh(x)^(5/2), x)

Mupad [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \left (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}\right ) \, dx=\frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (2\,{\mathrm {e}}^{2\,x}-x+x\,{\mathrm {e}}^{2\,x}+2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \]

[In]

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

[Out]

(4*exp(x)*(exp(-x)/2 + exp(x)/2)^(1/2)*(2*exp(2*x) - x + x*exp(2*x) + 2))/(3*(exp(2*x) + 1)^2)