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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3315} \[ \frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)} \]

Antiderivative was successfully verified.

[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 3315

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

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Mathematica [A]  time = 0.08, size = 16, normalized size = 0.67 \[ \frac {2 (x \tanh (x)+2)}{3 \sqrt {\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2 + x*Tanh[x]))/(3*Sqrt[Cosh[x]])

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fricas [B]  time = 0.50, size = 109, normalized size = 4.54 \[ \frac {4 \, {\left ({\left (x + 2\right )} \cosh \relax (x)^{3} + 3 \, {\left (x + 2\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (x + 2\right )} \sinh \relax (x)^{3} - {\left (x - 2\right )} \cosh \relax (x) + {\left (3 \, {\left (x + 2\right )} \cosh \relax (x)^{2} - x + 2\right )} \sinh \relax (x)\right )} \sqrt {\cosh \relax (x)}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\cosh \relax (x)}} + \frac {x}{\cosh \relax (x)^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cosh \relax (x )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cosh \relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{3 \, \sqrt {\cosh \relax (x)}} + \frac {x}{\cosh \relax (x)^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.97, size = 42, normalized size = 1.75 \[ \frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (2\,{\mathrm {e}}^{2\,x}-x+x\,{\mathrm {e}}^{2\,x}+2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {3 x}{\cosh ^{\frac {5}{2}}{\relax (x )}}\right )\, dx + \int \frac {x}{\sqrt {\cosh {\relax (x )}}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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