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]

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

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Rubi [A]  time = 0.0511449, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}

Mathematica [A]  time = 0.0750161, 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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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cosh \left ( x \right ) }}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\cosh \left (x\right )}} + \frac{x}{\cosh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.79833, size = 374, normalized size = 15.58 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x}{3 \, \sqrt{\cosh \left (x\right )}} + \frac{x}{\cosh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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)