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

Optimal. Leaf size=24 \[ \frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\cos (x)}} \]

[Out]

-4/(3*Sqrt[Cos[x]]) + (2*x*Sin[x])/(3*Cos[x]^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0459401, 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} \[ \frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(3*Sqrt[Cos[x]]) + (2*x*Sin[x])/(3*Cos[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}{\cos ^{\frac{5}{2}}(x)}-\frac{x}{3 \sqrt{\cos (x)}}\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\cos (x)}} \, dx\right )+\int \frac{x}{\cos ^{\frac{5}{2}}(x)} \, dx\\ &=-\frac{4}{3 \sqrt{\cos (x)}}+\frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0751972, size = 17, normalized size = 0.71 \[ -\frac{8-4 x \tan (x)}{6 \sqrt{\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(8 - 4*x*Tan[x])/(6*Sqrt[Cos[x]])

________________________________________________________________________________________

Maple [F]  time = 0.213, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cos \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.59847, size = 54, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (x \sin \left (x\right ) - 2 \, \cos \left (x\right )\right )}}{3 \, \cos \left (x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x*sin(x) - 2*cos(x))/cos(x)^(3/2)

________________________________________________________________________________________

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/cos(x)**(5/2)-1/3*x/cos(x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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