3.1.91 \(\int (\frac {x}{\cos ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cos (x)}}) \, dx\) [91]

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

[Out]

2/3*x*sin(x)/cos(x)^(3/2)-4/3/cos(x)^(1/2)

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Rubi [A]
time = 0.03, 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 = {3396} \begin {gather*} \frac {2 x \sin (x)}{3 \cos ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\cos (x)}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.04, size = 17, normalized size = 0.71 \begin {gather*} -\frac {8-4 x \tan (x)}{6 \sqrt {\cos (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(8 - 4*x*Tan[x])/Sqrt[Cos[x]]

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (x \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cos \left (x \right )}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.35, size = 15, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (x \sin \left (x\right ) - 2 \, \cos \left (x\right )\right )}}{3 \, \cos \left (x\right )^{\frac {3}{2}}} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \left (- \frac {3 x}{\cos ^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\cos {\left (x \right )}}}\, dx}{3} \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.14, size = 16, normalized size = 0.67 \begin {gather*} -\frac {4\,\cos \left (x\right )-2\,x\,\sin \left (x\right )}{3\,{\cos \left (x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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