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

Optimal. Leaf size=24 \[ \frac{4}{25 \sec ^{\frac{5}{2}}(x)}+\frac{2 x \sin (x)}{5 \sec ^{\frac{3}{2}}(x)} \]

[Out]

4/(25*Sec[x]^(5/2)) + (2*x*Sin[x])/(5*Sec[x]^(3/2))

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Rubi [A]  time = 0.0791018, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ \frac{4}{25 \sec ^{\frac{5}{2}}(x)}+\frac{2 x \sin (x)}{5 \sec ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(25*Sec[x]^(5/2)) + (2*x*Sin[x])/(5*Sec[x]^(3/2))

Rule 4187

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

Rule 4189

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \left (\frac{x}{\sec ^{\frac{5}{2}}(x)}-\frac{3 x}{5 \sqrt{\sec (x)}}\right ) \, dx &=-\left (\frac{3}{5} \int \frac{x}{\sqrt{\sec (x)}} \, dx\right )+\int \frac{x}{\sec ^{\frac{5}{2}}(x)} \, dx\\ &=\frac{4}{25 \sec ^{\frac{5}{2}}(x)}+\frac{2 x \sin (x)}{5 \sec ^{\frac{3}{2}}(x)}+\frac{3}{5} \int \frac{x}{\sqrt{\sec (x)}} \, dx-\frac{1}{5} \left (3 \sqrt{\cos (x)} \sqrt{\sec (x)}\right ) \int x \sqrt{\cos (x)} \, dx\\ &=\frac{4}{25 \sec ^{\frac{5}{2}}(x)}+\frac{2 x \sin (x)}{5 \sec ^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.139029, size = 17, normalized size = 0.71 \[ \frac{2 (5 x \tan (x)+2)}{25 \sec ^{\frac{5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2 + 5*x*Tan[x]))/(25*Sec[x]^(5/2))

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Maple [F]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sec \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{3\,x}{5}{\frac{1}{\sqrt{\sec \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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