3.82 \(\int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx\)

Optimal. Leaf size=61 \[ -\frac{9}{8} \text{Unintegrable}\left (\frac{\cos ^{\frac{3}{2}}(x)}{x},x\right )+\frac{3}{8} \text{Unintegrable}\left (\frac{1}{x \sqrt{\cos (x)}},x\right )-\frac{\cos ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3 \sin (x) \sqrt{\cos (x)}}{4 x} \]

[Out]

-Cos[x]^(3/2)/(2*x^2) + (3*Sqrt[Cos[x]]*Sin[x])/(4*x) + (3*Unintegrable[1/(x*Sqrt[Cos[x]]), x])/8 - (9*Uninteg
rable[Cos[x]^(3/2)/x, x])/8

________________________________________________________________________________________

Rubi [A]  time = 0.0768357, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cos[x]^(3/2)/x^3,x]

[Out]

-Cos[x]^(3/2)/(2*x^2) + (3*Sqrt[Cos[x]]*Sin[x])/(4*x) + (3*Defer[Int][1/(x*Sqrt[Cos[x]]), x])/8 - (9*Defer[Int
][Cos[x]^(3/2)/x, x])/8

Rubi steps

\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx &=-\frac{\cos ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3 \sqrt{\cos (x)} \sin (x)}{4 x}+\frac{3}{8} \int \frac{1}{x \sqrt{\cos (x)}} \, dx-\frac{9}{8} \int \frac{\cos ^{\frac{3}{2}}(x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 5.12425, size = 0, normalized size = 0. \[ \int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[x]^(3/2)/x^3,x]

[Out]

Integrate[Cos[x]^(3/2)/x^3, x]

________________________________________________________________________________________

Maple [A]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \cos \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^(3/2)/x^3,x)

[Out]

int(cos(x)^(3/2)/x^3,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^(3/2)/x^3,x, algorithm="maxima")

[Out]

integrate(cos(x)^(3/2)/x^3, x)

________________________________________________________________________________________

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(cos(x)^(3/2)/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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(cos(x)**(3/2)/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^(3/2)/x^3,x, algorithm="giac")

[Out]

integrate(cos(x)^(3/2)/x^3, x)