\(\int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx\) [82]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=-\frac {\cos ^{\frac {3}{2}}(x)}{2 x^2}+\frac {3 \sqrt {\cos (x)} \sin (x)}{4 x}+\frac {3}{8} \text {Int}\left (\frac {1}{x \sqrt {\cos (x)}},x\right )-\frac {9}{8} \text {Int}\left (\frac {\cos ^{\frac {3}{2}}(x)}{x},x\right ) \]

[Out]

-1/2*cos(x)^(3/2)/x^2+3/4*sin(x)*cos(x)^(1/2)/x-9/8*Unintegrable(cos(x)^(3/2)/x,x)+3/8*Unintegrable(1/x/cos(x)
^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx \]

[In]

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

[Out]

-1/2*Cos[x]^(3/2)/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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 6.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx \]

[In]

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

[Out]

Integrate[Cos[x]^(3/2)/x^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80

\[\int \frac {\cos ^{\frac {3}{2}}\left (x \right )}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [N/A]

Not integrable

Time = 92.97 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int \frac {\cos ^{\frac {3}{2}}{\left (x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(cos(x)**(3/2)/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int { \frac {\cos \left (x\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(cos(x)^(3/2)/x^3, x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int { \frac {\cos \left (x\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(cos(x)^(3/2)/x^3, x)

Mupad [N/A]

Not integrable

Time = 13.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx=\int \frac {{\cos \left (x\right )}^{3/2}}{x^3} \,d x \]

[In]

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

[Out]

int(cos(x)^(3/2)/x^3, x)