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

Optimal. Leaf size=63 \[ -\frac {\cos ^{\frac {3}{2}}(x)}{2 x^2}+\frac {3 \sqrt {\cos (x)} \sin (x)}{4 x}+\frac {3}{8} \text {Int}\left (\frac {1}{x \sqrt {\cos (x)}},x\right )-\frac {9}{8} \text {Int}\left (\frac {\cos ^{\frac {3}{2}}(x)}{x},x\right ) \]

[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)

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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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*}

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Mathematica [A]
time = 3.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{\frac {3}{2}}(x)}{x^3} \, dx \end {gather*}

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]

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{\frac {3}{2}}\left (x \right )}{x^{3}}\, dx\]

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)

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Maxima [A]
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(cos(x)^(3/2)/x^3,x, algorithm="maxima")

[Out]

integrate(cos(x)^(3/2)/x^3, x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)**(3/2)/x**3, x)

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Giac [A]
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(cos(x)^(3/2)/x^3,x, algorithm="giac")

[Out]

integrate(cos(x)^(3/2)/x^3, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\cos \left (x\right )}^{3/2}}{x^3} \,d x \end {gather*}

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)

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