3.44 \(\int \frac{\sin ^3(a)}{x} \, dx\)

Optimal. Leaf size=7 \[ \sin ^3(a) \log (x) \]

[Out]

Log[x]*Sin[a]^3

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Rubi [A]  time = 0.0047063, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {12, 29} \[ \sin ^3(a) \log (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[a]^3/x,x]

[Out]

Log[x]*Sin[a]^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\sin ^3(a)}{x} \, dx &=\sin ^3(a) \int \frac{1}{x} \, dx\\ &=\log (x) \sin ^3(a)\\ \end{align*}

Mathematica [A]  time = 0.0007848, size = 7, normalized size = 1. \[ \sin ^3(a) \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a]^3/x,x]

[Out]

Log[x]*Sin[a]^3

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Maple [A]  time = 0.013, size = 8, normalized size = 1.1 \begin{align*} \ln \left ( x \right ) \left ( \sin \left ( a \right ) \right ) ^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a)^3/x,x)

[Out]

ln(x)*sin(a)^3

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Maxima [A]  time = 0.994203, size = 9, normalized size = 1.29 \begin{align*} \log \left (x\right ) \sin \left (a\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^3/x,x, algorithm="maxima")

[Out]

log(x)*sin(a)^3

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Fricas [A]  time = 0.432228, size = 42, normalized size = 6. \begin{align*} -{\left (\cos \left (a\right )^{2} - 1\right )} \log \left (x\right ) \sin \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^3/x,x, algorithm="fricas")

[Out]

-(cos(a)^2 - 1)*log(x)*sin(a)

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Sympy [A]  time = 0.389422, size = 7, normalized size = 1. \begin{align*} \log{\left (x \right )} \sin ^{3}{\left (a \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)**3/x,x)

[Out]

log(x)*sin(a)**3

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Giac [A]  time = 1.1071, size = 11, normalized size = 1.57 \begin{align*} \log \left ({\left | x \right |}\right ) \sin \left (a\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^3/x,x, algorithm="giac")

[Out]

log(abs(x))*sin(a)^3