3.37 \(\int \frac {\sin ^2(a)}{x} \, dx\)

Optimal. Leaf size=7 \[ \sin ^2(a) \log (x) \]

[Out]

ln(x)*sin(a)^2

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Rubi [A]  time = 0.01, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 29} \[ \sin ^2(a) \log (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[a]^2/x,x]

[Out]

Log[x]*Sin[a]^2

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 ^2(a)}{x} \, dx &=\sin ^2(a) \int \frac {1}{x} \, dx\\ &=\log (x) \sin ^2(a)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \[ \sin ^2(a) \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a]^2/x,x]

[Out]

Log[x]*Sin[a]^2

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fricas [A]  time = 0.40, size = 10, normalized size = 1.43 \[ -{\left (\cos \relax (a)^{2} - 1\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^2/x,x, algorithm="fricas")

[Out]

-(cos(a)^2 - 1)*log(x)

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giac [A]  time = 0.16, size = 8, normalized size = 1.14 \[ \log \left ({\left | x \right |}\right ) \sin \relax (a)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^2/x,x, algorithm="giac")

[Out]

log(abs(x))*sin(a)^2

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maple [A]  time = 0.00, size = 8, normalized size = 1.14 \[ \ln \relax (x ) \left (\sin ^{2}\relax (a )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a)^2/x,x)

[Out]

ln(x)*sin(a)^2

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maxima [A]  time = 0.31, size = 7, normalized size = 1.00 \[ \log \relax (x) \sin \relax (a)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)^2/x,x, algorithm="maxima")

[Out]

log(x)*sin(a)^2

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mupad [B]  time = 0.02, size = 7, normalized size = 1.00 \[ {\sin \relax (a)}^2\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a)^2/x,x)

[Out]

sin(a)^2*log(x)

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sympy [A]  time = 0.05, size = 7, normalized size = 1.00 \[ \log {\relax (x )} \sin ^{2}{\relax (a )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a)**2/x,x)

[Out]

log(x)*sin(a)**2

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