3.1.37 \(\int \frac {\sin ^2(a)}{x} \, dx\) [37]

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

[Out]

ln(x)*sin(a)^2

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Rubi [A]
time = 0.00, 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} \begin {gather*} \sin ^2(a) \log (x) \end {gather*}

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 \begin {gather*} \log (x) \sin ^2(a) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]*Sin[a]^2

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Maple [A]
time = 0.02, size = 8, normalized size = 1.14

method	result	size
default	\(\ln \left (x \right ) \left (\sin ^{2}\left (a \right )\right )\)	\(8\)
norman	\(\ln \left (x \right ) \left (\sin ^{2}\left (a \right )\right )\)	\(8\)
risch	\(\ln \left (x \right ) \left (\sin ^{2}\left (a \right )\right )\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a)^2/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)*sin(a)^2

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Maxima [A]
time = 0.27, size = 7, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \sin \left (a\right )^{2} \end {gather*}

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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Fricas [A]
time = 1.47, size = 10, normalized size = 1.43 \begin {gather*} -{\left (\cos \left (a\right )^{2} - 1\right )} \log \left (x\right ) \end {gather*}

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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Sympy [A]
time = 0.01, size = 7, normalized size = 1.00 \begin {gather*} \log {\left (x \right )} \sin ^{2}{\left (a \right )} \end {gather*}

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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Giac [A]
time = 0.42, size = 8, normalized size = 1.14 \begin {gather*} \log \left ({\left | x \right |}\right ) \sin \left (a\right )^{2} \end {gather*}

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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Mupad [B]
time = 0.02, size = 7, normalized size = 1.00 \begin {gather*} {\sin \left (a\right )}^2\,\ln \left (x\right ) \end {gather*}

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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