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

Optimal result

Integrand size = 8, antiderivative size = 7 \[ \int \frac {\sin ^2(a)}{x} \, dx=\log (x) \sin ^2(a) \]

[Out]

ln(x)*sin(a)^2

Rubi [A] (verified)

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(a)}{x} \, dx=\log (x) \sin ^2(a) \]

[In]

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

[Out]

Log[x]*Sin[a]^2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14

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

[In]

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

[Out]

ln(x)*sin(a)^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^2(a)}{x} \, dx=-{\left (\cos \left (a\right )^{2} - 1\right )} \log \left (x\right ) \]

[In]

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

[Out]

-(cos(a)^2 - 1)*log(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(a)}{x} \, dx=\log {\left (x \right )} \sin ^{2}{\left (a \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(a)}{x} \, dx=\log \left (x\right ) \sin \left (a\right )^{2} \]

[In]

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

[Out]

log(x)*sin(a)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^2(a)}{x} \, dx=\log \left ({\left | x \right |}\right ) \sin \left (a\right )^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(a)}{x} \, dx={\sin \left (a\right )}^2\,\ln \left (x\right ) \]

[In]

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

[Out]

sin(a)^2*log(x)