3.589 \(\int \frac {\sin ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{a^2 x}+\frac {\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))} \]

[Out]

1/a^2/x+sin(a*x)/a^2/x/(a*x*cos(a*x)-sin(a*x))

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {4596} \[ \frac {1}{a^2 x}+\frac {\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a*x]^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

1/(a^2*x) + Sin[a*x]/(a^2*x*(a*x*Cos[a*x] - Sin[a*x]))

Rule 4596

Int[Sin[(a_.)*(x_)]^2/(Cos[(a_.)*(x_)]*(d_.)*(x_) + (c_.)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[1/(d^2*x), x]
+ Simp[Sin[a*x]/(a*d*x*(d*x*Cos[a*x] + c*Sin[a*x])), x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 24, normalized size = 0.69 \[ \frac {\cos (a x)}{a^2 x \cos (a x)-a \sin (a x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a*x]^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]

[Out]

Cos[a*x]/(a^2*x*Cos[a*x] - a*Sin[a*x])

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fricas [A]  time = 1.95, size = 24, normalized size = 0.69 \[ \frac {\cos \left (a x\right )}{a^{2} x \cos \left (a x\right ) - a \sin \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(a*x)/(a^2*x*cos(a*x) - a*sin(a*x))

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giac [A]  time = 0.19, size = 39, normalized size = 1.11 \[ \frac {\tan \left (\frac {1}{2} \, a x\right )^{2} - 1}{a^{2} x \tan \left (\frac {1}{2} \, a x\right )^{2} - a^{2} x + 2 \, a \tan \left (\frac {1}{2} \, a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(1/2*a*x)^2 - 1)/(a^2*x*tan(1/2*a*x)^2 - a^2*x + 2*a*tan(1/2*a*x))

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maple [B]  time = 1.47, size = 77, normalized size = 2.20 \[ \frac {\frac {\tan ^{4}\left (\frac {a x}{2}\right )}{a}+\frac {\tan ^{6}\left (\frac {a x}{2}\right )}{a}-\frac {1}{a}-\frac {\tan ^{2}\left (\frac {a x}{2}\right )}{a}}{\left (1+\tan ^{2}\left (\frac {a x}{2}\right )\right )^{2} \left (a x \left (\tan ^{2}\left (\frac {a x}{2}\right )\right )-a x +2 \tan \left (\frac {a x}{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a*x)^2/(a*x*cos(a*x)-sin(a*x))^2,x)

[Out]

(1/a*tan(1/2*a*x)^4+1/a*tan(1/2*a*x)^6-1/a-1/a*tan(1/2*a*x)^2)/(1+tan(1/2*a*x)^2)^2/(a*x*tan(1/2*a*x)^2-a*x+2*
tan(1/2*a*x))

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maxima [B]  time = 0.32, size = 114, normalized size = 3.26 \[ \frac {a x \cos \left (2 \, a x\right )^{2} + a x \sin \left (2 \, a x\right )^{2} + 2 \, a x \cos \left (2 \, a x\right ) + a x - 2 \, \sin \left (2 \, a x\right )}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} - 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} + 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*cos(2*a*x)^2 + a*x*sin(2*a*x)^2 + 2*a*x*cos(2*a*x) + a*x - 2*sin(2*a*x))/((a^2*x^2 + (a^2*x^2 + 1)*cos(2*
a*x)^2 - 4*a*x*sin(2*a*x) + (a^2*x^2 + 1)*sin(2*a*x)^2 + 2*(a^2*x^2 - 1)*cos(2*a*x) + 1)*a)

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mupad [B]  time = 3.03, size = 24, normalized size = 0.69 \[ -\frac {\cos \left (a\,x\right )}{a\,\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a*x)^2/(sin(a*x) - a*x*cos(a*x))^2,x)

[Out]

-cos(a*x)/(a*(sin(a*x) - a*x*cos(a*x)))

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sympy [A]  time = 3.42, size = 20, normalized size = 0.57 \[ \frac {\cos {\left (a x \right )}}{a^{2} x \cos {\left (a x \right )} - a \sin {\left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(a*x)/(a**2*x*cos(a*x) - a*sin(a*x))

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