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 = 0.0241799, antiderivative size = 35, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.297996, 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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Maple [B]  time = 0.454, size = 77, normalized size = 2.2 \begin{align*}{ \left ({\frac{1}{a} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{4}}+{\frac{1}{a} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{6}}-{a}^{-1}-{\frac{1}{a} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2} \right ) ^{-2} \left ( ax \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}-ax+2\,\tan \left ( 1/2\,ax \right ) \right ) ^{-1}} \end{align*}

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 = 1.06221, size = 154, normalized size = 4.4 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 2.08256, size = 54, normalized size = 1.54 \begin{align*} \frac{\cos \left (a x\right )}{a^{2} x \cos \left (a x\right ) - a \sin \left (a x\right )} \end{align*}

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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Sympy [A]  time = 3.59789, size = 20, normalized size = 0.57 \begin{align*} \frac{\cos{\left (a x \right )}}{a^{2} x \cos{\left (a x \right )} - a \sin{\left (a x \right )}} \end{align*}

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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Giac [A]  time = 1.14914, size = 53, normalized size = 1.51 \begin{align*} \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 )} \end{align*}

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))