\(\int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx\) [591]

Optimal result

Integrand size = 20, antiderivative size = 35 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=-\frac {\cot (a x)}{a^3}+\frac {x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))} \]

[Out]

-cot(a*x)/a^3+x*csc(a*x)/a^2/(a*x*cos(a*x)-sin(a*x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4690, 3852, 8} \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {\cot (a x)}{a^3} \]

[In]

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

[Out]

-(Cot[a*x]/a^3) + (x*Csc[a*x])/(a^2*(a*x*Cos[a*x] - Sin[a*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4690

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {\int \csc ^2(a x) \, dx}{a^2} \\ & = \frac {x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (a x))}{a^3} \\ & = -\frac {\cot (a x)}{a^3}+\frac {x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {\cos (a x)+a x \sin (a x)}{a^3 (a x \cos (a x)-\sin (a x))} \]

[In]

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

[Out]

(Cos[a*x] + a*x*Sin[a*x])/(a^3*(a*x*Cos[a*x] - Sin[a*x]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {2 i \left (a x -i\right )}{a^{3} \left (a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right )}\)	\(39\)
parallelrisch	\(\frac {-1-2 \tan \left (\frac {a x}{2}\right ) a x +\tan \left (\frac {a x}{2}\right )^{2}}{a^{3} \left (a x \tan \left (\frac {a x}{2}\right )^{2}-a x +2 \tan \left (\frac {a x}{2}\right )\right )}\)	\(47\)
norman	\(\frac {\frac {\tan \left (\frac {a x}{2}\right )^{2}}{a^{3}}-\frac {1}{a^{3}}-\frac {2 x \tan \left (\frac {a x}{2}\right )}{a^{2}}}{a x \tan \left (\frac {a x}{2}\right )^{2}-a x +2 \tan \left (\frac {a x}{2}\right )}\)	\(54\)

[In]

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

[Out]

2*I*(a*x-I)/a^3/(a*x*exp(2*I*a*x)+I*exp(2*I*a*x)+a*x-I)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {a x \sin \left (a x\right ) + \cos \left (a x\right )}{a^{4} x \cos \left (a x\right ) - a^{3} \sin \left (a x\right )} \]

[In]

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

[Out]

(a*x*sin(a*x) + cos(a*x))/(a^4*x*cos(a*x) - a^3*sin(a*x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (31) = 62\).

Time = 2.92 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.20 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=- \frac {2 a x \tan {\left (\frac {a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} + \frac {\tan ^{2}{\left (\frac {a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} - \frac {1}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} \]

[In]

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

[Out]

-2*a*x*tan(a*x/2)/(a**4*x*tan(a*x/2)**2 - a**4*x + 2*a**3*tan(a*x/2)) + tan(a*x/2)**2/(a**4*x*tan(a*x/2)**2 -
a**4*x + 2*a**3*tan(a*x/2)) - 1/(a**4*x*tan(a*x/2)**2 - a**4*x + 2*a**3*tan(a*x/2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (35) = 70\).

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {2 \, {\left (2 \, a x \cos \left (2 \, a x\right ) + {\left (a^{2} x^{2} - 1\right )} \sin \left (2 \, a x\right )\right )}}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} - 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} + 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a^{3}} \]

[In]

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

[Out]

2*(2*a*x*cos(2*a*x) + (a^2*x^2 - 1)*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^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=-\frac {2 \, a x \tan \left (\frac {1}{2} \, a x\right ) - \tan \left (\frac {1}{2} \, a x\right )^{2} + 1}{a^{4} x \tan \left (\frac {1}{2} \, a x\right )^{2} - a^{4} x + 2 \, a^{3} \tan \left (\frac {1}{2} \, a x\right )} \]

[In]

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

[Out]

-(2*a*x*tan(1/2*a*x) - tan(1/2*a*x)^2 + 1)/(a^4*x*tan(1/2*a*x)^2 - a^4*x + 2*a^3*tan(1/2*a*x))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {x^2}{{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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