\(\int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx\) [600]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^3} \]

[Out]

-x*sec(a*x)/a^2/(cos(a*x)+a*x*sin(a*x))+tan(a*x)/a^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4691, 3852, 8} \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\tan (a x)}{a^3}-\frac {x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]

[In]

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

[Out]

-((x*Sec[a*x])/(a^2*(Cos[a*x] + a*x*Sin[a*x]))) + Tan[a*x]/a^3

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 4691

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

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

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-a x \cos (a x)+\sin (a x)}{a^3 (\cos (a x)+a x \sin (a x))} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.89 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21

method	result	size
risch	\(-\frac {2 i \left (a x -i\right )}{a^{3} \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right )}\)	\(40\)
parallelrisch	\(\frac {-a x \tan \left (\frac {a x}{2}\right )^{2}+a x -2 \tan \left (\frac {a x}{2}\right )}{a^{3} \left (-1-2 \tan \left (\frac {a x}{2}\right ) a x +\tan \left (\frac {a x}{2}\right )^{2}\right )}\)	\(47\)
norman	\(\frac {\frac {x \tan \left (\frac {a x}{2}\right )^{2}}{a^{2}}+\frac {2 \tan \left (\frac {a x}{2}\right )}{a^{3}}-\frac {x}{a^{2}}}{1+2 \tan \left (\frac {a x}{2}\right ) a x -\tan \left (\frac {a x}{2}\right )^{2}}\)	\(53\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {a x \cos \left (a x\right ) - \sin \left (a x\right )}{a^{4} x \sin \left (a x\right ) + a^{3} \cos \left (a x\right )} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (31) = 62\).

Time = 2.41 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.30 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \tan ^{2}{\left (\frac {a x}{2} \right )}}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} - \frac {a x}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} + \frac {2 \tan {\left (\frac {a x}{2} \right )}}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (33) = 66\).

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.03 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, {\left (2 \, a x \cos \left (2 \, a x\right ) + {\left (a^{2} x^{2} - 1\right )} \sin \left (2 \, a x\right )\right )}}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} + 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a^{3}} \]

[In]

integrate(x^2/(cos(a*x)+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 = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \tan \left (\frac {1}{2} \, a x\right )^{2} - a x + 2 \, \tan \left (\frac {1}{2} \, a x\right )}{2 \, a^{4} x \tan \left (\frac {1}{2} \, a x\right ) - a^{3} \tan \left (\frac {1}{2} \, a x\right )^{2} + a^{3}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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