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

Optimal. Leaf size=33 \[ \frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]

[Out]

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

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Rubi [A]  time = 0.0379801, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4595, 3767, 8} \[ \frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[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 4595

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]

Rule 3767

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 8

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

Rubi steps

\begin{align*} \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{\int \sec ^2(a x) \, dx}{a^2}\\ &=-\frac{x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}-\frac{\operatorname{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]  time = 0.417018, size = 31, normalized size = 0.94 \[ \frac{\sin (a x)-a x \cos (a x)}{a^3 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[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]))

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Maple [A]  time = 0.372, size = 53, normalized size = 1.6 \begin{align*}{ \left ({\frac{x}{{a}^{2}} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}}-{\frac{x}{{a}^{2}}}+2\,{\frac{\tan \left ( 1/2\,ax \right ) }{{a}^{3}}} \right ) \left ( 2\,\tan \left ( 1/2\,ax \right ) xa- \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02277, size = 135, normalized size = 4.09 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.97615, size = 81, normalized size = 2.45 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [B]  time = 5.1095, size = 109, normalized size = 3.3 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.15183, size = 70, normalized size = 2.12 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)