\(\int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx\) [602]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 i x^2}{a^3}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right )}{a^5}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3} \]

[Out]

-2*I*x^2/a^3+4*x*ln(1+exp(2*I*a*x))/a^4-2*I*polylog(2,-exp(2*I*a*x))/a^5-x*sec(a*x)^2/a^4-x^3*sec(a*x)^3/a^2/(
cos(a*x)+a*x*sin(a*x))+tan(a*x)/a^5+2*x^2*tan(a*x)/a^3+x^2*sec(a*x)^2*tan(a*x)/a^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4697, 4271, 3852, 8, 4269, 3800, 2221, 2317, 2438} \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right )}{a^5}+\frac {\tan (a x)}{a^5}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {x \sec ^2(a x)}{a^4}-\frac {2 i x^2}{a^3}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \tan (a x) \sec ^2(a x)}{a^3}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]

[In]

Int[(x^4*Sec[a*x]^2)/(Cos[a*x] + a*x*Sin[a*x])^2,x]

[Out]

((-2*I)*x^2)/a^3 + (4*x*Log[1 + E^((2*I)*a*x)])/a^4 - ((2*I)*PolyLog[2, -E^((2*I)*a*x)])/a^5 - (x*Sec[a*x]^2)/
a^4 - (x^3*Sec[a*x]^3)/(a^2*(Cos[a*x] + a*x*Sin[a*x])) + Tan[a*x]/a^5 + (2*x^2*Tan[a*x])/a^3 + (x^2*Sec[a*x]^2
*Tan[a*x])/a^3

Rule 8

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

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 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4697

Int[(((b_.)*(x_))^(m_.)*Sec[(a_.)*(x_)]^(n_.))/(Cos[(a_.)*(x_)]*(c_.) + (d_.)*(x_)*Sin[(a_.)*(x_)])^2, x_Symbo
l] :> Simp[(-b)*(b*x)^(m - 1)*(Sec[a*x]^(n + 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x]))), x] + Dist[b^2*((n + 1)/d^2
), Int[(b*x)^(m - 2)*Sec[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, n +
2]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {3 \int x^2 \sec ^4(a x) \, dx}{a^2} \\ & = -\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3}+\frac {\int \sec ^2(a x) \, dx}{a^4}+\frac {2 \int x^2 \sec ^2(a x) \, dx}{a^2} \\ & = -\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3}-\frac {\text {Subst}(\int 1 \, dx,x,-\tan (a x))}{a^5}-\frac {4 \int x \tan (a x) \, dx}{a^3} \\ & = -\frac {2 i x^2}{a^3}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3}+\frac {(8 i) \int \frac {e^{2 i a x} x}{1+e^{2 i a x}} \, dx}{a^3} \\ & = -\frac {2 i x^2}{a^3}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3}-\frac {4 \int \log \left (1+e^{2 i a x}\right ) \, dx}{a^4} \\ & = -\frac {2 i x^2}{a^3}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3}+\frac {(2 i) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i a x}\right )}{a^5} \\ & = -\frac {2 i x^2}{a^3}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right )}{a^5}-\frac {x \sec ^2(a x)}{a^4}-\frac {x^3 \sec ^3(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^5}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \sec ^2(a x) \tan (a x)}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-a x \left (1+2 i a x+a^2 x^2-4 \log \left (1+e^{2 i a x}\right )\right )+\left (1+2 a^2 x^2-2 i a^3 x^3+4 a^2 x^2 \log \left (1+e^{2 i a x}\right )\right ) \tan (a x)+a^3 x^3 \tan ^2(a x)-2 i \operatorname {PolyLog}\left (2,-e^{2 i a x}\right ) (1+a x \tan (a x))}{a^5 (1+a x \tan (a x))} \]

[In]

Integrate[(x^4*Sec[a*x]^2)/(Cos[a*x] + a*x*Sin[a*x])^2,x]

[Out]

(-(a*x*(1 + (2*I)*a*x + a^2*x^2 - 4*Log[1 + E^((2*I)*a*x)])) + (1 + 2*a^2*x^2 - (2*I)*a^3*x^3 + 4*a^2*x^2*Log[
1 + E^((2*I)*a*x)])*Tan[a*x] + a^3*x^3*Tan[a*x]^2 - (2*I)*PolyLog[2, -E^((2*I)*a*x)]*(1 + a*x*Tan[a*x]))/(a^5*
(1 + a*x*Tan[a*x]))

Maple [A] (verified)

Time = 3.54 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {2 i \left (-2 i a^{2} x^{2} {\mathrm e}^{2 i a x}+2 a^{3} x^{3}-2 i a^{2} x^{2}+a x \,{\mathrm e}^{2 i a x}-i {\mathrm e}^{2 i a x}+a x -i\right )}{\left (1+{\mathrm e}^{2 i a x}\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right ) a^{5}}-\frac {4 i x^{2}}{a^{3}}+\frac {4 x \ln \left (1+{\mathrm e}^{2 i a x}\right )}{a^{4}}-\frac {2 i \operatorname {polylog}\left (2, -{\mathrm e}^{2 i a x}\right )}{a^{5}}\)	\(141\)

[In]

int(x^4*sec(a*x)^2/(cos(a*x)+a*x*sin(a*x))^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*(-2*I*a^2*x^2*exp(2*I*a*x)+2*a^3*x^3-2*I*a^2*x^2+a*x*exp(2*I*a*x)-I*exp(2*I*a*x)+a*x-I)/(1+exp(2*I*a*x))/
(a*x*exp(2*I*a*x)-a*x+I*exp(2*I*a*x)+I)/a^5-4*I/a^3*x^2+4*x*ln(1+exp(2*I*a*x))/a^4-2*I*polylog(2,-exp(2*I*a*x)
)/a^5

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (113) = 226\).

Time = 0.28 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.08 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a^{3} x^{3} - {\left (2 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{2} + {\left (2 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right ) \sin \left (a x\right ) - 2 \, {\left (-i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - 2 \, {\left (i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) - 2 \, {\left (i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - 2 \, {\left (-i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right )}{a^{6} x \cos \left (a x\right ) \sin \left (a x\right ) + a^{5} \cos \left (a x\right )^{2}} \]

[In]

integrate(x^4*sec(a*x)^2/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="fricas")

[Out]

(a^3*x^3 - (2*a^3*x^3 + a*x)*cos(a*x)^2 + (2*a^2*x^2 + 1)*cos(a*x)*sin(a*x) - 2*(-I*a*x*cos(a*x)*sin(a*x) - I*
cos(a*x)^2)*dilog(I*cos(a*x) + sin(a*x)) - 2*(I*a*x*cos(a*x)*sin(a*x) + I*cos(a*x)^2)*dilog(I*cos(a*x) - sin(a
*x)) - 2*(I*a*x*cos(a*x)*sin(a*x) + I*cos(a*x)^2)*dilog(-I*cos(a*x) + sin(a*x)) - 2*(-I*a*x*cos(a*x)*sin(a*x)
- I*cos(a*x)^2)*dilog(-I*cos(a*x) - sin(a*x)) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(I*cos(a*x)
+ sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(I*cos(a*x) - sin(a*x) + 1) + 2*(a^2*x^2*c
os(a*x)*sin(a*x) + a*x*cos(a*x)^2)*log(-I*cos(a*x) + sin(a*x) + 1) + 2*(a^2*x^2*cos(a*x)*sin(a*x) + a*x*cos(a*
x)^2)*log(-I*cos(a*x) - sin(a*x) + 1))/(a^6*x*cos(a*x)*sin(a*x) + a^5*cos(a*x)^2)

Sympy [F]

\[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^{4} \sec ^{2}{\left (a x \right )}}{\left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \]

[In]

integrate(x**4*sec(a*x)**2/(cos(a*x)+a*x*sin(a*x))**2,x)

[Out]

Integral(x**4*sec(a*x)**2/(a*x*sin(a*x) + cos(a*x))**2, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (113) = 226\).

Time = 0.32 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.00 \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, {\left (a x + 2 \, {\left (a^{2} x^{2} - 2 i \, a x \cos \left (2 \, a x\right ) + 2 \, a x \sin \left (2 \, a x\right ) - i \, a x - {\left (a^{2} x^{2} + i \, a x\right )} \cos \left (4 \, a x\right ) + {\left (-i \, a^{2} x^{2} + a x\right )} \sin \left (4 \, a x\right )\right )} \arctan \left (\sin \left (2 \, a x\right ), \cos \left (2 \, a x\right ) + 1\right ) + 2 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \cos \left (4 \, a x\right ) + {\left (2 i \, a^{2} x^{2} + a x - i\right )} \cos \left (2 \, a x\right ) - {\left (a x - {\left (a x + i\right )} \cos \left (4 \, a x\right ) - {\left (i \, a x - 1\right )} \sin \left (4 \, a x\right ) - 2 i \, \cos \left (2 \, a x\right ) + 2 \, \sin \left (2 \, a x\right ) - i\right )} {\rm Li}_2\left (-e^{\left (2 i \, a x\right )}\right ) + {\left (-i \, a^{2} x^{2} - 2 \, a x \cos \left (2 \, a x\right ) - 2 i \, a x \sin \left (2 \, a x\right ) - a x + {\left (i \, a^{2} x^{2} - a x\right )} \cos \left (4 \, a x\right ) - {\left (a^{2} x^{2} + i \, a x\right )} \sin \left (4 \, a x\right )\right )} \log \left (\cos \left (2 \, a x\right )^{2} + \sin \left (2 \, a x\right )^{2} + 2 \, \cos \left (2 \, a x\right ) + 1\right ) + 2 \, {\left (i \, a^{3} x^{3} - a^{2} x^{2}\right )} \sin \left (4 \, a x\right ) - {\left (2 \, a^{2} x^{2} - i \, a x - 1\right )} \sin \left (2 \, a x\right ) - i\right )}}{{\left (i \, a x + {\left (-i \, a x + 1\right )} \cos \left (4 \, a x\right ) + {\left (a x + i\right )} \sin \left (4 \, a x\right ) + 2 \, \cos \left (2 \, a x\right ) + 2 i \, \sin \left (2 \, a x\right ) + 1\right )} a^{5}} \]

[In]

integrate(x^4*sec(a*x)^2/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="maxima")

[Out]

-2*(a*x + 2*(a^2*x^2 - 2*I*a*x*cos(2*a*x) + 2*a*x*sin(2*a*x) - I*a*x - (a^2*x^2 + I*a*x)*cos(4*a*x) + (-I*a^2*
x^2 + a*x)*sin(4*a*x))*arctan2(sin(2*a*x), cos(2*a*x) + 1) + 2*(a^3*x^3 + I*a^2*x^2)*cos(4*a*x) + (2*I*a^2*x^2
+ a*x - I)*cos(2*a*x) - (a*x - (a*x + I)*cos(4*a*x) - (I*a*x - 1)*sin(4*a*x) - 2*I*cos(2*a*x) + 2*sin(2*a*x)
- I)*dilog(-e^(2*I*a*x)) + (-I*a^2*x^2 - 2*a*x*cos(2*a*x) - 2*I*a*x*sin(2*a*x) - a*x + (I*a^2*x^2 - a*x)*cos(4
*a*x) - (a^2*x^2 + I*a*x)*sin(4*a*x))*log(cos(2*a*x)^2 + sin(2*a*x)^2 + 2*cos(2*a*x) + 1) + 2*(I*a^3*x^3 - a^2
*x^2)*sin(4*a*x) - (2*a^2*x^2 - I*a*x - 1)*sin(2*a*x) - I)/((I*a*x + (-I*a*x + 1)*cos(4*a*x) + (a*x + I)*sin(4
*a*x) + 2*cos(2*a*x) + 2*I*sin(2*a*x) + 1)*a^5)

Giac [F]

\[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int { \frac {x^{4} \sec \left (a x\right )^{2}}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )}^{2}} \,d x } \]

[In]

integrate(x^4*sec(a*x)^2/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="giac")

[Out]

integrate(x^4*sec(a*x)^2/(a*x*sin(a*x) + cos(a*x))^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^4}{{\cos \left (a\,x\right )}^2\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]

[In]

int(x^4/(cos(a*x)^2*(cos(a*x) + a*x*sin(a*x))^2),x)

[Out]

int(x^4/(cos(a*x)^2*(cos(a*x) + a*x*sin(a*x))^2), x)