\(\int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx\) [544]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 i x \arctan \left (e^{i a x}\right )}{a^3}+\frac {i \operatorname {PolyLog}\left (2,-i e^{i a x}\right )}{a^4}-\frac {i \operatorname {PolyLog}\left (2,i e^{i a x}\right )}{a^4}-\frac {\sec (a x)}{a^4}-\frac {x^2 \sec ^2(a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {x \sec (a x) \tan (a x)}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {-a x \log \left (1-i e^{i a x}\right )+a x \log \left (1+i e^{i a x}\right )+\sec (a x)+a^2 x^2 \sec (a x)-a^2 x^2 \log \left (1-i e^{i a x}\right ) \tan (a x)+a^2 x^2 \log \left (1+i e^{i a x}\right ) \tan (a x)-i \operatorname {PolyLog}\left (2,-i e^{i a x}\right ) (1+a x \tan (a x))+i \operatorname {PolyLog}\left (2,i e^{i a x}\right ) (1+a x \tan (a x))}{a^4 (1+a x \tan (a x))} \] Input:

Integrate[(x^3*Sec[a*x])/(Cos[a*x] + a*x*Sin[a*x])^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5112, 3042, 4673, 3042, 4669, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \sec (a x)}{(a x \sin (a x)+\cos (a x))^2} \, dx\)

\(\Big \downarrow \) 5112

\(\displaystyle \frac {2 \int x \sec ^3(a x)dx}{a^2}-\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \int x \csc \left (a x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {2 \left (\frac {1}{2} \int x \sec (a x)dx-\frac {\sec (a x)}{2 a^2}+\frac {x \tan (a x) \sec (a x)}{2 a}\right )}{a^2}-\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{2} \int x \csc \left (a x+\frac {\pi }{2}\right )dx-\frac {\sec (a x)}{2 a^2}+\frac {x \tan (a x) \sec (a x)}{2 a}\right )}{a^2}-\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {2 \left (\frac {1}{2} \left (-\frac {\int \log \left (1-i e^{i a x}\right )dx}{a}+\frac {\int \log \left (1+i e^{i a x}\right )dx}{a}-\frac {2 i x \arctan \left (e^{i a x}\right )}{a}\right )-\frac {\sec (a x)}{2 a^2}+\frac {x \tan (a x) \sec (a x)}{2 a}\right )}{a^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {2 \left (\frac {1}{2} \left (\frac {i \int e^{-i a x} \log \left (1-i e^{i a x}\right )de^{i a x}}{a^2}-\frac {i \int e^{-i a x} \log \left (1+i e^{i a x}\right )de^{i a x}}{a^2}-\frac {2 i x \arctan \left (e^{i a x}\right )}{a}\right )-\frac {\sec (a x)}{2 a^2}+\frac {x \tan (a x) \sec (a x)}{2 a}\right )}{a^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {2 \left (\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,-i e^{i a x}\right )}{a^2}-\frac {i \operatorname {PolyLog}\left (2,i e^{i a x}\right )}{a^2}-\frac {2 i x \arctan \left (e^{i a x}\right )}{a}\right )-\frac {\sec (a x)}{2 a^2}+\frac {x \tan (a x) \sec (a x)}{2 a}\right )}{a^2}\)

Input:

Int[(x^3*Sec[a*x])/(Cos[a*x] + a*x*Sin[a*x])^2,x]
 

Output:

-((x^2*Sec[a*x]^2)/(a^2*(Cos[a*x] + a*x*Sin[a*x]))) + (2*((((-2*I)*x*ArcTa 
n[E^(I*a*x)])/a + (I*PolyLog[2, (-I)*E^(I*a*x)])/a^2 - (I*PolyLog[2, I*E^( 
I*a*x)])/a^2)/2 - Sec[a*x]/(2*a^2) + (x*Sec[a*x]*Tan[a*x])/(2*a)))/a^2
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4673

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

rule 5112

Int[(((b_.)*(x_))^(m_.)*Sec[(a_.)*(x_)]^(n_.))/(Cos[(a_.)*(x_)]*(c_.) + (d_ 
.)*(x_)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[(-b)*(b*x)^(m - 1)*(Sec[a*x]^ 
(n + 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x]))), x] + Simp[b^2*((n + 1)/d^2)   I 
nt[(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]
 

Maple [F]

\[\int \frac {x^{3} \sec \left (a x \right )}{\left (\cos \left (a x \right )+a x \sin \left (a x \right )\right )^{2}}d x\]

Input:

int(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x)
 

Output:

int(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x)
 

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. 290 vs. \(2 (89) = 178\).

Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.64 \[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, a^{2} x^{2} - {\left (-i \, a x \sin \left (a x\right ) - i \, \cos \left (a x\right )\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - {\left (-i \, a x \sin \left (a x\right ) - i \, \cos \left (a x\right )\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) - {\left (i \, a x \sin \left (a x\right ) + i \, \cos \left (a x\right )\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) - {\left (i \, a x \sin \left (a x\right ) + i \, \cos \left (a x\right )\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) - {\left (a^{2} x^{2} \sin \left (a x\right ) + a x \cos \left (a x\right )\right )} \log \left (i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \sin \left (a x\right ) + a x \cos \left (a x\right )\right )} \log \left (i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right ) - {\left (a^{2} x^{2} \sin \left (a x\right ) + a x \cos \left (a x\right )\right )} \log \left (-i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \sin \left (a x\right ) + a x \cos \left (a x\right )\right )} \log \left (-i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right ) + 2}{2 \, {\left (a^{5} x \sin \left (a x\right ) + a^{4} \cos \left (a x\right )\right )}} \] Input:

integrate(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^{3} \sec {\left (a x \right )}}{\left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \] Input:

integrate(x**3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))**2,x)
 

Output:

Integral(x**3*sec(a*x)/(a*x*sin(a*x) + cos(a*x))**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int { \frac {x^{3} \sec \left (a x\right )}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )}^{2}} \,d x } \] Input:

integrate(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x, algorithm="giac")
 

Output:

integrate(x^3*sec(a*x)/(a*x*sin(a*x) + cos(a*x))^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^3}{\cos \left (a\,x\right )\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \] Input:

int(x^3/(cos(a*x)*(cos(a*x) + a*x*sin(a*x))^2),x)
 

Output:

int(x^3/(cos(a*x)*(cos(a*x) + a*x*sin(a*x))^2), x)
 

Reduce [F]

\[ \int \frac {x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx =\text {Too large to display} \] Input:

int(x^3*sec(a*x)/(cos(a*x)+a*x*sin(a*x))^2,x)
 

Output:

( - 16*cos(a*x)*int(x**3/(tan((a*x)/2)**6 - 4*tan((a*x)/2)**5*a*x + 4*tan( 
(a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 8*tan((a*x)/2)**3*a*x - 4*tan( 
(a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 4*tan((a*x)/2)*a*x - 1),x)*a** 
4 - 24*cos(a*x)*int((tan((a*x)/2)**2*x)/(tan((a*x)/2)**6 - 4*tan((a*x)/2)* 
*5*a*x + 4*tan((a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 8*tan((a*x)/2)* 
*3*a*x - 4*tan((a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 4*tan((a*x)/2)* 
a*x - 1),x)*a**2 + 48*cos(a*x)*int((tan((a*x)/2)*x**2)/(tan((a*x)/2)**6 - 
4*tan((a*x)/2)**5*a*x + 4*tan((a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 
8*tan((a*x)/2)**3*a*x - 4*tan((a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 
4*tan((a*x)/2)*a*x - 1),x)*a**3 + 24*cos(a*x)*int(x/(tan((a*x)/2)**6 - 4*t 
an((a*x)/2)**5*a*x + 4*tan((a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 8*t 
an((a*x)/2)**3*a*x - 4*tan((a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 4*t 
an((a*x)/2)*a*x - 1),x)*a**2 + 6*cos(a*x)*log(tan((a*x)/2)**2 + 1) - 6*cos 
(a*x)*log(tan((a*x)/2) - 1) - 6*cos(a*x)*log(tan((a*x)/2) + 1) + 9*cos(a*x 
)*a**2*x**2 - 16*int(x**3/(tan((a*x)/2)**6 - 4*tan((a*x)/2)**5*a*x + 4*tan 
((a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 8*tan((a*x)/2)**3*a*x - 4*tan 
((a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 4*tan((a*x)/2)*a*x - 1),x)*si 
n(a*x)*a**5*x - 24*int((tan((a*x)/2)**2*x)/(tan((a*x)/2)**6 - 4*tan((a*x)/ 
2)**5*a*x + 4*tan((a*x)/2)**4*a**2*x**2 - 3*tan((a*x)/2)**4 + 8*tan((a*x)/ 
2)**3*a*x - 4*tan((a*x)/2)**2*a**2*x**2 + 3*tan((a*x)/2)**2 - 4*tan((a*...