Optimal. Leaf size=124 \[ -\frac {2 i \text {Li}_2\left (-e^{2 i a x}\right )}{a^5}+\frac {\tan (a x)}{a^5}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}-\frac {x \sec ^2(a x)}{a^4}-\frac {2 i x^2}{a^3}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \tan (a x) \sec ^2(a x)}{a^3}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4601, 4186, 3767, 8, 4184, 3719, 2190, 2279, 2391} \[ -\frac {2 i \text {PolyLog}\left (2,-e^{2 i a x}\right )}{a^5}-\frac {2 i x^2}{a^3}+\frac {2 x^2 \tan (a x)}{a^3}+\frac {x^2 \tan (a x) \sec ^2(a x)}{a^3}-\frac {x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}+\frac {4 x \log \left (1+e^{2 i a x}\right )}{a^4}+\frac {\tan (a x)}{a^5}-\frac {x \sec ^2(a x)}{a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 3767
Rule 4184
Rule 4186
Rule 4601
Rubi steps
\begin {align*} \int \frac {x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx &=-\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 {\operatorname {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) \operatorname {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 \text {Li}_2\left (-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] time = 1.07, size = 130, normalized size = 1.05 \[ \frac {a^3 x^3 \tan ^2(a x)-a x \left (a^2 x^2+2 i a x-4 \log \left (1+e^{2 i a x}\right )+1\right )+\left (-2 i a^3 x^3+2 a^2 x^2+4 a^2 x^2 \log \left (1+e^{2 i a x}\right )+1\right ) \tan (a x)-2 i \text {Li}_2\left (-e^{2 i a x}\right ) (a x \tan (a x)+1)}{a^5 (a x \tan (a x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.88, size = 378, normalized size = 3.05 \[ \frac {a^{3} x^{3} - {\left (2 \, a^{3} x^{3} + a x\right )} \cos \left (a x\right )^{2} + {\left (2 \, a^{2} x^{2} + 1\right )} \cos \left (a x\right ) \sin \left (a x\right ) + {\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) + {\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) + {\left (-2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) - 2 i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) + \sin \left (a x\right )\right ) + {\left (2 i \, a x \cos \left (a x\right ) \sin \left (a x\right ) + 2 i \, \cos \left (a x\right )^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (a x\right ) - \sin \left (a x\right )\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) + \sin \left (a x\right ) + 1\right ) + 2 \, {\left (a^{2} x^{2} \cos \left (a x\right ) \sin \left (a x\right ) + a x \cos \left (a x\right )^{2}\right )} \log \left (-i \, \cos \left (a x\right ) - \sin \left (a x\right ) + 1\right )}{a^{6} x \cos \left (a x\right ) \sin \left (a x\right ) + a^{5} \cos \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sec \left (a x\right )^{2}}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.24, size = 141, normalized size = 1.14 \[ -\frac {2 i \left (-2 i a^{2} x^{2} {\mathrm e}^{2 i a x}+2 x^{3} a^{3}-2 i a^{2} x^{2}+a x \,{\mathrm e}^{2 i a x}-i {\mathrm e}^{2 i a x}+a x -i\right )}{\left (1+{\mathrm e}^{2 i a x}\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right ) a^{5}}-\frac {4 i x^{2}}{a^{3}}+\frac {4 x \ln \left (1+{\mathrm e}^{2 i a x}\right )}{a^{4}}-\frac {2 i \polylog \left (2, -{\mathrm e}^{2 i a x}\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.46, size = 381, normalized size = 3.07 \[ -\frac {2 \, a x + {\left (4 \, a^{2} x^{2} - 8 i \, a x \cos \left (2 \, a x\right ) + 8 \, a x \sin \left (2 \, a x\right ) - 4 i \, a x - {\left (4 \, a^{2} x^{2} + 4 i \, a x\right )} \cos \left (4 \, a x\right ) + 4 \, {\left (-i \, a^{2} x^{2} + a x\right )} \sin \left (4 \, a x\right )\right )} \arctan \left (\sin \left (2 \, a x\right ), \cos \left (2 \, a x\right ) + 1\right ) + 4 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \cos \left (4 \, a x\right ) - {\left (-4 i \, a^{2} x^{2} - 2 \, a x + 2 i\right )} \cos \left (2 \, a x\right ) - {\left (2 \, a x - {\left (2 \, a x + 2 i\right )} \cos \left (4 \, a x\right ) - 2 \, {\left (i \, a x - 1\right )} \sin \left (4 \, a x\right ) - 4 i \, \cos \left (2 \, a x\right ) + 4 \, \sin \left (2 \, a x\right ) - 2 i\right )} {\rm Li}_2\left (-e^{\left (2 i \, a x\right )}\right ) - {\left (2 i \, a^{2} x^{2} + 4 \, a x \cos \left (2 \, a x\right ) + 4 i \, a x \sin \left (2 \, a x\right ) + 2 \, a x - 2 \, {\left (i \, a^{2} x^{2} - a x\right )} \cos \left (4 \, a x\right ) + {\left (2 \, a^{2} x^{2} + 2 i \, a x\right )} \sin \left (4 \, a x\right )\right )} \log \left (\cos \left (2 \, a x\right )^{2} + \sin \left (2 \, a x\right )^{2} + 2 \, \cos \left (2 \, a x\right ) + 1\right ) - {\left (-4 i \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sin \left (4 \, a x\right ) - {\left (4 \, a^{2} x^{2} - 2 i \, a x - 2\right )} \sin \left (2 \, a x\right ) - 2 i}{{\left (i \, a x + {\left (-i \, a x + 1\right )} \cos \left (4 \, a x\right ) + {\left (a x + i\right )} \sin \left (4 \, a x\right ) + 2 \, \cos \left (2 \, a x\right ) + 2 i \, \sin \left (2 \, a x\right ) + 1\right )} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\cos \left (a\,x\right )}^2\,{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sec ^{2}{\left (a x \right )}}{\left (a x \sin {\left (a x \right )} + \cos {\left (a x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________