Integrand size = 6, antiderivative size = 59 \[ \int x \tan ^3(x) \, dx=\frac {x}{2}-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {\tan (x)}{2}+\frac {1}{2} x \tan ^2(x) \] Output:
1/2*x-1/2*I*x^2+x*ln(1+exp(2*I*x))-1/2*I*polylog(2,-exp(2*I*x))-1/2*tan(x) +1/2*x*tan(x)^2
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int x \tan ^3(x) \, dx=-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2} \] Input:
Integrate[x*Tan[x]^3,x]
Output:
(-1/2*I)*x^2 + x*Log[1 + E^((2*I)*x)] - (I/2)*PolyLog[2, -E^((2*I)*x)] + ( x*Sec[x]^2)/2 - Tan[x]/2
Time = 0.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 4203, 3042, 3954, 24, 4202, 2620, 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.
\(\displaystyle \int x \tan ^3(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x \tan (x)^3dx\) |
\(\Big \downarrow \) 4203 |
\(\displaystyle -\frac {1}{2} \int \tan ^2(x)dx-\int x \tan (x)dx+\frac {1}{2} x \tan ^2(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int x \tan (x)dx-\frac {1}{2} \int \tan (x)^2dx+\frac {1}{2} x \tan ^2(x)\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {1}{2} (\int 1dx-\tan (x))-\int x \tan (x)dx+\frac {1}{2} x \tan ^2(x)\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\int x \tan (x)dx+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} (x-\tan (x))\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle 2 i \int \frac {e^{2 i x} x}{1+e^{2 i x}}dx-\frac {i x^2}{2}+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} (x-\tan (x))\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (\frac {1}{2} i \int \log \left (1+e^{2 i x}\right )dx-\frac {1}{2} i x \log \left (1+e^{2 i x}\right )\right )-\frac {i x^2}{2}+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} (x-\tan (x))\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 i \left (\frac {1}{4} \int e^{-2 i x} \log \left (1+e^{2 i x}\right )de^{2 i x}-\frac {1}{2} i x \log \left (1+e^{2 i x}\right )\right )-\frac {i x^2}{2}+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} (x-\tan (x))\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i x \log \left (1+e^{2 i x}\right )\right )-\frac {i x^2}{2}+\frac {1}{2} x \tan ^2(x)+\frac {1}{2} (x-\tan (x))\) |
Input:
Int[x*Tan[x]^3,x]
Output:
(-1/2*I)*x^2 + (2*I)*((-1/2*I)*x*Log[1 + E^((2*I)*x)] - PolyLog[2, -E^((2* I)*x)]/4) + (x - Tan[x])/2 + (x*Tan[x]^2)/2
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] - Si mp[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]
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]
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]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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] && IGt Q[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {i x^{2}}{2}+\frac {2 x \,{\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}-i}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}+x \ln \left ({\mathrm e}^{2 i x}+1\right )-\frac {i \operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )}{2}\) | \(59\) |
Input:
int(x*sin(x)^3/cos(x)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*I*x^2+(2*x*exp(2*I*x)-I*exp(2*I*x)-I)/(exp(2*I*x)+1)^2+x*ln(exp(2*I*x )+1)-1/2*I*polylog(2,-exp(2*I*x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (38) = 76\).
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.34 \[ \int x \tan ^3(x) \, dx=\frac {x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - \cos \left (x\right ) \sin \left (x\right ) + x}{2 \, \cos \left (x\right )^{2}} \] Input:
integrate(x*sin(x)^3/cos(x)^3,x, algorithm="fricas")
Output:
1/2*(x*cos(x)^2*log(I*cos(x) + sin(x) + 1) + x*cos(x)^2*log(I*cos(x) - sin (x) + 1) + x*cos(x)^2*log(-I*cos(x) + sin(x) + 1) + x*cos(x)^2*log(-I*cos( x) - sin(x) + 1) + I*cos(x)^2*dilog(I*cos(x) + sin(x)) - I*cos(x)^2*dilog( I*cos(x) - sin(x)) - I*cos(x)^2*dilog(-I*cos(x) + sin(x)) + I*cos(x)^2*dil og(-I*cos(x) - sin(x)) - cos(x)*sin(x) + x)/cos(x)^2
\[ \int x \tan ^3(x) \, dx=\int \frac {x \sin ^{3}{\left (x \right )}}{\cos ^{3}{\left (x \right )}}\, dx \] Input:
integrate(x*sin(x)**3/cos(x)**3,x)
Output:
Integral(x*sin(x)**3/cos(x)**3, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (38) = 76\).
Time = 0.12 (sec) , antiderivative size = 210, normalized size of antiderivative = 3.56 \[ \int x \tan ^3(x) \, dx=-\frac {x^{2} \cos \left (4 \, x\right ) + i \, x^{2} \sin \left (4 \, x\right ) + x^{2} - 2 \, {\left (x \cos \left (4 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) + i \, x \sin \left (4 \, x\right ) + 2 i \, x \sin \left (2 \, x\right ) + x\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + 2 \, {\left (x^{2} + 2 i \, x + 1\right )} \cos \left (2 \, x\right ) + {\left (\cos \left (4 \, x\right ) + 2 \, \cos \left (2 \, x\right ) + i \, \sin \left (4 \, x\right ) + 2 i \, \sin \left (2 \, x\right ) + 1\right )} {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) - {\left (-i \, x \cos \left (4 \, x\right ) - 2 i \, x \cos \left (2 \, x\right ) + x \sin \left (4 \, x\right ) + 2 \, x \sin \left (2 \, x\right ) - i \, x\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + 2 \, {\left (i \, x^{2} - 2 \, x + i\right )} \sin \left (2 \, x\right ) + 2}{-2 i \, \cos \left (4 \, x\right ) - 4 i \, \cos \left (2 \, x\right ) + 2 \, \sin \left (4 \, x\right ) + 4 \, \sin \left (2 \, x\right ) - 2 i} \] Input:
integrate(x*sin(x)^3/cos(x)^3,x, algorithm="maxima")
Output:
-(x^2*cos(4*x) + I*x^2*sin(4*x) + x^2 - 2*(x*cos(4*x) + 2*x*cos(2*x) + I*x *sin(4*x) + 2*I*x*sin(2*x) + x)*arctan2(sin(2*x), cos(2*x) + 1) + 2*(x^2 + 2*I*x + 1)*cos(2*x) + (cos(4*x) + 2*cos(2*x) + I*sin(4*x) + 2*I*sin(2*x) + 1)*dilog(-e^(2*I*x)) - (-I*x*cos(4*x) - 2*I*x*cos(2*x) + x*sin(4*x) + 2* x*sin(2*x) - I*x)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) + 2*(I*x^2 - 2*x + I)*sin(2*x) + 2)/(-2*I*cos(4*x) - 4*I*cos(2*x) + 2*sin(4*x) + 4*s in(2*x) - 2*I)
\[ \int x \tan ^3(x) \, dx=\int { \frac {x \sin \left (x\right )^{3}}{\cos \left (x\right )^{3}} \,d x } \] Input:
integrate(x*sin(x)^3/cos(x)^3,x, algorithm="giac")
Output:
integrate(x*sin(x)^3/cos(x)^3, x)
Timed out. \[ \int x \tan ^3(x) \, dx=\int \frac {x\,{\sin \left (x\right )}^3}{{\cos \left (x\right )}^3} \,d x \] Input:
int((x*sin(x)^3)/cos(x)^3,x)
Output:
int((x*sin(x)^3)/cos(x)^3, x)
\[ \int x \tan ^3(x) \, dx=\frac {\cos \left (x \right ) \sin \left (x \right )-2 \cos \left (x \right ) x +8 \left (\int \frac {\tan \left (\frac {x}{2}\right ) x}{\tan \left (\frac {x}{2}\right )^{6}-3 \tan \left (\frac {x}{2}\right )^{4}+3 \tan \left (\frac {x}{2}\right )^{2}-1}d x \right ) \sin \left (x \right )^{2}-8 \left (\int \frac {\tan \left (\frac {x}{2}\right ) x}{\tan \left (\frac {x}{2}\right )^{6}-3 \tan \left (\frac {x}{2}\right )^{4}+3 \tan \left (\frac {x}{2}\right )^{2}-1}d x \right )+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right )-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right )-x}{\sin \left (x \right )^{2}-1} \] Input:
int(x*sin(x)^3/cos(x)^3,x)
Output:
(cos(x)*sin(x) - 2*cos(x)*x + 8*int((tan(x/2)*x)/(tan(x/2)**6 - 3*tan(x/2) **4 + 3*tan(x/2)**2 - 1),x)*sin(x)**2 - 8*int((tan(x/2)*x)/(tan(x/2)**6 - 3*tan(x/2)**4 + 3*tan(x/2)**2 - 1),x) + 2*log(tan(x/2) - 1)*sin(x)**2 - 2* log(tan(x/2) - 1) - 2*log(tan(x/2) + 1)*sin(x)**2 + 2*log(tan(x/2) + 1) - x)/(sin(x)**2 - 1)