Integrand size = 8, antiderivative size = 104 \[ \int x^3 \tan ^4(x) \, dx=-\frac {x^2}{2}+\frac {4 i x^3}{3}+\frac {x^4}{4}-4 x^2 \log \left (1+e^{2 i x}\right )+\log (\cos (x))+4 i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-2 \operatorname {PolyLog}\left (3,-e^{2 i x}\right )+x \tan (x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)+\frac {1}{3} x^3 \tan ^3(x) \]
-1/2*x^2+4/3*I*x^3+1/4*x^4-4*x^2*ln(1+exp(2*I*x))+ln(cos(x))+4*I*x*polylog (2,-exp(2*I*x))-2*polylog(3,-exp(2*I*x))+x*tan(x)-x^3*tan(x)-1/2*x^2*tan(x )^2+1/3*x^3*tan(x)^3
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int x^3 \tan ^4(x) \, dx=\frac {4 i x^3}{3}+\frac {x^4}{4}-4 x^2 \log \left (1+e^{2 i x}\right )+\log (\cos (x))+4 i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-2 \operatorname {PolyLog}\left (3,-e^{2 i x}\right )-\frac {1}{2} x^2 \sec ^2(x)+x \tan (x)-\frac {4}{3} x^3 \tan (x)+\frac {1}{3} x^3 \sec ^2(x) \tan (x) \]
((4*I)/3)*x^3 + x^4/4 - 4*x^2*Log[1 + E^((2*I)*x)] + Log[Cos[x]] + (4*I)*x *PolyLog[2, -E^((2*I)*x)] - 2*PolyLog[3, -E^((2*I)*x)] - (x^2*Sec[x]^2)/2 + x*Tan[x] - (4*x^3*Tan[x])/3 + (x^3*Sec[x]^2*Tan[x])/3
Time = 0.76 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.875, Rules used = {3042, 4203, 3042, 4203, 15, 3042, 4202, 2620, 3011, 2720, 4203, 15, 3042, 3956, 7143}
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.
| \(\displaystyle \int x^3 \tan ^4(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \tan (x)^4dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\int x^3 \tan ^2(x)dx-\int x^2 \tan ^3(x)dx+\frac {1}{3} x^3 \tan ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int x^3 \tan (x)^2dx-\int x^2 \tan (x)^3dx+\frac {1}{3} x^3 \tan ^3(x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^3dx+4 \int x^2 \tan (x)dx+\int x \tan ^2(x)dx+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 4 \int x^2 \tan (x)dx+\int x \tan ^2(x)dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \int x^2 \tan (x)dx+\int x \tan (x)^2dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \int \frac {e^{2 i x} x^2}{1+e^{2 i x}}dx\right )+\int x \tan (x)^2dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \int x \log \left (1+e^{2 i x}\right )dx-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )+\int x \tan (x)^2dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,-e^{2 i x}\right )dx\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )+\int x \tan (x)^2dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )+\int x \tan (x)^2dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\int xdx-\int \tan (x)dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\int \tan (x)dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {x^2}{2}-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\int \tan (x)dx+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {x^2}{2}-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {x^2}{2}-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)+\log (\cos (x))\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 i x}\right )\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )+\frac {x^4}{4}+\frac {1}{3} x^3 \tan ^3(x)-x^3 \tan (x)-\frac {x^2}{2}-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)+\log (\cos (x))\) |
-1/2*x^2 + x^4/4 + Log[Cos[x]] + 4*((I/3)*x^3 - (2*I)*((-1/2*I)*x^2*Log[1 + E^((2*I)*x)] + I*((I/2)*x*PolyLog[2, -E^((2*I)*x)] - PolyLog[3, -E^((2*I )*x)]/4))) + x*Tan[x] - x^3*Tan[x] - (x^2*Tan[x]^2)/2 + (x^3*Tan[x]^3)/3
3.2.28.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {x^{4}}{4}-\frac {2 i x \left (6 x^{2} {\mathrm e}^{4 i x}+6 x^{2} {\mathrm e}^{2 i x}-3 \,{\mathrm e}^{4 i x}-3 i x \,{\mathrm e}^{4 i x}+4 x^{2}-6 \,{\mathrm e}^{2 i x}-3 i x \,{\mathrm e}^{2 i x}-3\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}+\ln \left ({\mathrm e}^{2 i x}+1\right )-2 \ln \left ({\mathrm e}^{i x}\right )+\frac {8 i x^{3}}{3}-4 x^{2} \ln \left ({\mathrm e}^{2 i x}+1\right )+4 i x \,\operatorname {Li}_{2}\left (-{\mathrm e}^{2 i x}\right )-2 \,\operatorname {Li}_{3}\left (-{\mathrm e}^{2 i x}\right )\) | \(138\) |
1/4*x^4-2/3*I*x*(6*x^2*exp(4*I*x)+6*x^2*exp(2*I*x)-3*exp(4*I*x)-3*I*x*exp( 4*I*x)+4*x^2-6*exp(2*I*x)-3*I*x*exp(2*I*x)-3)/(exp(2*I*x)+1)^3+ln(exp(2*I* x)+1)-2*ln(exp(I*x))+8/3*I*x^3-4*x^2*ln(exp(2*I*x)+1)+4*I*x*polylog(2,-exp (2*I*x))-2*polylog(3,-exp(2*I*x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.75 \[ \int x^3 \tan ^4(x) \, dx=\frac {1}{3} \, x^{3} \tan \left (x\right )^{3} + \frac {1}{4} \, x^{4} - \frac {1}{2} \, x^{2} \tan \left (x\right )^{2} - \frac {1}{2} \, x^{2} - 2 i \, x {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + 2 i \, x {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac {1}{2} \, {\left (4 \, x^{2} - 1\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{2} \, {\left (4 \, x^{2} - 1\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) - {\left (x^{3} - x\right )} \tan \left (x\right ) - {\rm polylog}\left (3, \frac {\tan \left (x\right )^{2} + 2 i \, \tan \left (x\right ) - 1}{\tan \left (x\right )^{2} + 1}\right ) - {\rm polylog}\left (3, \frac {\tan \left (x\right )^{2} - 2 i \, \tan \left (x\right ) - 1}{\tan \left (x\right )^{2} + 1}\right ) \]
1/3*x^3*tan(x)^3 + 1/4*x^4 - 1/2*x^2*tan(x)^2 - 1/2*x^2 - 2*I*x*dilog(2*(I *tan(x) - 1)/(tan(x)^2 + 1) + 1) + 2*I*x*dilog(2*(-I*tan(x) - 1)/(tan(x)^2 + 1) + 1) - 1/2*(4*x^2 - 1)*log(-2*(I*tan(x) - 1)/(tan(x)^2 + 1)) - 1/2*( 4*x^2 - 1)*log(-2*(-I*tan(x) - 1)/(tan(x)^2 + 1)) - (x^3 - x)*tan(x) - pol ylog(3, (tan(x)^2 + 2*I*tan(x) - 1)/(tan(x)^2 + 1)) - polylog(3, (tan(x)^2 - 2*I*tan(x) - 1)/(tan(x)^2 + 1))
\[ \int x^3 \tan ^4(x) \, dx=\int x^{3} \tan ^{4}{\left (x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (80) = 160\).
Time = 0.32 (sec) , antiderivative size = 491, normalized size of antiderivative = 4.72 \[ \int x^3 \tan ^4(x) \, dx =\text {Too large to display} \]
-(3*I*x^4 + 12*(4*x^2 + (4*x^2 - 1)*cos(6*x) + 3*(4*x^2 - 1)*cos(4*x) + 3* (4*x^2 - 1)*cos(2*x) - (-4*I*x^2 + I)*sin(6*x) - 3*(-4*I*x^2 + I)*sin(4*x) - 3*(-4*I*x^2 + I)*sin(2*x) - 1)*arctan2(sin(2*x), cos(2*x) + 1) + (3*I*x ^4 - 32*x^3 + 24*x)*cos(6*x) - 3*(-3*I*x^4 + 16*x^3 + 8*I*x^2 - 16*x)*cos( 4*x) - 3*(-3*I*x^4 + 16*x^3 + 8*I*x^2 - 8*x)*cos(2*x) - 48*(x*cos(6*x) + 3 *x*cos(4*x) + 3*x*cos(2*x) + I*x*sin(6*x) + 3*I*x*sin(4*x) + 3*I*x*sin(2*x ) + x)*dilog(-e^(2*I*x)) - 6*(4*I*x^2 + (4*I*x^2 - I)*cos(6*x) + 3*(4*I*x^ 2 - I)*cos(4*x) + 3*(4*I*x^2 - I)*cos(2*x) - (4*x^2 - 1)*sin(6*x) - 3*(4*x ^2 - 1)*sin(4*x) - 3*(4*x^2 - 1)*sin(2*x) - I)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) - 24*(I*cos(6*x) + 3*I*cos(4*x) + 3*I*cos(2*x) - sin(6* x) - 3*sin(4*x) - 3*sin(2*x) + I)*polylog(3, -e^(2*I*x)) - (3*x^4 + 32*I*x ^3 - 24*I*x)*sin(6*x) - 3*(3*x^4 + 16*I*x^3 - 8*x^2 - 16*I*x)*sin(4*x) - 3 *(3*x^4 + 16*I*x^3 - 8*x^2 - 8*I*x)*sin(2*x))/(-12*I*cos(6*x) - 36*I*cos(4 *x) - 36*I*cos(2*x) + 12*sin(6*x) + 36*sin(4*x) + 36*sin(2*x) - 12*I)
\[ \int x^3 \tan ^4(x) \, dx=\int { x^{3} \tan \left (x\right )^{4} \,d x } \]
Timed out. \[ \int x^3 \tan ^4(x) \, dx=\int x^3\,{\mathrm {tan}\left (x\right )}^4 \,d x \]
\[ \int x^3 \tan ^4(x) \, dx=4 \left (\int \tan \left (x \right ) x^{2}d x \right )-\frac {\mathrm {log}\left (\tan \left (x \right )^{2}+1\right )}{2}+\frac {\tan \left (x \right )^{3} x^{3}}{3}-\frac {\tan \left (x \right )^{2} x^{2}}{2}-\tan \left (x \right ) x^{3}+\tan \left (x \right ) x +\frac {x^{4}}{4}-\frac {x^{2}}{2} \]