Integrand size = 18, antiderivative size = 143 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=-\frac {i x^4 \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}}+\frac {x^3 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt {a \sec ^4(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt {a \sec ^4(x)}}+\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i x}\right ) \sec ^2(x)}{4 \sqrt {a \sec ^4(x)}} \]
-1/4*I*x^4*sec(x)^2/(a*sec(x)^4)^(1/2)+x^3*ln(1-exp(2*I*x))*sec(x)^2/(a*se c(x)^4)^(1/2)-3/2*I*x^2*polylog(2,exp(2*I*x))*sec(x)^2/(a*sec(x)^4)^(1/2)+ 3/2*x*polylog(3,exp(2*I*x))*sec(x)^2/(a*sec(x)^4)^(1/2)+3/4*I*polylog(4,ex p(2*I*x))*sec(x)^2/(a*sec(x)^4)^(1/2)
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=-\frac {i \left (\pi ^4-16 x^4+64 i x^3 \log \left (1-e^{-2 i x}\right )-96 x^2 \operatorname {PolyLog}\left (2,e^{-2 i x}\right )+96 i x \operatorname {PolyLog}\left (3,e^{-2 i x}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i x}\right )\right ) \sec ^2(x)}{64 \sqrt {a \sec ^4(x)}} \]
((-1/64*I)*(Pi^4 - 16*x^4 + (64*I)*x^3*Log[1 - E^((-2*I)*x)] - 96*x^2*Poly Log[2, E^((-2*I)*x)] + (96*I)*x*PolyLog[3, E^((-2*I)*x)] + 48*PolyLog[4, E ^((-2*I)*x)])*Sec[x]^2)/Sqrt[a*Sec[x]^4]
Time = 0.94 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {7271, 3042, 25, 4200, 25, 2620, 3011, 7163, 2720, 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.
| \(\displaystyle \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \frac {\sec ^2(x) \int x^3 \cot (x)dx}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^2(x) \int -x^3 \tan \left (x+\frac {\pi }{2}\right )dx}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sec ^2(x) \int x^3 \tan \left (x+\frac {\pi }{2}\right )dx}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -\frac {\sec ^2(x) \left (\frac {i x^4}{4}-2 i \int -\frac {e^{2 i x} x^3}{1-e^{2 i x}}dx\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \int \frac {e^{2 i x} x^3}{1-e^{2 i x}}dx+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \left (\frac {1}{2} i x^3 \log \left (1-e^{2 i x}\right )-\frac {3}{2} i \int x^2 \log \left (1-e^{2 i x}\right )dx\right )+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \left (\frac {1}{2} i x^3 \log \left (1-e^{2 i x}\right )-\frac {3}{2} i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-i \int x \operatorname {PolyLog}\left (2,e^{2 i x}\right )dx\right )\right )+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \left (\frac {1}{2} i x^3 \log \left (1-e^{2 i x}\right )-\frac {3}{2} i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-i \left (\frac {1}{2} i \int \operatorname {PolyLog}\left (3,e^{2 i x}\right )dx-\frac {1}{2} i x \operatorname {PolyLog}\left (3,e^{2 i x}\right )\right )\right )\right )+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \left (\frac {1}{2} i x^3 \log \left (1-e^{2 i x}\right )-\frac {3}{2} i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-i \left (\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (3,e^{2 i x}\right )de^{2 i x}-\frac {1}{2} i x \operatorname {PolyLog}\left (3,e^{2 i x}\right )\right )\right )\right )+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {\sec ^2(x) \left (2 i \left (\frac {1}{2} i x^3 \log \left (1-e^{2 i x}\right )-\frac {3}{2} i \left (\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-i \left (\frac {1}{4} \operatorname {PolyLog}\left (4,e^{2 i x}\right )-\frac {1}{2} i x \operatorname {PolyLog}\left (3,e^{2 i x}\right )\right )\right )\right )+\frac {i x^4}{4}\right )}{\sqrt {a \sec ^4(x)}}\) |
-((((I/4)*x^4 + (2*I)*((I/2)*x^3*Log[1 - E^((2*I)*x)] - ((3*I)/2)*((I/2)*x ^2*PolyLog[2, E^((2*I)*x)] - I*((-1/2*I)*x*PolyLog[3, E^((2*I)*x)] + PolyL og[4, E^((2*I)*x)]/4))))*Sec[x]^2)/Sqrt[a*Sec[x]^4])
3.9.73.3.1 Defintions of rubi rules used
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Time = 0.99 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{2 i x} x^{4}}{4 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {2 i {\mathrm e}^{2 i x} \left (-\frac {x^{4}}{4}-\frac {i x^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{2}-\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )}{2}-3 i x \operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )+3 \operatorname {polylog}\left (4, -{\mathrm e}^{i x}\right )-\frac {i x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{2}-\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )}{2}-3 i x \operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )+3 \operatorname {polylog}\left (4, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}\) | \(181\) |
1/4*I/(a*exp(4*I*x)/(exp(2*I*x)+1)^4)^(1/2)/(exp(2*I*x)+1)^2*exp(2*I*x)*x^ 4+2*I/(a*exp(4*I*x)/(exp(2*I*x)+1)^4)^(1/2)/(exp(2*I*x)+1)^2*exp(2*I*x)*(- 1/4*x^4-1/2*I*x^3*ln(exp(I*x)+1)-3/2*x^2*polylog(2,-exp(I*x))-3*I*x*polylo g(3,-exp(I*x))+3*polylog(4,-exp(I*x))-1/2*I*x^3*ln(1-exp(I*x))-3/2*x^2*pol ylog(2,exp(I*x))-3*I*x*polylog(3,exp(I*x))+3*polylog(4,exp(I*x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.49 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=\frac {6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (4, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (4, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (4, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2} {\rm polylog}\left (4, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + {\left (x^{3} \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{3} \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + x^{3} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{3} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - 3 i \, x^{2} \cos \left (x\right )^{2} {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 3 i \, x^{2} \cos \left (x\right )^{2} {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 3 i \, x^{2} \cos \left (x\right )^{2} {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 3 i \, x^{2} \cos \left (x\right )^{2} {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4}}}}{2 \, a} \]
1/2*(6*x*sqrt(a/cos(x)^4)*cos(x)^2*polylog(3, cos(x) + I*sin(x)) + 6*x*sqr t(a/cos(x)^4)*cos(x)^2*polylog(3, cos(x) - I*sin(x)) + 6*x*sqrt(a/cos(x)^4 )*cos(x)^2*polylog(3, -cos(x) + I*sin(x)) + 6*x*sqrt(a/cos(x)^4)*cos(x)^2* polylog(3, -cos(x) - I*sin(x)) + 6*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, cos(x) + I*sin(x)) - 6*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, cos(x) - I*s in(x)) - 6*I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, -cos(x) + I*sin(x)) + 6* I*sqrt(a/cos(x)^4)*cos(x)^2*polylog(4, -cos(x) - I*sin(x)) + (x^3*cos(x)^2 *log(cos(x) + I*sin(x) + 1) + x^3*cos(x)^2*log(cos(x) - I*sin(x) + 1) + x^ 3*cos(x)^2*log(-cos(x) + I*sin(x) + 1) + x^3*cos(x)^2*log(-cos(x) - I*sin( x) + 1) - 3*I*x^2*cos(x)^2*dilog(cos(x) + I*sin(x)) + 3*I*x^2*cos(x)^2*dil og(cos(x) - I*sin(x)) + 3*I*x^2*cos(x)^2*dilog(-cos(x) + I*sin(x)) - 3*I*x ^2*cos(x)^2*dilog(-cos(x) - I*sin(x)))*sqrt(a/cos(x)^4))/a
\[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=\int \frac {x^{3} \csc {\left (x \right )} \sec {\left (x \right )}}{\sqrt {a \sec ^{4}{\left (x \right )}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=\frac {-i \, x^{4} + 4 i \, x^{3} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 4 i \, x^{3} \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + 2 \, x^{3} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + 2 \, x^{3} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 12 i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 12 i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 24 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) + 24 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) + 24 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) + 24 i \, {\rm Li}_{4}(e^{\left (i \, x\right )})}{4 \, \sqrt {a}} \]
1/4*(-I*x^4 + 4*I*x^3*arctan2(sin(x), cos(x) + 1) - 4*I*x^3*arctan2(sin(x) , -cos(x) + 1) + 2*x^3*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 2*x^3*log (cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 12*I*x^2*dilog(-e^(I*x)) - 12*I*x^2 *dilog(e^(I*x)) + 24*x*polylog(3, -e^(I*x)) + 24*x*polylog(3, e^(I*x)) + 2 4*I*polylog(4, -e^(I*x)) + 24*I*polylog(4, e^(I*x)))/sqrt(a)
\[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=\int { \frac {x^{3} \csc \left (x\right ) \sec \left (x\right )}{\sqrt {a \sec \left (x\right )^{4}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^4(x)}} \, dx=\int \frac {x^3}{\cos \left (x\right )\,\sin \left (x\right )\,\sqrt {\frac {a}{{\cos \left (x\right )}^4}}} \,d x \]