Integrand size = 18, antiderivative size = 128 \[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {2 x^2 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 i x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 i x \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {2 \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {2 \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}} \]
-2*x^2*arctanh(exp(I*x))*sec(x)/(a*sec(x)^2)^(1/2)+2*I*x*polylog(2,-exp(I* x))*sec(x)/(a*sec(x)^2)^(1/2)-2*I*x*polylog(2,exp(I*x))*sec(x)/(a*sec(x)^2 )^(1/2)-2*polylog(3,-exp(I*x))*sec(x)/(a*sec(x)^2)^(1/2)+2*polylog(3,exp(I *x))*sec(x)/(a*sec(x)^2)^(1/2)
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\frac {\left (x^2 \log \left (1-e^{i x}\right )-x^2 \log \left (1+e^{i x}\right )+2 i x \operatorname {PolyLog}\left (2,-e^{i x}\right )-2 i x \operatorname {PolyLog}\left (2,e^{i x}\right )-2 \operatorname {PolyLog}\left (3,-e^{i x}\right )+2 \operatorname {PolyLog}\left (3,e^{i x}\right )\right ) \sec (x)}{\sqrt {a \sec ^2(x)}} \]
((x^2*Log[1 - E^(I*x)] - x^2*Log[1 + E^(I*x)] + (2*I)*x*PolyLog[2, -E^(I*x )] - (2*I)*x*PolyLog[2, E^(I*x)] - 2*PolyLog[3, -E^(I*x)] + 2*PolyLog[3, E ^(I*x)])*Sec[x])/Sqrt[a*Sec[x]^2]
Time = 0.82 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7271, 3042, 4671, 3011, 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.
\(\displaystyle \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \frac {\sec (x) \int x^2 \csc (x)dx}{\sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (x) \int x^2 \csc (x)dx}{\sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle \frac {\sec (x) \left (-2 \int x \log \left (1-e^{i x}\right )dx+2 \int x \log \left (1+e^{i x}\right )dx-2 x^2 \text {arctanh}\left (e^{i x}\right )\right )}{\sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\sec (x) \left (2 \left (i x \operatorname {PolyLog}\left (2,-e^{i x}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i x}\right )dx\right )-2 \left (i x \operatorname {PolyLog}\left (2,e^{i x}\right )-i \int \operatorname {PolyLog}\left (2,e^{i x}\right )dx\right )-2 x^2 \text {arctanh}\left (e^{i x}\right )\right )}{\sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\sec (x) \left (2 \left (i x \operatorname {PolyLog}\left (2,-e^{i x}\right )-\int e^{-i x} \operatorname {PolyLog}\left (2,-e^{i x}\right )de^{i x}\right )-2 \left (i x \operatorname {PolyLog}\left (2,e^{i x}\right )-\int e^{-i x} \operatorname {PolyLog}\left (2,e^{i x}\right )de^{i x}\right )-2 x^2 \text {arctanh}\left (e^{i x}\right )\right )}{\sqrt {a \sec ^2(x)}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\sec (x) \left (-2 x^2 \text {arctanh}\left (e^{i x}\right )+2 \left (i x \operatorname {PolyLog}\left (2,-e^{i x}\right )-\operatorname {PolyLog}\left (3,-e^{i x}\right )\right )-2 \left (i x \operatorname {PolyLog}\left (2,e^{i x}\right )-\operatorname {PolyLog}\left (3,e^{i x}\right )\right )\right )}{\sqrt {a \sec ^2(x)}}\) |
((-2*x^2*ArcTanh[E^(I*x)] + 2*(I*x*PolyLog[2, -E^(I*x)] - PolyLog[3, -E^(I *x)]) - 2*(I*x*PolyLog[2, E^(I*x)] - PolyLog[3, E^(I*x)]))*Sec[x])/Sqrt[a* Sec[x]^2]
3.9.69.3.1 Defintions of rubi rules used
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[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[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[(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 = 0.94 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{i x} \left (\frac {x^{2} \ln \left ({\mathrm e}^{i x}+1\right )}{2}-i x \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )+\operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )-\frac {x^{2} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+i x \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )-\operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}\) | \(106\) |
-2/(a*exp(2*I*x)/(exp(2*I*x)+1)^2)^(1/2)/(exp(2*I*x)+1)*exp(I*x)*(1/2*x^2* ln(exp(I*x)+1)-I*x*polylog(2,-exp(I*x))+polylog(3,-exp(I*x))-1/2*x^2*ln(1- exp(I*x))+I*x*polylog(2,exp(I*x))-polylog(3,exp(I*x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (97) = 194\).
Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.77 \[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\frac {2 \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 2 \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - 2 \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 2 \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - {\left (x^{2} \cos \left (x\right ) \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{2} \cos \left (x\right ) \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - x^{2} \cos \left (x\right ) \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - x^{2} \cos \left (x\right ) \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + 2 i \, x \cos \left (x\right ) {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 2 i \, x \cos \left (x\right ) {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 i \, x \cos \left (x\right ) {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 2 i \, x \cos \left (x\right ) {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{2}}}}{2 \, a} \]
1/2*(2*sqrt(a/cos(x)^2)*cos(x)*polylog(3, cos(x) + I*sin(x)) + 2*sqrt(a/co s(x)^2)*cos(x)*polylog(3, cos(x) - I*sin(x)) - 2*sqrt(a/cos(x)^2)*cos(x)*p olylog(3, -cos(x) + I*sin(x)) - 2*sqrt(a/cos(x)^2)*cos(x)*polylog(3, -cos( x) - I*sin(x)) - (x^2*cos(x)*log(cos(x) + I*sin(x) + 1) + x^2*cos(x)*log(c os(x) - I*sin(x) + 1) - x^2*cos(x)*log(-cos(x) + I*sin(x) + 1) - x^2*cos(x )*log(-cos(x) - I*sin(x) + 1) + 2*I*x*cos(x)*dilog(cos(x) + I*sin(x)) - 2* I*x*cos(x)*dilog(cos(x) - I*sin(x)) + 2*I*x*cos(x)*dilog(-cos(x) + I*sin(x )) - 2*I*x*cos(x)*dilog(-cos(x) - I*sin(x)))*sqrt(a/cos(x)^2))/a
\[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int \frac {x^{2} \csc {\left (x \right )} \sec {\left (x \right )}}{\sqrt {a \sec ^{2}{\left (x \right )}}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {2 i \, x^{2} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + 2 i \, x^{2} \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x^{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - x^{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 4 i \, x {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + 4 i \, x {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 4 \, {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 4 \, {\rm Li}_{3}(e^{\left (i \, x\right )})}{2 \, \sqrt {a}} \]
-1/2*(2*I*x^2*arctan2(sin(x), cos(x) + 1) + 2*I*x^2*arctan2(sin(x), -cos(x ) + 1) + x^2*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) - x^2*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 4*I*x*dilog(-e^(I*x)) + 4*I*x*dilog(e^(I*x)) + 4*polylog(3, -e^(I*x)) - 4*polylog(3, e^(I*x)))/sqrt(a)
\[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int { \frac {x^{2} \csc \left (x\right ) \sec \left (x\right )}{\sqrt {a \sec \left (x\right )^{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int \frac {x^2}{\cos \left (x\right )\,\sin \left (x\right )\,\sqrt {\frac {a}{{\cos \left (x\right )}^2}}} \,d x \]