Integrand size = 16, antiderivative size = 142 \[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\frac {1}{2} x \cos ^2(x) \sqrt {a \sec ^4(x)}-2 x \text {arctanh}\left (e^{2 i x}\right ) \cos ^2(x) \sqrt {a \sec ^4(x)}+\frac {1}{2} i \cos ^2(x) \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} i \cos ^2(x) \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \sqrt {a \sec ^4(x)}-\frac {1}{2} \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {1}{2} x \sqrt {a \sec ^4(x)} \sin ^2(x) \] Output:
1/2*x*cos(x)^2*(a*sec(x)^4)^(1/2)-2*x*arctanh(exp(2*I*x))*cos(x)^2*(a*sec( x)^4)^(1/2)+1/2*I*cos(x)^2*polylog(2,-exp(2*I*x))*(a*sec(x)^4)^(1/2)-1/2*I *cos(x)^2*polylog(2,exp(2*I*x))*(a*sec(x)^4)^(1/2)-1/2*cos(x)*(a*sec(x)^4) ^(1/2)*sin(x)+1/2*x*(a*sec(x)^4)^(1/2)*sin(x)^2
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.60 \[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\frac {1}{2} \cos ^2(x) \sqrt {a \sec ^4(x)} \left (2 x \log \left (1-e^{2 i x}\right )-2 x \log \left (1+e^{2 i x}\right )+i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+x \sec ^2(x)-\tan (x)\right ) \] Input:
Integrate[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]
Output:
(Cos[x]^2*Sqrt[a*Sec[x]^4]*(2*x*Log[1 - E^((2*I)*x)] - 2*x*Log[1 + E^((2*I )*x)] + I*PolyLog[2, -E^((2*I)*x)] - I*PolyLog[2, E^((2*I)*x)] + x*Sec[x]^ 2 - Tan[x]))/2
Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.56, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {7271, 4920, 2009}
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.
\(\displaystyle \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cos ^2(x) \sqrt {a \sec ^4(x)} \int x \csc (x) \sec ^3(x)dx\) |
\(\Big \downarrow \) 4920 |
\(\displaystyle \cos ^2(x) \sqrt {a \sec ^4(x)} \left (-\int \left (\frac {\tan ^2(x)}{2}+\log (\tan (x))\right )dx+\frac {1}{2} x \tan ^2(x)+x \log (\tan (x))\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \cos ^2(x) \sqrt {a \sec ^4(x)} \left (-2 x \text {arctanh}\left (e^{2 i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {x}{2}+\frac {1}{2} x \tan ^2(x)-\frac {\tan (x)}{2}\right )\) |
Input:
Int[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]
Output:
Cos[x]^2*Sqrt[a*Sec[x]^4]*(x/2 - 2*x*ArcTanh[E^((2*I)*x)] + (I/2)*PolyLog[ 2, -E^((2*I)*x)] - (I/2)*PolyLog[2, E^((2*I)*x)] - Tan[x]/2 + (x*Tan[x]^2) /2)
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(p_.), x_Symbol] :> Module[{u = IntHide[Csc[a + b*x]^n*Sec[a + b* x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1)*u, x ], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
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 = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{4}}}\, \left (-i+2 x -i {\mathrm e}^{-2 i x}\right )-4 i \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{4}}}\, {\mathrm e}^{-2 i x} \left (1+{\mathrm e}^{2 i x}\right )^{2} \left (-\frac {i x \ln \left (1+{\mathrm e}^{2 i x}\right )}{4}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )}{8}+\frac {i x \ln \left (1-{\mathrm e}^{i x}\right )}{4}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )}{4}+\frac {i x \ln \left ({\mathrm e}^{i x}+1\right )}{4}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )}{4}\right )\) | \(140\) |
Input:
int(x*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
(a*exp(4*I*x)/(1+exp(2*I*x))^4)^(1/2)*(-I+2*x-I*exp(-2*I*x))-4*I*(a*exp(4* I*x)/(1+exp(2*I*x))^4)^(1/2)*exp(-2*I*x)*(1+exp(2*I*x))^2*(-1/4*I*x*ln(1+e xp(2*I*x))-1/8*polylog(2,-exp(2*I*x))+1/4*I*x*ln(1-exp(I*x))+1/4*polylog(2 ,exp(I*x))+1/4*I*x*ln(exp(I*x)+1)+1/4*polylog(2,-exp(I*x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (105) = 210\).
Time = 0.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.90 \[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\frac {1}{2} \, {\left (x \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) - x \cos \left (x\right )^{2} \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) - x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) - x \cos \left (x\right )^{2} \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) + i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - i \, \cos \left (x\right )^{2} {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \cos \left (x\right ) \sin \left (x\right ) + x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4}}} \] Input:
integrate(x*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x, algorithm="fricas")
Output:
1/2*(x*cos(x)^2*log(cos(x) + I*sin(x) + 1) + x*cos(x)^2*log(cos(x) - I*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) - x*cos(x)^2*log(- I*cos(x) - sin(x) + 1) + x*cos(x)^2*log(-cos(x) + I*sin(x) + 1) + x*cos(x) ^2*log(-cos(x) - I*sin(x) + 1) - I*cos(x)^2*dilog(cos(x) + I*sin(x)) + I*c os(x)^2*dilog(cos(x) - I*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*dilog(-I*cos(x) + sin(x)) - I*cos(x)^2*dilog(-I*cos(x) - sin(x)) + I*cos(x)^2*dilog(-cos(x) + I*sin( x)) - I*cos(x)^2*dilog(-cos(x) - I*sin(x)) - cos(x)*sin(x) + x)*sqrt(a/cos (x)^4)
\[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\int x \sqrt {a \sec ^{4}{\left (x \right )}} \csc {\left (x \right )} \sec {\left (x \right )}\, dx \] Input:
integrate(x*csc(x)*sec(x)*(a*sec(x)**4)**(1/2),x)
Output:
Integral(x*sqrt(a*sec(x)**4)*csc(x)*sec(x), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (105) = 210\).
Time = 0.16 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.98 \[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx =\text {Too large to display} \] Input:
integrate(x*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x, algorithm="maxima")
Output:
-(2*(x*cos(4*x) + 2*x*cos(2*x) + I*x*sin(4*x) + 2*I*x*sin(2*x) + x)*arctan 2(sin(2*x), cos(2*x) + 1) - 2*(x*cos(4*x) + 2*x*cos(2*x) + I*x*sin(4*x) + 2*I*x*sin(2*x) + x)*arctan2(sin(x), cos(x) + 1) + 2*(x*cos(4*x) + 2*x*cos( 2*x) + I*x*sin(4*x) + 2*I*x*sin(2*x) + x)*arctan2(sin(x), -cos(x) + 1) - 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)) + 2*(cos(4*x) + 2*cos(2*x) + I*sin(4*x) + 2*I*sin (2*x) + 1)*dilog(-e^(I*x)) + 2*(cos(4*x) + 2*cos(2*x) + I*sin(4*x) + 2*I*s in(2*x) + 1)*dilog(e^(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) + (I* x*cos(4*x) + 2*I*x*cos(2*x) - x*sin(4*x) - 2*x*sin(2*x) + I*x)*log(cos(x)^ 2 + sin(x)^2 + 2*cos(x) + 1) + (I*x*cos(4*x) + 2*I*x*cos(2*x) - x*sin(4*x) - 2*x*sin(2*x) + I*x)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) - 2*(2*x - I)*sin(2*x) + 2)*sqrt(a)/(-2*I*cos(4*x) - 4*I*cos(2*x) + 2*sin(4*x) + 4*si n(2*x) - 2*I)
\[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\int { \sqrt {a \sec \left (x\right )^{4}} x \csc \left (x\right ) \sec \left (x\right ) \,d x } \] Input:
integrate(x*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*sec(x)^4)*x*csc(x)*sec(x), x)
Timed out. \[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\int \frac {x\,\sqrt {\frac {a}{{\cos \left (x\right )}^4}}}{\cos \left (x\right )\,\sin \left (x\right )} \,d x \] Input:
int((x*(a/cos(x)^4)^(1/2))/(cos(x)*sin(x)),x)
Output:
int((x*(a/cos(x)^4)^(1/2))/(cos(x)*sin(x)), x)
\[ \int x \csc (x) \sec (x) \sqrt {a \sec ^4(x)} \, dx=\sqrt {a}\, \left (\int \csc \left (x \right ) \sec \left (x \right )^{3} x d x \right ) \] Input:
int(x*csc(x)*sec(x)*(a*sec(x)^4)^(1/2),x)
Output:
sqrt(a)*int(csc(x)*sec(x)**3*x,x)