Integrand size = 18, antiderivative size = 126 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=-\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b} \]
-3*x*arctanh(exp(I*(b*x+a)))/b-arctanh(sin(b*x+a))/b^2-1/2*csc(b*x+a)/b^2+ 3/2*I*polylog(2,-exp(I*(b*x+a)))/b^2-3/2*I*polylog(2,exp(I*(b*x+a)))/b^2+3 /2*x*sec(b*x+a)/b-1/2*x*csc(b*x+a)^2*sec(b*x+a)/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(282\) vs. \(2(126)=252\).
Time = 3.69 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.24 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {8 b x-2 \cot \left (\frac {1}{2} (a+b x)\right )-b x \csc ^2\left (\frac {1}{2} (a+b x)\right )+12 (a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )+8 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )-8 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )-12 a \log \left (\tan \left (\frac {1}{2} (a+b x)\right )\right )+12 i \left (\operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )\right )+b x \sec ^2\left (\frac {1}{2} (a+b x)\right )+\frac {8 b x \sin \left (\frac {1}{2} (a+b x)\right )}{\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )}-\frac {8 b x \sin \left (\frac {1}{2} (a+b x)\right )}{\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )}-2 \tan \left (\frac {1}{2} (a+b x)\right )}{8 b^2} \]
(8*b*x - 2*Cot[(a + b*x)/2] - b*x*Csc[(a + b*x)/2]^2 + 12*(a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + 8*Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - 8*Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]] - 12*a*Log [Tan[(a + b*x)/2]] + (12*I)*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^( I*(a + b*x))]) + b*x*Sec[(a + b*x)/2]^2 + (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*x)/2] - Sin[(a + b*x)/2]) - (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*x)/2 ] + Sin[(a + b*x)/2]) - 2*Tan[(a + b*x)/2])/(8*b^2)
Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
| \(\displaystyle \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4920 |
| \(\displaystyle -\int \left (-\frac {\sec (a+b x) \csc ^2(a+b x)}{2 b}-\frac {3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 \sec (a+b x)}{2 b}\right )dx-\frac {3 x \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
(-3*x*ArcTanh[E^(I*(a + b*x))])/b - ArcTanh[Sin[a + b*x]]/b^2 - Csc[a + b* x]/(2*b^2) + (((3*I)/2)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (((3*I)/2)*Pol yLog[2, E^(I*(a + b*x))])/b^2 + (3*x*Sec[a + b*x])/(2*b) - (x*Csc[a + b*x] ^2*Sec[a + b*x])/(2*b)
3.3.87.3.1 Defintions of rubi rules used
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (108 ) = 216\).
Time = 0.70 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.12
| method | result | size |
| risch | \(\frac {2 i {\mathrm e}^{i \left (x b +a \right )}+6 x b \,{\mathrm e}^{5 i \left (x b +a \right )}-4 x b \,{\mathrm e}^{3 i \left (x b +a \right )}+6 x b \,{\mathrm e}^{i \left (x b +a \right )}-3 i {\mathrm e}^{4 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{4 i \left (x b +a \right )}-3 i {\mathrm e}^{4 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i {\mathrm e}^{6 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i {\mathrm e}^{6 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )+4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{6 i \left (x b +a \right )}+3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{4 i \left (x b +a \right )} a -3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{6 i \left (x b +a \right )} a +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{2 i \left (x b +a \right )} a -4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{2 i \left (x b +a \right )}-3 i {\mathrm e}^{2 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )-3 i {\mathrm e}^{2 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) b x +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{2 i \left (x b +a \right )} b x -3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{6 i \left (x b +a \right )} b x +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{4 i \left (x b +a \right )} b x -2 i {\mathrm e}^{5 i \left (x b +a \right )}+4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-3 a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}\) | \(519\) |
1/2*(4*I*arctan(exp(I*(b*x+a)))+3*I*dilog(exp(I*(b*x+a)))+3*I*dilog(exp(I* (b*x+a))+1)-3*a*ln(exp(I*(b*x+a))-1)-3*I*exp(4*I*(b*x+a))*dilog(exp(I*(b*x +a))+1)-4*I*arctan(exp(I*(b*x+a)))*exp(4*I*(b*x+a))-3*I*exp(4*I*(b*x+a))*d ilog(exp(I*(b*x+a)))+3*I*exp(6*I*(b*x+a))*dilog(exp(I*(b*x+a)))+3*I*exp(6* I*(b*x+a))*dilog(exp(I*(b*x+a))+1)+4*I*arctan(exp(I*(b*x+a)))*exp(6*I*(b*x +a))+3*ln(exp(I*(b*x+a))-1)*exp(4*I*(b*x+a))*a-3*ln(exp(I*(b*x+a))-1)*exp( 6*I*(b*x+a))*a+3*ln(exp(I*(b*x+a))-1)*exp(2*I*(b*x+a))*a-4*I*arctan(exp(I* (b*x+a)))*exp(2*I*(b*x+a))-3*I*exp(2*I*(b*x+a))*dilog(exp(I*(b*x+a)))-3*I* exp(2*I*(b*x+a))*dilog(exp(I*(b*x+a))+1)-3*ln(exp(I*(b*x+a))+1)*b*x-2*I*ex p(5*I*(b*x+a))+2*I*exp(I*(b*x+a))+3*ln(exp(I*(b*x+a))+1)*exp(2*I*(b*x+a))* b*x-3*ln(exp(I*(b*x+a))+1)*exp(6*I*(b*x+a))*b*x+3*ln(exp(I*(b*x+a))+1)*exp (4*I*(b*x+a))*b*x+6*x*b*exp(5*I*(b*x+a))-4*x*b*exp(3*I*(b*x+a))+6*x*b*exp( I*(b*x+a)))/b^2/(exp(2*I*(b*x+a))-1)^2/(exp(2*I*(b*x+a))+1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (104) = 208\).
Time = 0.26 (sec) , antiderivative size = 531, normalized size of antiderivative = 4.21 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {6 \, b x \cos \left (b x + a\right )^{2} - 4 \, b x - 3 \, {\left (i \, \cos \left (b x + a\right )^{3} - i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, \cos \left (b x + a\right )^{3} + i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (i \, \cos \left (b x + a\right )^{3} - i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, \cos \left (b x + a\right )^{3} + i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (b x \cos \left (b x + a\right )^{3} - b x \cos \left (b x + a\right )\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b x \cos \left (b x + a\right )^{3} - b x \cos \left (b x + a\right )\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (a \cos \left (b x + a\right )^{3} - a \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, {\left (a \cos \left (b x + a\right )^{3} - a \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right )^{3} - {\left (b x + a\right )} \cos \left (b x + a\right )\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right )^{3} - {\left (b x + a\right )} \cos \left (b x + a\right )\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right )}{4 \, {\left (b^{2} \cos \left (b x + a\right )^{3} - b^{2} \cos \left (b x + a\right )\right )}} \]
1/4*(6*b*x*cos(b*x + a)^2 - 4*b*x - 3*(I*cos(b*x + a)^3 - I*cos(b*x + a))* dilog(cos(b*x + a) + I*sin(b*x + a)) - 3*(-I*cos(b*x + a)^3 + I*cos(b*x + a))*dilog(cos(b*x + a) - I*sin(b*x + a)) - 3*(I*cos(b*x + a)^3 - I*cos(b*x + a))*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 3*(-I*cos(b*x + a)^3 + I*co s(b*x + a))*dilog(-cos(b*x + a) - I*sin(b*x + a)) - 3*(b*x*cos(b*x + a)^3 - b*x*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x + a) + 1) - 3*(b*x*cos(b* x + a)^3 - b*x*cos(b*x + a))*log(cos(b*x + a) - I*sin(b*x + a) + 1) - 3*(a *cos(b*x + a)^3 - a*cos(b*x + a))*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - 3*(a*cos(b*x + a)^3 - a*cos(b*x + a))*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) + 3*((b*x + a)*cos(b*x + a)^3 - (b*x + a)*cos(b* x + a))*log(-cos(b*x + a) + I*sin(b*x + a) + 1) + 3*((b*x + a)*cos(b*x + a )^3 - (b*x + a)*cos(b*x + a))*log(-cos(b*x + a) - I*sin(b*x + a) + 1) - 2* (cos(b*x + a)^3 - cos(b*x + a))*log(sin(b*x + a) + 1) + 2*(cos(b*x + a)^3 - cos(b*x + a))*log(-sin(b*x + a) + 1) + 2*cos(b*x + a)*sin(b*x + a))/(b^2 *cos(b*x + a)^3 - b^2*cos(b*x + a))
\[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int x \csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (104) = 208\).
Time = 0.49 (sec) , antiderivative size = 1173, normalized size of antiderivative = 9.31 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]
(8*I*b*x*cos(3*b*x + 3*a) - 8*b*x*sin(3*b*x + 3*a) - 4*(cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + I*sin(6*b*x + 6*a) - I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a) + 1)*arctan2(2*(cos(b*x + 2*a)*cos(a) + sin(b*x + 2*a)*sin(a))/(cos(b*x + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b*x + 2*a) + si n(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin(a) + sin(a)^2), (cos(b*x + 2*a)^2 - cos(a)^2 + sin(b*x + 2*a)^2 - sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)^2 + 2*c os(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin(a) + sin(a) ^2)) - 6*(b*x*cos(6*b*x + 6*a) - b*x*cos(4*b*x + 4*a) - b*x*cos(2*b*x + 2* a) + I*b*x*sin(6*b*x + 6*a) - I*b*x*sin(4*b*x + 4*a) - I*b*x*sin(2*b*x + 2 *a) + b*x)*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 6*(b*x*cos(6*b*x + 6* a) - b*x*cos(4*b*x + 4*a) - b*x*cos(2*b*x + 2*a) + I*b*x*sin(6*b*x + 6*a) - I*b*x*sin(4*b*x + 4*a) - I*b*x*sin(2*b*x + 2*a) + b*x)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(3*I*b*x + 1)*cos(5*b*x + 5*a) - 4*(3*I*b*x - 1)*cos(b*x + a) + 6*(cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a ) + I*sin(6*b*x + 6*a) - I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a) + 1)*dilo g(-e^(I*b*x + I*a)) - 6*(cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + I*sin(6*b*x + 6*a) - I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a) + 1)* dilog(e^(I*b*x + I*a)) - 3*(-I*b*x*cos(6*b*x + 6*a) + I*b*x*cos(4*b*x + 4* a) + I*b*x*cos(2*b*x + 2*a) + b*x*sin(6*b*x + 6*a) - b*x*sin(4*b*x + 4*a) - b*x*sin(2*b*x + 2*a) - I*b*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2...
\[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { x \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int \frac {x}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]