Integrand size = 24, antiderivative size = 305 \[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {4 i d^2 x \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {3 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {2 c d \text {arctanh}(\sin (a+b x))}{b^2}-\frac {c d \csc (a+b x)}{b^2}-\frac {d^2 x \csc (a+b x)}{b^2}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b} \]
4*I*d^2*x*arctan(exp(I*(b*x+a)))/b^2-3*(d*x+c)^2*arctanh(exp(I*(b*x+a)))/b -d^2*arctanh(cos(b*x+a))/b^3-2*c*d*arctanh(sin(b*x+a))/b^2-c*d*csc(b*x+a)/ b^2-d^2*x*csc(b*x+a)/b^2+3*I*d*(d*x+c)*polylog(2,-exp(I*(b*x+a)))/b^2-2*I* d^2*polylog(2,-I*exp(I*(b*x+a)))/b^3+2*I*d^2*polylog(2,I*exp(I*(b*x+a)))/b ^3-3*I*d*(d*x+c)*polylog(2,exp(I*(b*x+a)))/b^2-3*d^2*polylog(3,-exp(I*(b*x +a)))/b^3+3*d^2*polylog(3,exp(I*(b*x+a)))/b^3+3/2*(d*x+c)^2*sec(b*x+a)/b-1 /2*(d*x+c)^2*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. \(889\) vs. \(2(305)=610\).
Time = 7.93 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.91 \[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \csc ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {3 b^2 c^2 \log \left (1-e^{i (a+b x)}\right )+2 d^2 \log \left (1-e^{i (a+b x)}\right )+6 b^2 c d x \log \left (1-e^{i (a+b x)}\right )+3 b^2 d^2 x^2 \log \left (1-e^{i (a+b x)}\right )-3 b^2 c^2 \log \left (1+e^{i (a+b x)}\right )-2 d^2 \log \left (1+e^{i (a+b x)}\right )-6 b^2 c d x \log \left (1+e^{i (a+b x)}\right )-3 b^2 d^2 x^2 \log \left (1+e^{i (a+b x)}\right )+6 i b d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-6 i b d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )-6 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )+6 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{2 b^3}+\frac {\left (c^2+2 c d x+d^2 x^2\right ) \sec ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {(c+d x) \csc (a) \sec (a) (-d \cos (a)+b c \sin (a)+b d x \sin (a))}{b^2}-\frac {4 i c d \arctan \left (\frac {-i \sin (a)-i \cos (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{b^2 \sqrt {\cos ^2(a)+\sin ^2(a)}}-\frac {2 d^2 \left (-\frac {\csc (a) \left ((b x-\arctan (\cot (a))) \left (\log \left (1-e^{i (b x-\arctan (\cot (a)))}\right )-\log \left (1+e^{i (b x-\arctan (\cot (a)))}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (b x-\arctan (\cot (a)))}\right )-\operatorname {PolyLog}\left (2,e^{i (b x-\arctan (\cot (a)))}\right )\right )\right )}{\sqrt {1+\cot ^2(a)}}+\frac {2 \arctan (\cot (a)) \text {arctanh}\left (\frac {\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{b^3}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c d \sin \left (\frac {b x}{2}\right )-d^2 x \sin \left (\frac {b x}{2}\right )\right )}{2 b^2}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sin \left (\frac {b x}{2}\right )+d^2 x \sin \left (\frac {b x}{2}\right )\right )}{2 b^2}+\frac {c^2 \sin \left (\frac {b x}{2}\right )+2 c d x \sin \left (\frac {b x}{2}\right )+d^2 x^2 \sin \left (\frac {b x}{2}\right )}{b \left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {-c^2 \sin \left (\frac {b x}{2}\right )-2 c d x \sin \left (\frac {b x}{2}\right )-d^2 x^2 \sin \left (\frac {b x}{2}\right )}{b \left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )} \]
((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a/2 + (b*x)/2]^2)/(8*b) + (3*b^2*c^2*Log[1 - E^(I*(a + b*x))] + 2*d^2*Log[1 - E^(I*(a + b*x))] + 6*b^2*c*d*x*Log[1 - E^(I*(a + b*x))] + 3*b^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] - 3*b^2*c^2*Log [1 + E^(I*(a + b*x))] - 2*d^2*Log[1 + E^(I*(a + b*x))] - 6*b^2*c*d*x*Log[1 + E^(I*(a + b*x))] - 3*b^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] + (6*I)*b*d*( c + d*x)*PolyLog[2, -E^(I*(a + b*x))] - (6*I)*b*d*(c + d*x)*PolyLog[2, E^( I*(a + b*x))] - 6*d^2*PolyLog[3, -E^(I*(a + b*x))] + 6*d^2*PolyLog[3, E^(I *(a + b*x))])/(2*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a/2 + (b*x)/2]^2)/( 8*b) + ((c + d*x)*Csc[a]*Sec[a]*(-(d*Cos[a]) + b*c*Sin[a] + b*d*x*Sin[a])) /b^2 - ((4*I)*c*d*ArcTan[((-I)*Sin[a] - I*Cos[a]*Tan[(b*x)/2])/Sqrt[Cos[a] ^2 + Sin[a]^2]])/(b^2*Sqrt[Cos[a]^2 + Sin[a]^2]) - (2*d^2*(-((Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*( b*x - ArcTan[Cot[a]]))]) + I*(PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))])))/Sqrt[1 + Cot[a]^2]) + (2*ArcTa n[Cot[a]]*ArcTanh[(Sin[a] + Cos[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2] ])/Sqrt[Cos[a]^2 + Sin[a]^2]))/b^3 + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c*d*S in[(b*x)/2]) - d^2*x*Sin[(b*x)/2]))/(2*b^2) + (Csc[a/2]*Csc[a/2 + (b*x)/2] *(c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2]))/(2*b^2) + (c^2*Sin[(b*x)/2] + 2* c*d*x*Sin[(b*x)/2] + d^2*x^2*Sin[(b*x)/2])/(b*(Cos[a/2] - Sin[a/2])*(Cos[a /2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-(c^2*Sin[(b*x)/2]) - 2*c*d*x*S...
Time = 1.14 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4920, 27, 7292, 27, 7293, 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.
| \(\displaystyle \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4920 |
| \(\displaystyle -2 d \int -\frac {1}{2} (c+d x) \left (\frac {\sec (a+b x) \csc ^2(a+b x)}{b}+\frac {3 \text {arctanh}(\cos (a+b x))}{b}-\frac {3 \sec (a+b x)}{b}\right )dx-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \int (c+d x) \left (\frac {\sec (a+b x) \csc ^2(a+b x)}{b}+\frac {3 \text {arctanh}(\cos (a+b x))}{b}-\frac {3 \sec (a+b x)}{b}\right )dx-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle d \int \frac {(c+d x) \left (\sec (a+b x) \csc ^2(a+b x)+3 \text {arctanh}(\cos (a+b x))-3 \sec (a+b x)\right )}{b}dx-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int (c+d x) \left (\sec (a+b x) \csc ^2(a+b x)+3 \text {arctanh}(\cos (a+b x))-3 \sec (a+b x)\right )dx}{b}-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {d \int \left (3 (c+d x) \text {arctanh}(\cos (a+b x))+(c+d x) \left (\csc ^2(a+b x)-3\right ) \sec (a+b x)\right )dx}{b}-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (\frac {4 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}-\frac {3 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{d}+\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 d}-\frac {2 c \text {arctanh}(\sin (a+b x))}{b}-\frac {2 i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {2 i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {3 d \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^2}+\frac {3 d \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^2}+\frac {3 i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b}-\frac {3 i (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b}-\frac {c \csc (a+b x)}{b}-\frac {d x \csc (a+b x)}{b}\right )}{b}-\frac {3 (c+d x)^2 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac {(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}\) |
(-3*(c + d*x)^2*ArcTanh[Cos[a + b*x]])/(2*b) + (d*(((4*I)*d*x*ArcTan[E^(I* (a + b*x))])/b - (3*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/d - (d*ArcTanh[C os[a + b*x]])/b^2 + (3*(c + d*x)^2*ArcTanh[Cos[a + b*x]])/(2*d) - (2*c*Arc Tanh[Sin[a + b*x]])/b - (c*Csc[a + b*x])/b - (d*x*Csc[a + b*x])/b + ((3*I) *(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b - ((2*I)*d*PolyLog[2, (-I)*E^(I *(a + b*x))])/b^2 + ((2*I)*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - ((3*I)*( c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b - (3*d*PolyLog[3, -E^(I*(a + b*x)) ])/b^2 + (3*d*PolyLog[3, E^(I*(a + b*x))])/b^2))/b + (3*(c + d*x)^2*Sec[a + b*x])/(2*b) - ((c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)
3.3.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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. 801 vs. \(2 (278 ) = 556\).
Time = 2.01 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.63
| method | result | size |
| risch | \(-\frac {3 i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {3 i d c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {3 d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {3 d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {3 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{2 b}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{2 b}+\frac {3 d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b^{3}}+\frac {3 d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}+\frac {3 c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{3}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 i d^{2} \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{2 b^{3}}-\frac {3 d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b}+\frac {3 x^{2} d^{2} b \,{\mathrm e}^{5 i \left (x b +a \right )}+6 c d x b \,{\mathrm e}^{5 i \left (x b +a \right )}+3 c^{2} b \,{\mathrm e}^{5 i \left (x b +a \right )}-2 x^{2} d^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}-4 c d x b \,{\mathrm e}^{3 i \left (x b +a \right )}-2 i d^{2} x \,{\mathrm e}^{5 i \left (x b +a \right )}-2 c^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+3 x^{2} d^{2} b \,{\mathrm e}^{i \left (x b +a \right )}-2 i c d \,{\mathrm e}^{5 i \left (x b +a \right )}+6 c d x b \,{\mathrm e}^{i \left (x b +a \right )}+3 c^{2} b \,{\mathrm e}^{i \left (x b +a \right )}+2 i d^{2} x \,{\mathrm e}^{i \left (x b +a \right )}+2 i d c \,{\mathrm e}^{i \left (x b +a \right )}}{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 )}+\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}+\frac {2 i d^{2} \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {4 i d^{2} a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 c d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}+\frac {3 i c d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {3 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {4 i d c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}\) | \(802\) |
3/2/b^3*d^2*a^2*ln(exp(I*(b*x+a))-1)+4*I/b^2*d*c*arctan(exp(I*(b*x+a)))-4* I/b^3*d^2*a*arctan(exp(I*(b*x+a)))-3/2/b*c^2*ln(exp(I*(b*x+a))+1)+3/2/b*c^ 2*ln(exp(I*(b*x+a))-1)-3/2/b^3*d^2*ln(1-exp(I*(b*x+a)))*a^2+3/2/b*d^2*ln(1 -exp(I*(b*x+a)))*x^2-3/2/b*d^2*ln(exp(I*(b*x+a))+1)*x^2+2*I/b^3*d^2*dilog( 1-I*exp(I*(b*x+a)))-3/b*d*c*ln(exp(I*(b*x+a))+1)*x+3/b^2*d*c*ln(1-exp(I*(b *x+a)))*a-3/b^2*c*d*a*ln(exp(I*(b*x+a))-1)+3/b*d*c*ln(1-exp(I*(b*x+a)))*x+ 2/b^2*d^2*ln(1+I*exp(I*(b*x+a)))*x+2/b^3*d^2*ln(1+I*exp(I*(b*x+a)))*a-2/b^ 2*d^2*ln(1-I*exp(I*(b*x+a)))*x-2/b^3*d^2*ln(1-I*exp(I*(b*x+a)))*a-2*I/b^3* d^2*dilog(1+I*exp(I*(b*x+a)))+1/b^2/(exp(2*I*(b*x+a))-1)^2/(exp(2*I*(b*x+a ))+1)*(3*x^2*d^2*b*exp(5*I*(b*x+a))+6*c*d*x*b*exp(5*I*(b*x+a))+3*c^2*b*exp (5*I*(b*x+a))-2*x^2*d^2*b*exp(3*I*(b*x+a))-4*c*d*x*b*exp(3*I*(b*x+a))-2*I* d^2*x*exp(5*I*(b*x+a))-2*c^2*b*exp(3*I*(b*x+a))+3*x^2*d^2*b*exp(I*(b*x+a)) -2*I*c*d*exp(5*I*(b*x+a))+6*c*d*x*b*exp(I*(b*x+a))+3*c^2*b*exp(I*(b*x+a))+ 2*I*d^2*x*exp(I*(b*x+a))+2*I*d*c*exp(I*(b*x+a)))-3*d^2*polylog(3,-exp(I*(b *x+a)))/b^3+3*d^2*polylog(3,exp(I*(b*x+a)))/b^3-1/b^3*d^2*ln(exp(I*(b*x+a) )+1)+1/b^3*d^2*ln(exp(I*(b*x+a))-1)-3*I/b^2*c*d*polylog(2,exp(I*(b*x+a)))+ 3*I/b^2*c*d*polylog(2,-exp(I*(b*x+a)))+3*I/b^2*d^2*polylog(2,-exp(I*(b*x+a )))*x-3*I/b^2*d^2*polylog(2,exp(I*(b*x+a)))*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1801 vs. \(2 (267) = 534\).
Time = 0.34 (sec) , antiderivative size = 1801, normalized size of antiderivative = 5.90 \[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]
-1/4*(4*b^2*d^2*x^2 + 8*b^2*c*d*x + 4*b^2*c^2 - 6*(b^2*d^2*x^2 + 2*b^2*c*d *x + b^2*c^2)*cos(b*x + a)^2 - 4*(b*d^2*x + b*c*d)*cos(b*x + a)*sin(b*x + a) + 6*((I*b*d^2*x + I*b*c*d)*cos(b*x + a)^3 + (-I*b*d^2*x - I*b*c*d)*cos( b*x + a))*dilog(cos(b*x + a) + I*sin(b*x + a)) + 6*((-I*b*d^2*x - I*b*c*d) *cos(b*x + a)^3 + (I*b*d^2*x + I*b*c*d)*cos(b*x + a))*dilog(cos(b*x + a) - I*sin(b*x + a)) + 4*(-I*d^2*cos(b*x + a)^3 + I*d^2*cos(b*x + a))*dilog(I* cos(b*x + a) + sin(b*x + a)) + 4*(-I*d^2*cos(b*x + a)^3 + I*d^2*cos(b*x + a))*dilog(I*cos(b*x + a) - sin(b*x + a)) + 4*(I*d^2*cos(b*x + a)^3 - I*d^2 *cos(b*x + a))*dilog(-I*cos(b*x + a) + sin(b*x + a)) + 4*(I*d^2*cos(b*x + a)^3 - I*d^2*cos(b*x + a))*dilog(-I*cos(b*x + a) - sin(b*x + a)) + 6*((I*b *d^2*x + I*b*c*d)*cos(b*x + a)^3 + (-I*b*d^2*x - I*b*c*d)*cos(b*x + a))*di log(-cos(b*x + a) + I*sin(b*x + a)) + 6*((-I*b*d^2*x - I*b*c*d)*cos(b*x + a)^3 + (I*b*d^2*x + I*b*c*d)*cos(b*x + a))*dilog(-cos(b*x + a) - I*sin(b*x + a)) + ((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*d^2)*cos(b*x + a)^3 - (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*d^2)*cos(b*x + a))*log(cos (b*x + a) + I*sin(b*x + a) + 1) + 4*((b*c*d - a*d^2)*cos(b*x + a)^3 - (b*c *d - a*d^2)*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x + a) + I) + ((3*b^2 *d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*d^2)*cos(b*x + a)^3 - (3*b^2*d^2*x^ 2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*d^2)*cos(b*x + a))*log(cos(b*x + a) - I*si n(b*x + a) + 1) - 4*((b*c*d - a*d^2)*cos(b*x + a)^3 - (b*c*d - a*d^2)*c...
\[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int \left (c + d x\right )^{2} \csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3814 vs. \(2 (267) = 534\).
Time = 0.99 (sec) , antiderivative size = 3814, normalized size of antiderivative = 12.50 \[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]
1/4*(c^2*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log (cos(b*x + a) + 1) + 3*log(cos(b*x + a) - 1)) - 2*a*c*d*(2*(3*cos(b*x + a) ^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log(cos(b*x + a) + 1) + 3*log( cos(b*x + a) - 1))/b + a^2*d^2*(2*(3*cos(b*x + a)^2 - 2)/(cos(b*x + a)^3 - cos(b*x + a)) - 3*log(cos(b*x + a) + 1) + 3*log(cos(b*x + a) - 1))/b^2 + 4*(8*(b*c*d + (b*x + a)*d^2 - a*d^2 + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos( 6*b*x + 6*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) + (I*b*c*d + I*(b*x + a)*d^2 - I* a*d^2)*sin(6*b*x + 6*a) + (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(4*b*x + 4*a) + (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(2*b*x + 2*a))*arctan2 (cos(b*x + a), sin(b*x + a) + 1) + 8*(b*c*d + (b*x + a)*d^2 - a*d^2 + (b*c *d + (b*x + a)*d^2 - a*d^2)*cos(6*b*x + 6*a) - (b*c*d + (b*x + a)*d^2 - a* d^2)*cos(4*b*x + 4*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) + (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(6*b*x + 6*a) + (-I*b*c*d - I*(b *x + a)*d^2 + I*a*d^2)*sin(4*b*x + 4*a) + (-I*b*c*d - I*(b*x + a)*d^2 + I* a*d^2)*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) - 2*(3*( b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2 + (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(6*b*x + 6*a) - (3*(b*x + a)^2*d ^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(4*b*x + 4*a) - (3*(b*x + a)^ 2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(2*b*x + 2*a) - (-3*I*(...
\[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Hanged} \]