Integrand size = 20, antiderivative size = 131 \[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}-\frac {d x \text {arctanh}(\sin (a+b x))}{b}+\frac {(c+d x) \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x) \csc (a+b x)}{b}+\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2} \]
-2*I*d*x*arctan(exp(I*(b*x+a)))/b-d*arctanh(cos(b*x+a))/b^2-d*x*arctanh(si n(b*x+a))/b+(d*x+c)*arctanh(sin(b*x+a))/b-(d*x+c)*csc(b*x+a)/b+I*d*polylog (2,-I*exp(I*(b*x+a)))/b^2-I*d*polylog(2,I*exp(I*(b*x+a)))/b^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.83 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.95 \[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=\frac {d \left (a \cos \left (\frac {1}{2} (a+b x)\right )-(a+b x) \cos \left (\frac {1}{2} (a+b x)\right )\right ) \csc \left (\frac {1}{2} (a+b x)\right )}{2 b^2}-\frac {c \csc (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\sin ^2(a+b x)\right )}{b}-\frac {d \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b^2}+\frac {d \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b^2}-\frac {d x \left (a \log \left (1-\tan \left (\frac {1}{2} (a+b x)\right )\right )-a \log \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )-i \left (\log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)-(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )+i \left (\log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )-i \left (\log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)+(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )+i \left (\log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1-i)+(1+i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )\right )}{b \left (a-i \log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right )+i \log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )}+\frac {d \sec \left (\frac {1}{2} (a+b x)\right ) \left (a \sin \left (\frac {1}{2} (a+b x)\right )-(a+b x) \sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b^2} \]
(d*(a*Cos[(a + b*x)/2] - (a + b*x)*Cos[(a + b*x)/2])*Csc[(a + b*x)/2])/(2* b^2) - (c*Csc[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[a + b*x]^2])/b - (d*Log[Cos[(a + b*x)/2]])/b^2 + (d*Log[Sin[(a + b*x)/2]])/b^2 - (d*x*(a* Log[1 - Tan[(a + b*x)/2]] - a*Log[1 + Tan[(a + b*x)/2]] - I*(Log[1 + I*Tan [(a + b*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(1 /2 + I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])]) - I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log [1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] + PolyL og[2, ((1 - I) + (1 + I)*Tan[(a + b*x)/2])/2])))/(b*(a - I*Log[1 - I*Tan[( a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]])) + (d*Sec[(a + b*x)/2]*(a*Si n[(a + b*x)/2] - (a + b*x)*Sin[(a + b*x)/2]))/(2*b^2)
Time = 0.32 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
\(\displaystyle \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx\) |
\(\Big \downarrow \) 4920 |
\(\displaystyle -d \int \left (\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}\right )dx+\frac {(c+d x) \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x) \csc (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -d \left (\frac {2 i x \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {\text {arctanh}(\cos (a+b x))}{b^2}+\frac {x \text {arctanh}(\sin (a+b x))}{b}-\frac {i \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}\right )+\frac {(c+d x) \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x) \csc (a+b x)}{b}\) |
((c + d*x)*ArcTanh[Sin[a + b*x]])/b - ((c + d*x)*Csc[a + b*x])/b - d*(((2* I)*x*ArcTan[E^(I*(a + b*x))])/b + ArcTanh[Cos[a + b*x]]/b^2 + (x*ArcTanh[S in[a + b*x]])/b - (I*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 + (I*PolyLog[2, I*E^(I*(a + b*x))])/b^2)
3.3.37.3.1 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]
Time = 1.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.79
method | result | size |
risch | \(-\frac {2 i \left (d x +c \right ) {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}-\frac {d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {2 i c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {i d \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {i d \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 i a d \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{2}}+\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}-\frac {d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}\) | \(235\) |
-2*I*(d*x+c)*exp(I*(b*x+a))/b/(exp(2*I*(b*x+a))-1)-1/b*d*ln(1+I*exp(I*(b*x +a)))*x+1/b*d*ln(1-I*exp(I*(b*x+a)))*x-2*I/b*c*arctan(exp(I*(b*x+a)))+I/b^ 2*d*dilog(1+I*exp(I*(b*x+a)))-I/b^2*d*dilog(1-I*exp(I*(b*x+a)))+2*I/b^2*a* d*arctan(exp(I*(b*x+a)))-d/b^2*ln(exp(I*(b*x+a))+1)+d/b^2*ln(exp(I*(b*x+a) )-1)-1/b^2*d*ln(1+I*exp(I*(b*x+a)))*a+1/b^2*d*ln(1-I*exp(I*(b*x+a)))*a
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (113) = 226\).
Time = 0.29 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.31 \[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 \, b d x + i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) + {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) + d \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - d \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) - {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) + {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) + 2 \, b c}{2 \, b^{2} \sin \left (b x + a\right )} \]
-1/2*(2*b*d*x + I*d*dilog(I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + I* d*dilog(I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) - I*d*dilog(-I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) - I*d*dilog(-I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) - (b*c - a*d)*log(cos(b*x + a) + I*sin(b*x + a) + I)*sin (b*x + a) + (b*c - a*d)*log(cos(b*x + a) - I*sin(b*x + a) + I)*sin(b*x + a ) + d*log(1/2*cos(b*x + a) + 1/2)*sin(b*x + a) - (b*d*x + a*d)*log(I*cos(b *x + a) + sin(b*x + a) + 1)*sin(b*x + a) + (b*d*x + a*d)*log(I*cos(b*x + a ) - sin(b*x + a) + 1)*sin(b*x + a) - (b*d*x + a*d)*log(-I*cos(b*x + a) + s in(b*x + a) + 1)*sin(b*x + a) + (b*d*x + a*d)*log(-I*cos(b*x + a) - sin(b* x + a) + 1)*sin(b*x + a) - d*log(-1/2*cos(b*x + a) + 1/2)*sin(b*x + a) - ( b*c - a*d)*log(-cos(b*x + a) + I*sin(b*x + a) + I)*sin(b*x + a) + (b*c - a *d)*log(-cos(b*x + a) - I*sin(b*x + a) + I)*sin(b*x + a) + 2*b*c)/(b^2*sin (b*x + a))
\[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=\int \left (c + d x\right ) \csc ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]
\[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=\int { {\left (d x + c\right )} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right ) \,d x } \]
-1/2*(4*(b*d*x + b*c)*cos(b*x + a)*sin(2*b*x + 2*a) - 4*(b*d*x + b*c)*cos( 2*b*x + 2*a)*sin(b*x + a) - 4*(b^2*d*cos(2*b*x + 2*a)^2 + b^2*d*sin(2*b*x + 2*a)^2 - 2*b^2*d*cos(2*b*x + 2*a) + b^2*d)*integrate((x*cos(2*b*x + 2*a) *cos(b*x + a) + x*sin(2*b*x + 2*a)*sin(b*x + a) + x*cos(b*x + a))/(cos(2*b *x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1), x) - (b*c*cos( 2*b*x + 2*a)^2 + b*c*sin(2*b*x + 2*a)^2 - 2*b*c*cos(2*b*x + 2*a) + b*c)*lo g(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + (b*c*cos(2*b*x + 2*a)^2 + b*c*sin(2*b*x + 2*a)^2 - 2*b*c*cos(2*b*x + 2*a) + b*c)*log(cos(b *x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + (d*cos(2*b*x + 2*a)^2 + d*sin(2*b*x + 2*a)^2 - 2*d*cos(2*b*x + 2*a) + d)*log(cos(b*x)^2 + 2*cos(b *x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - (d*co s(2*b*x + 2*a)^2 + d*sin(2*b*x + 2*a)^2 - 2*d*cos(2*b*x + 2*a) + d)*log(co s(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*(b*d*x + b*c)*sin(b*x + a))/(b^2*cos(2*b*x + 2*a)^2 + b^2*s in(2*b*x + 2*a)^2 - 2*b^2*cos(2*b*x + 2*a) + b^2)
\[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=\int { {\left (d x + c\right )} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right ) \,d x } \]
Timed out. \[ \int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx=\text {Hanged} \]