Integrand size = 26, antiderivative size = 169 \[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2} \]
2*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d-(f*x+e)*cot(1/2*c+1/4*Pi+1/2*d*x)/a/ d-(f*x+e)*cot(d*x+c)/a/d+2*f*ln(sin(1/2*c+1/4*Pi+1/2*d*x))/a/d^2+f*ln(sin( d*x+c))/a/d^2-I*f*polylog(2,-exp(I*(d*x+c)))/a/d^2+I*f*polylog(2,exp(I*(d* x+c)))/a/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(404\) vs. \(2(169)=338\).
Time = 6.96 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.39 \[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-d (e+f x) \cos \left (\frac {1}{2} (c+d x)\right ) \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )+4 d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right )-2 f (c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 d e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 f (\log (\cos (c+d x))+\log (\tan (c+d x))) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 a d^2 (1+\sin (c+d x))} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-(d*(e + f*x)*Cos[(c + d*x)/2]*(1 + Cot[(c + d*x)/2])) + 4*d*(e + f*x)*Sin[(c + d*x)/2] - 2*f*(c + d*x)*(Cos [(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d* x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*d*e*Log[Tan[(c + d*x)/2]] *(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*c*f*Log[Tan[(c + d*x)/2]]*(Cos[ (c + d*x)/2] + Sin[(c + d*x)/2]) + 2*f*(Log[Cos[c + d*x]] + Log[Tan[c + d* x]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*f*((c + d*x)*(Log[1 - E^(I* (c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + d* (e + f*x)*Sin[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])))/(2*a*d^2*(1 + Sin[c + d*x]))
Time = 1.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {5046, 3042, 4672, 3042, 25, 3956, 5046, 3042, 3799, 3042, 4671, 2715, 2838, 4672, 3042, 25, 3956}
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.
| \(\displaystyle \int \frac {(e+f x) \csc ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\int (e+f x) \csc ^2(c+d x)dx}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x) \csc (c+d x)^2dx}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {f \int \cot (c+d x)dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x) \cot (c+d x)}{d}}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}-\int \frac {(e+f x) \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \int \frac {e+f x}{\sin (c+d x) a+a}dx-\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {e+f x}{\sin (c+d x) a+a}dx-\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\int (e+f x) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}-\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {\int (e+f x) \csc (c+d x)dx}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {2 f \int \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f \int -\tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2 f \int \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}-\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}}{a}+\frac {\frac {4 f \log \left (-\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\pi }{4}\right )\right )}{d^2}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}}{a}\) |
((-2*(e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/d + (4*f*Log[-Cos[c/2 - Pi/4 + ( d*x)/2]])/d^2)/(2*a) + (-(((e + f*x)*Cot[c + d*x])/d) + (f*Log[-Sin[c + d* x]])/d^2)/a - ((-2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/d + (I*f*PolyLog[2, -E^(I*(c + d*x))])/d^2 - (I*f*PolyLog[2, E^(I*(c + d*x))])/d^2)/a
3.3.5.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(Csc[c + d*x]^(n - 1)/(a + b*S in[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ [n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (149 ) = 298\).
Time = 0.43 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.18
| method | result | size |
| risch | \(-\frac {2 \left (-2 f x +i {\mathrm e}^{i \left (d x +c \right )} f x -2 e +i {\mathrm e}^{i \left (d x +c \right )} e +f x \,{\mathrm e}^{2 i \left (d x +c \right )}+e \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d a}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) x}{d a}-\frac {f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) x}{d a}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a \,d^{2}}+\frac {f \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a \,d^{2}}-\frac {2 i f \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {e \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}+\frac {e \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}+\frac {c f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d^{2} a}-\frac {f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) c}{d^{2} a}+\frac {i f \,\operatorname {Li}_{2}\left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {i f \,\operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {4 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}\) | \(369\) |
-2*(-2*f*x+I*exp(I*(d*x+c))*f*x-2*e+I*exp(I*(d*x+c))*e+f*x*exp(2*I*(d*x+c) )+e*exp(2*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)/(exp(I*(d*x+c))+I)/d/a+1/d/a*f* ln(exp(I*(d*x+c))+1)*x-1/d/a*f*ln(1-exp(I*(d*x+c)))*x+1/a/d^2*f*ln(exp(I*( d*x+c))-1)+1/a/d^2*f*ln(1+exp(2*I*(d*x+c)))+1/a/d^2*f*ln(exp(I*(d*x+c))+1) -2*I/a/d^2*f*arctan(exp(I*(d*x+c)))-1/d/a*e*ln(exp(I*(d*x+c))-1)+1/d/a*e*l n(exp(I*(d*x+c))+1)+1/d^2/a*c*f*ln(exp(I*(d*x+c))-1)-1/d^2/a*f*ln(1-exp(I* (d*x+c)))*c+I*f*polylog(2,exp(I*(d*x+c)))/a/d^2-I*f*polylog(2,-exp(I*(d*x+ c)))/a/d^2-4/d^2/a*f*ln(exp(I*(d*x+c)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (145) = 290\).
Time = 0.32 (sec) , antiderivative size = 858, normalized size of antiderivative = 5.08 \[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(2*d*f*x - 4*(d*f*x + d*e)*cos(d*x + c)^2 + 2*d*e - 2*(d*f*x + d*e)*c os(d*x + c) + (-I*f*cos(d*x + c)^2 + (I*f*cos(d*x + c) + I*f)*sin(d*x + c) + I*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) + (I*f*cos(d*x + c)^2 + (-I*f *cos(d*x + c) - I*f)*sin(d*x + c) - I*f)*dilog(cos(d*x + c) - I*sin(d*x + c)) + (-I*f*cos(d*x + c)^2 + (I*f*cos(d*x + c) + I*f)*sin(d*x + c) + I*f)* dilog(-cos(d*x + c) + I*sin(d*x + c)) + (I*f*cos(d*x + c)^2 + (-I*f*cos(d* x + c) - I*f)*sin(d*x + c) - I*f)*dilog(-cos(d*x + c) - I*sin(d*x + c)) + (d*f*x - (d*f*x + d*e + f)*cos(d*x + c)^2 + d*e + (d*f*x + d*e + (d*f*x + d*e + f)*cos(d*x + c) + f)*sin(d*x + c) + f)*log(cos(d*x + c) + I*sin(d*x + c) + 1) + (d*f*x - (d*f*x + d*e + f)*cos(d*x + c)^2 + d*e + (d*f*x + d*e + (d*f*x + d*e + f)*cos(d*x + c) + f)*sin(d*x + c) + f)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + ((d*e - (c + 1)*f)*cos(d*x + c)^2 - d*e + (c + 1)* f - (d*e - (c + 1)*f + (d*e - (c + 1)*f)*cos(d*x + c))*sin(d*x + c))*log(- 1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) + ((d*e - (c + 1)*f)*cos(d*x + c)^2 - d*e + (c + 1)*f - (d*e - (c + 1)*f + (d*e - (c + 1)*f)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) - (d*f *x - (d*f*x + c*f)*cos(d*x + c)^2 + c*f + (d*f*x + c*f + (d*f*x + c*f)*cos (d*x + c))*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + 1) - (d*f*x - (d*f*x + c*f)*cos(d*x + c)^2 + c*f + (d*f*x + c*f + (d*f*x + c*f)*cos(d* x + c))*sin(d*x + c))*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - 2*(f*co...
\[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f x \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e*csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f*x*csc(c + d*x)**2/(sin(c + d*x) + 1), x))/a
Exception generated. \[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \csc \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]