3.2.98 \(\int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx\) [198]

3.2.98.1 Optimal result

Integrand size = 26, antiderivative size = 249 \[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i (e+f x)^2}{a d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \]

output

I*(f*x+e)^2/a/d-2*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)^2*cot(1/2* 
c+1/4*Pi+1/2*d*x)/a/d-4*f*(f*x+e)*ln(1-I*exp(I*(d*x+c)))/a/d^2+2*I*f*(f*x+ 
e)*polylog(2,-exp(I*(d*x+c)))/a/d^2+4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/ 
d^3-2*I*f*(f*x+e)*polylog(2,exp(I*(d*x+c)))/a/d^2-2*f^2*polylog(3,-exp(I*( 
d*x+c)))/a/d^3+2*f^2*polylog(3,exp(I*(d*x+c)))/a/d^3
 

3.2.98.2 Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.33 \[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {(e+f x)^2 \log \left (1-e^{i (c+d x)}\right )-(e+f x)^2 \log \left (1+e^{i (c+d x)}\right )+\frac {2 i f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )+i f \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )\right )}{d^2}+\frac {2 f \left (-i d (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )+f \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )\right )}{d^2}+\frac {4 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{\cos (c)+i (1+\sin (c))}-\frac {2 (e+f x)^2 \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{a d} \]

input

Integrate[((e + f*x)^2*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]
 

output

((e + f*x)^2*Log[1 - E^(I*(c + d*x))] - (e + f*x)^2*Log[1 + E^(I*(c + d*x) 
)] + ((2*I)*f*(d*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))] + I*f*PolyLog[3, - 
E^(I*(c + d*x))]))/d^2 + (2*f*((-I)*d*(e + f*x)*PolyLog[2, E^(I*(c + d*x)) 
] + f*PolyLog[3, E^(I*(c + d*x))]))/d^2 + (4*f*(Cos[c] + I*Sin[c])*(((e + 
f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Si 
n[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - 
 Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) 
- (2*(e + f*x)^2*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + 
Sin[(c + d*x)/2])))/(a*d)
 

3.2.98.3 Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {5046, 3042, 3799, 3042, 4671, 3011, 2720, 4672, 3042, 25, 4202, 2620, 2715, 2838, 7143}

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.

\(\displaystyle \int \frac {(e+f x)^2 \csc (c+d x)}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 5046

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3799

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {\int (e+f x)^2 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)dx}{a}-\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\)

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {-\frac {2 f \int (e+f x) \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {2 f \int (e+f x) \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {4 f \int (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {4 f \int -\left ((e+f x) \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {4 f \int (e+f x) \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d^2}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

input

Int[((e + f*x)^2*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]
 

output

-1/2*((-2*(e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/d - (4*f*(((I/2)*(e + f*x 
)^2)/f - (2*I)*(((-I)*(e + f*x)*Log[1 + E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d 
 - (f*PolyLog[2, -E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d^2)))/d)/a + ((-2*(e + 
 f*x)^2*ArcTanh[E^(I*(c + d*x))])/d + (2*f*((I*(e + f*x)*PolyLog[2, -E^(I* 
(c + d*x))])/d - (f*PolyLog[3, -E^(I*(c + d*x))])/d^2))/d - (2*f*((I*(e + 
f*x)*PolyLog[2, E^(I*(c + d*x))])/d - (f*PolyLog[3, E^(I*(c + d*x))])/d^2) 
)/d)/a
 

3.2.98.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

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]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

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]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3799

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])
 

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4671

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]
 

rule 4672

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]
 

rule 5046

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.2.98.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (223 ) = 446\).

Time = 0.38 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.58

method	result	size
risch	\(\frac {4 c \,f^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{3} a}-\frac {4 e f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{2} a}-\frac {2 e f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) x}{d a}+\frac {2 e f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) x}{d a}+\frac {2 e f \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) c}{d^{2} a}-\frac {2 c e f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d^{2} a}-\frac {2 i e f \,\operatorname {Li}_{2}\left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}+\frac {2 i e f \,\operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}-\frac {2 i f^{2} \operatorname {Li}_{2}\left ({\mathrm e}^{i \left (d x +c \right )}\right ) x}{d^{2} a}+\frac {2 i f^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{i \left (d x +c \right )}\right ) x}{d^{2} a}+\frac {e^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {e^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}-\frac {2 f^{2} \operatorname {Li}_{3}\left (-{\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 f^{2} \operatorname {Li}_{3}\left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {f^{2} \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) c^{2}}{d^{3} a}+\frac {2 x^{2} f^{2}+4 f e x +2 e^{2}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {c^{2} f^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d^{3} a}-\frac {f^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) x^{2}}{d a}+\frac {f^{2} \ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{d a}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {4 i c \,f^{2} x}{d^{2} a}+\frac {4 f e \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {4 i f^{2} \operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 i f^{2} x^{2}}{a d}\)	\(643\)

input

int((f*x+e)^2*csc(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

output

-4/d^2/a*e*f*ln(exp(I*(d*x+c))+I)+1/d^3/a*c^2*f^2*ln(exp(I*(d*x+c))-1)-1/d 
/a*f^2*ln(exp(I*(d*x+c))+1)*x^2+1/d/a*f^2*ln(1-exp(I*(d*x+c)))*x^2+4*I/a/d 
^2*f^2*c*x+4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3+4/a/d^2*f*e*ln(exp(I* 
(d*x+c)))-4/a/d^2*f^2*ln(1-I*exp(I*(d*x+c)))*x-4/a/d^3*f^2*ln(1-I*exp(I*(d 
*x+c)))*c-4/a/d^3*f^2*c*ln(exp(I*(d*x+c)))+2*I/a/d*f^2*x^2+2*I/a/d^3*f^2*c 
^2+2*(f^2*x^2+2*e*f*x+e^2)/d/a/(exp(I*(d*x+c))+I)+1/d/a*e^2*ln(exp(I*(d*x+ 
c))-1)-1/d/a*e^2*ln(exp(I*(d*x+c))+1)-1/d^3/a*f^2*ln(1-exp(I*(d*x+c)))*c^2 
+4/d^3/a*c*f^2*ln(exp(I*(d*x+c))+I)-2/d/a*e*f*ln(exp(I*(d*x+c))+1)*x+2/d/a 
*e*f*ln(1-exp(I*(d*x+c)))*x+2/d^2/a*e*f*ln(1-exp(I*(d*x+c)))*c-2/d^2/a*c*e 
*f*ln(exp(I*(d*x+c))-1)-2*I/d^2/a*e*f*polylog(2,exp(I*(d*x+c)))+2*I/d^2/a* 
e*f*polylog(2,-exp(I*(d*x+c)))-2*I/d^2/a*f^2*polylog(2,exp(I*(d*x+c)))*x+2 
*I/d^2/a*f^2*polylog(2,-exp(I*(d*x+c)))*x-2*f^2*polylog(3,-exp(I*(d*x+c))) 
/a/d^3+2*f^2*polylog(3,exp(I*(d*x+c)))/a/d^3
 

3.2.98.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (214) = 428\).

Time = 0.34 (sec) , antiderivative size = 1642, normalized size of antiderivative = 6.59 \[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

input

integrate((f*x+e)^2*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

output

1/2*(2*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^2*e^2 + 2*(d^2*f^2*x^2 + 2*d^2*e*f* 
x + d^2*e^2)*cos(d*x + c) - 2*(I*d*f^2*x + I*d*e*f + (I*d*f^2*x + I*d*e*f) 
*cos(d*x + c) + (I*d*f^2*x + I*d*e*f)*sin(d*x + c))*dilog(cos(d*x + c) + I 
*sin(d*x + c)) - 2*(-I*d*f^2*x - I*d*e*f + (-I*d*f^2*x - I*d*e*f)*cos(d*x 
+ c) + (-I*d*f^2*x - I*d*e*f)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x 
 + c)) - 4*(-I*f^2*cos(d*x + c) - I*f^2*sin(d*x + c) - I*f^2)*dilog(I*cos( 
d*x + c) - sin(d*x + c)) - 4*(I*f^2*cos(d*x + c) + I*f^2*sin(d*x + c) + I* 
f^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 2*(I*d*f^2*x + I*d*e*f + (I*d 
*f^2*x + I*d*e*f)*cos(d*x + c) + (I*d*f^2*x + I*d*e*f)*sin(d*x + c))*dilog 
(-cos(d*x + c) + I*sin(d*x + c)) - 2*(-I*d*f^2*x - I*d*e*f + (-I*d*f^2*x - 
 I*d*e*f)*cos(d*x + c) + (-I*d*f^2*x - I*d*e*f)*sin(d*x + c))*dilog(-cos(d 
*x + c) - I*sin(d*x + c)) - (d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 + (d^2*f^ 
2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c) + (d^2*f^2*x^2 + 2*d^2*e*f*x + 
 d^2*e^2)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + 1) - 4*(d*e*f 
- c*f^2 + (d*e*f - c*f^2)*cos(d*x + c) + (d*e*f - c*f^2)*sin(d*x + c))*log 
(cos(d*x + c) + I*sin(d*x + c) + I) - (d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 
 + (d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c) + (d^2*f^2*x^2 + 2*d 
^2*e*f*x + d^2*e^2)*sin(d*x + c))*log(cos(d*x + c) - I*sin(d*x + c) + 1) - 
 4*(d*f^2*x + c*f^2 + (d*f^2*x + c*f^2)*cos(d*x + c) + (d*f^2*x + c*f^2)*s 
in(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 4*(d*f^2*x + c*f^...
 

3.2.98.6 Sympy [F]

\[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

input

integrate((f*x+e)**2*csc(d*x+c)/(a+a*sin(d*x+c)),x)
 

output

(Integral(e**2*csc(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f**2*x**2*cs 
c(c + d*x)/(sin(c + d*x) + 1), x) + Integral(2*e*f*x*csc(c + d*x)/(sin(c + 
 d*x) + 1), x))/a
 

3.2.98.7 Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (214) = 428\).

Time = 0.37 (sec) , antiderivative size = 1418, normalized size of antiderivative = 5.69 \[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

input

integrate((f*x+e)^2*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

output

-(2*c*e*f*(2/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1)) + log(sin(d*x + c 
)/(cos(d*x + c) + 1))/(a*d)) - e^2*(log(sin(d*x + c)/(cos(d*x + c) + 1))/a 
 + 2/(a + a*sin(d*x + c)/(cos(d*x + c) + 1))) + (4*I*c^2*f^2 - 8*(-I*d*e*f 
 + I*c*f^2 - (d*e*f - c*f^2)*cos(d*x + c) + (-I*d*e*f + I*c*f^2)*sin(d*x + 
 c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 8*((d*x + c)*f^2*cos(d*x + 
c) + I*(d*x + c)*f^2*sin(d*x + c) + I*(d*x + c)*f^2)*arctan2(cos(d*x + c), 
 sin(d*x + c) + 1) - 2*(-I*(d*x + c)^2*f^2 - I*c^2*f^2 + 2*(-I*d*e*f + I*c 
*f^2)*(d*x + c) - ((d*x + c)^2*f^2 + c^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c) 
)*cos(d*x + c) + (-I*(d*x + c)^2*f^2 - I*c^2*f^2 + 2*(-I*d*e*f + I*c*f^2)* 
(d*x + c))*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) - 2*(c^2* 
f^2*cos(d*x + c) + I*c^2*f^2*sin(d*x + c) + I*c^2*f^2)*arctan2(sin(d*x + c 
), cos(d*x + c) - 1) - 2*(-I*(d*x + c)^2*f^2 + 2*(-I*d*e*f + I*c*f^2)*(d*x 
 + c) - ((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2)*(d*x + c))*cos(d*x + c) + (-I 
*(d*x + c)^2*f^2 + 2*(-I*d*e*f + I*c*f^2)*(d*x + c))*sin(d*x + c))*arctan2 
(sin(d*x + c), -cos(d*x + c) + 1) - 4*((d*x + c)^2*f^2 + 2*(d*e*f - c*f^2) 
*(d*x + c))*cos(d*x + c) - 8*(f^2*cos(d*x + c) + I*f^2*sin(d*x + c) + I*f^ 
2)*dilog(I*e^(I*d*x + I*c)) - 4*(I*d*e*f + I*(d*x + c)*f^2 - I*c*f^2 + (d* 
e*f + (d*x + c)*f^2 - c*f^2)*cos(d*x + c) + (I*d*e*f + I*(d*x + c)*f^2 - I 
*c*f^2)*sin(d*x + c))*dilog(-e^(I*d*x + I*c)) - 4*(-I*d*e*f - I*(d*x + c)* 
f^2 + I*c*f^2 - (d*e*f + (d*x + c)*f^2 - c*f^2)*cos(d*x + c) + (-I*d*e*...
 

3.2.98.8 Giac [F]

\[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \csc \left (d x + c\right )}{a \sin \left (d x + c\right ) + a} \,d x } \]

input

integrate((f*x+e)^2*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
 

output

integrate((f*x + e)^2*csc(d*x + c)/(a*sin(d*x + c) + a), x)
 

3.2.98.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]

input

int((e + f*x)^2/(sin(c + d*x)*(a + a*sin(c + d*x))),x)
 

output

\text{Hanged}