3.4.23 \(\int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx\) [323]

3.4.23.1 Optimal result

Integrand size = 24, antiderivative size = 318 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=-\frac {6 d^2 (c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b^3}-\frac {4 (c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{b^2}-\frac {2 (c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{b}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{2 b^4} \]

output

-6*d^2*(d*x+c)*arctanh(exp(2*I*(b*x+a)))/b^3-4*(d*x+c)^3*arctanh(exp(2*I*( 
b*x+a)))/b-3*d*(d*x+c)^2*csc(2*b*x+2*a)/b^2-2*(d*x+c)^3*cot(2*b*x+2*a)*csc 
(2*b*x+2*a)/b+3/2*I*d^3*polylog(2,-exp(2*I*(b*x+a)))/b^4+3*I*d*(d*x+c)^2*p 
olylog(2,-exp(2*I*(b*x+a)))/b^2-3/2*I*d^3*polylog(2,exp(2*I*(b*x+a)))/b^4- 
3*I*d*(d*x+c)^2*polylog(2,exp(2*I*(b*x+a)))/b^2-3*d^2*(d*x+c)*polylog(3,-e 
xp(2*I*(b*x+a)))/b^3+3*d^2*(d*x+c)*polylog(3,exp(2*I*(b*x+a)))/b^3-3/2*I*d 
^3*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*I*d^3*polylog(4,exp(2*I*(b*x+a)))/ 
b^4
 

3.4.23.2 Mathematica [A] (verified)

Time = 6.27 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.52 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=-\frac {8 b^3 c^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )+12 b c d^2 \text {arctanh}\left (e^{2 i (a+b x)}\right )+2 b^2 (c+d x)^2 (3 d+2 b (c+d x) \cot (2 (a+b x))) \csc (2 (a+b x))-12 b^3 c^2 d x \log \left (1-e^{2 i (a+b x)}\right )-6 b d^3 x \log \left (1-e^{2 i (a+b x)}\right )-12 b^3 c d^2 x^2 \log \left (1-e^{2 i (a+b x)}\right )-4 b^3 d^3 x^3 \log \left (1-e^{2 i (a+b x)}\right )+12 b^3 c^2 d x \log \left (1+e^{2 i (a+b x)}\right )+6 b d^3 x \log \left (1+e^{2 i (a+b x)}\right )+12 b^3 c d^2 x^2 \log \left (1+e^{2 i (a+b x)}\right )+4 b^3 d^3 x^3 \log \left (1+e^{2 i (a+b x)}\right )-3 i d \left (d^2+2 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )+3 i d \left (d^2+2 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )+3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )-3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{2 b^4} \]

input

Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^3,x]
 

output

-1/2*(8*b^3*c^3*ArcTanh[E^((2*I)*(a + b*x))] + 12*b*c*d^2*ArcTanh[E^((2*I) 
*(a + b*x))] + 2*b^2*(c + d*x)^2*(3*d + 2*b*(c + d*x)*Cot[2*(a + b*x)])*Cs 
c[2*(a + b*x)] - 12*b^3*c^2*d*x*Log[1 - E^((2*I)*(a + b*x))] - 6*b*d^3*x*L 
og[1 - E^((2*I)*(a + b*x))] - 12*b^3*c*d^2*x^2*Log[1 - E^((2*I)*(a + b*x)) 
] - 4*b^3*d^3*x^3*Log[1 - E^((2*I)*(a + b*x))] + 12*b^3*c^2*d*x*Log[1 + E^ 
((2*I)*(a + b*x))] + 6*b*d^3*x*Log[1 + E^((2*I)*(a + b*x))] + 12*b^3*c*d^2 
*x^2*Log[1 + E^((2*I)*(a + b*x))] + 4*b^3*d^3*x^3*Log[1 + E^((2*I)*(a + b* 
x))] - (3*I)*d*(d^2 + 2*b^2*(c + d*x)^2)*PolyLog[2, -E^((2*I)*(a + b*x))] 
+ (3*I)*d*(d^2 + 2*b^2*(c + d*x)^2)*PolyLog[2, E^((2*I)*(a + b*x))] + 6*b* 
c*d^2*PolyLog[3, -E^((2*I)*(a + b*x))] + 6*b*d^3*x*PolyLog[3, -E^((2*I)*(a 
 + b*x))] - 6*b*c*d^2*PolyLog[3, E^((2*I)*(a + b*x))] - 6*b*d^3*x*PolyLog[ 
3, E^((2*I)*(a + b*x))] + (3*I)*d^3*PolyLog[4, -E^((2*I)*(a + b*x))] - (3* 
I)*d^3*PolyLog[4, E^((2*I)*(a + b*x))])/b^4
 

3.4.23.3 Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {4919, 3042, 4674, 3042, 4671, 2715, 2838, 3011, 7163, 2720, 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.

\(\displaystyle \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx\)

\(\Big \downarrow \) 4919

\(\displaystyle 8 \int (c+d x)^3 \csc ^3(2 a+2 b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 \int (c+d x)^3 \csc (2 a+2 b x)^3dx\)

\(\Big \downarrow \) 4674

\(\displaystyle 8 \left (\frac {3 d^2 \int (c+d x) \csc (2 a+2 b x)dx}{4 b^2}+\frac {1}{2} \int (c+d x)^3 \csc (2 a+2 b x)dx-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 8 \left (\frac {3 d^2 \int (c+d x) \csc (2 a+2 b x)dx}{4 b^2}+\frac {1}{2} \int (c+d x)^3 \csc (2 a+2 b x)dx-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 4671

\(\displaystyle 8 \left (\frac {3 d^2 \left (-\frac {d \int \log \left (1-e^{2 i (a+b x)}\right )dx}{2 b}+\frac {d \int \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )}{4 b^2}+\frac {1}{2} \left (-\frac {3 d \int (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )dx}{2 b}+\frac {3 d \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle 8 \left (\frac {3 d^2 \left (\frac {i d \int e^{-2 i (a+b x)} \log \left (1-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i d \int e^{-2 i (a+b x)} \log \left (1+e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )}{4 b^2}+\frac {1}{2} \left (-\frac {3 d \int (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )dx}{2 b}+\frac {3 d \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle 8 \left (\frac {1}{2} \left (-\frac {3 d \int (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )dx}{2 b}+\frac {3 d \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )+\frac {3 d^2 \left (-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle 8 \left (\frac {1}{2} \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )+\frac {3 d^2 \left (-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 8 \left (\frac {1}{2} \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )+\frac {3 d^2 \left (-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle 8 \left (\frac {1}{2} \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )+\frac {3 d^2 \left (-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{4 b^2}\right )}{4 b^2}-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 8 \left (\frac {3 d^2 \left (-\frac {(c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{4 b^2}\right )}{4 b^2}+\frac {1}{2} \left (-\frac {(c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}\right )-\frac {3 d (c+d x)^2 \csc (2 a+2 b x)}{8 b^2}-\frac {(c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{4 b}\right )\)

input

Int[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^3,x]
 

output

8*((-3*d*(c + d*x)^2*Csc[2*a + 2*b*x])/(8*b^2) - ((c + d*x)^3*Cot[2*a + 2* 
b*x]*Csc[2*a + 2*b*x])/(4*b) + (3*d^2*(-(((c + d*x)*ArcTanh[E^((2*I)*(a + 
b*x))])/b) + ((I/4)*d*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - ((I/4)*d*Pol 
yLog[2, E^((2*I)*(a + b*x))])/b^2))/(4*b^2) + (-(((c + d*x)^3*ArcTanh[E^(( 
2*I)*(a + b*x))])/b) + (3*d*(((I/2)*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + 
b*x))])/b - (I*d*(((-1/2*I)*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))])/b 
+ (d*PolyLog[4, -E^((2*I)*(a + b*x))])/(4*b^2)))/b))/(2*b) - (3*d*(((I/2)* 
(c + d*x)^2*PolyLog[2, E^((2*I)*(a + b*x))])/b - (I*d*(((-1/2*I)*(c + d*x) 
*PolyLog[3, E^((2*I)*(a + b*x))])/b + (d*PolyLog[4, E^((2*I)*(a + b*x))])/ 
(4*b^2)))/b))/(2*b))/2)
 

3.4.23.3.1 Defintions of rubi rules used

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 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 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

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]
 

rule 7163

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

3.4.23.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1328 vs. \(2 (284 ) = 568\).

Time = 1.22 (sec) , antiderivative size = 1329, normalized size of antiderivative = 4.18

method	result	size
risch	\(\text {Expression too large to display}\)	\(1329\)

input

int((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
 

output

-3/2*I*d^3*polylog(4,-exp(2*I*(b*x+a)))/b^4-3*I*d^3*polylog(2,exp(I*(b*x+a 
)))/b^4-3/b^3*d^3*polylog(3,-exp(2*I*(b*x+a)))*x-3/b^3*c*d^2*polylog(3,-ex 
p(2*I*(b*x+a)))-2/b*d^3*ln(exp(2*I*(b*x+a))+1)*x^3-3/b^3*d^2*c*ln(exp(2*I* 
(b*x+a))+1)-3/b^3*d^3*ln(exp(2*I*(b*x+a))+1)*x+2/b^2/(exp(2*I*(b*x+a))+1)^ 
2/(exp(2*I*(b*x+a))-1)^2*(2*d^3*x^3*b*exp(6*I*(b*x+a))+6*c*d^2*x^2*b*exp(6 
*I*(b*x+a))+6*c^2*d*x*b*exp(6*I*(b*x+a))+2*c^3*b*exp(6*I*(b*x+a))-3*I*d^3* 
x^2*exp(6*I*(b*x+a))+2*b*d^3*x^3*exp(2*I*(b*x+a))-6*I*c*d^2*x*exp(6*I*(b*x 
+a))+6*b*c*d^2*x^2*exp(2*I*(b*x+a))-3*I*c^2*d*exp(6*I*(b*x+a))+6*b*c^2*d*x 
*exp(2*I*(b*x+a))+2*b*c^3*exp(2*I*(b*x+a))+3*I*d^3*x^2*exp(2*I*(b*x+a))+6* 
I*c*d^2*x*exp(2*I*(b*x+a))+3*I*c^2*d*exp(2*I*(b*x+a)))+3/2*I*d^3*polylog(2 
,-exp(2*I*(b*x+a)))/b^4-2/b*c^3*ln(exp(2*I*(b*x+a))+1)+6/b^3*c*d^2*a^2*ln( 
exp(I*(b*x+a))-1)-6/b^2*c^2*d*a*ln(exp(I*(b*x+a))-1)+6/b^2*d*c^2*ln(1-exp( 
I*(b*x+a)))*a-6/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2-12*I/b^2*c*d^2*polylog( 
2,-exp(I*(b*x+a)))*x+6*I/b^2*c*d^2*polylog(2,-exp(2*I*(b*x+a)))*x-12*I/b^2 
*c*d^2*polylog(2,exp(I*(b*x+a)))*x+2/b*c^3*ln(exp(I*(b*x+a))+1)+2/b*c^3*ln 
(exp(I*(b*x+a))-1)-6/b*c*d^2*ln(exp(2*I*(b*x+a))+1)*x^2-6/b*c^2*d*ln(exp(2 
*I*(b*x+a))+1)*x+3/b^3*c*d^2*ln(exp(I*(b*x+a))+1)+3/b^3*c*d^2*ln(exp(I*(b* 
x+a))-1)+3/b^3*d^3*ln(exp(I*(b*x+a))+1)*x+3/b^3*d^3*ln(1-exp(I*(b*x+a)))*x 
+12*I/b^4*d^3*polylog(4,-exp(I*(b*x+a)))+12*I/b^4*d^3*polylog(4,exp(I*(b*x 
+a)))-3*I/b^4*d^3*polylog(2,-exp(I*(b*x+a)))+6/b*d*c^2*ln(1-exp(I*(b*x+...
 

3.4.23.5 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. 4193 vs. \(2 (274) = 548\).

Time = 0.47 (sec) , antiderivative size = 4193, normalized size of antiderivative = 13.19 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^3,x, algorithm="fricas")
 

output

-1/2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3 - 2*(b^3*d^3 
*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(b*x + a)^2 - 3*(b^2* 
d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a)*sin(b*x + a) + 3*((2*I*b 
^2*d^3*x^2 + 4*I*b^2*c*d^2*x + 2*I*b^2*c^2*d + I*d^3)*cos(b*x + a)^4 + (-2 
*I*b^2*d^3*x^2 - 4*I*b^2*c*d^2*x - 2*I*b^2*c^2*d - I*d^3)*cos(b*x + a)^2)* 
dilog(cos(b*x + a) + I*sin(b*x + a)) + 3*((-2*I*b^2*d^3*x^2 - 4*I*b^2*c*d^ 
2*x - 2*I*b^2*c^2*d - I*d^3)*cos(b*x + a)^4 + (2*I*b^2*d^3*x^2 + 4*I*b^2*c 
*d^2*x + 2*I*b^2*c^2*d + I*d^3)*cos(b*x + a)^2)*dilog(cos(b*x + a) - I*sin 
(b*x + a)) + 3*((2*I*b^2*d^3*x^2 + 4*I*b^2*c*d^2*x + 2*I*b^2*c^2*d + I*d^3 
)*cos(b*x + a)^4 + (-2*I*b^2*d^3*x^2 - 4*I*b^2*c*d^2*x - 2*I*b^2*c^2*d - I 
*d^3)*cos(b*x + a)^2)*dilog(I*cos(b*x + a) + sin(b*x + a)) + 3*((-2*I*b^2* 
d^3*x^2 - 4*I*b^2*c*d^2*x - 2*I*b^2*c^2*d - I*d^3)*cos(b*x + a)^4 + (2*I*b 
^2*d^3*x^2 + 4*I*b^2*c*d^2*x + 2*I*b^2*c^2*d + I*d^3)*cos(b*x + a)^2)*dilo 
g(I*cos(b*x + a) - sin(b*x + a)) + 3*((-2*I*b^2*d^3*x^2 - 4*I*b^2*c*d^2*x 
- 2*I*b^2*c^2*d - I*d^3)*cos(b*x + a)^4 + (2*I*b^2*d^3*x^2 + 4*I*b^2*c*d^2 
*x + 2*I*b^2*c^2*d + I*d^3)*cos(b*x + a)^2)*dilog(-I*cos(b*x + a) + sin(b* 
x + a)) + 3*((2*I*b^2*d^3*x^2 + 4*I*b^2*c*d^2*x + 2*I*b^2*c^2*d + I*d^3)*c 
os(b*x + a)^4 + (-2*I*b^2*d^3*x^2 - 4*I*b^2*c*d^2*x - 2*I*b^2*c^2*d - I*d^ 
3)*cos(b*x + a)^2)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + 3*((-2*I*b^2*d^ 
3*x^2 - 4*I*b^2*c*d^2*x - 2*I*b^2*c^2*d - I*d^3)*cos(b*x + a)^4 + (2*I*...
 

3.4.23.6 Sympy [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=\text {Timed out} \]

input

integrate((d*x+c)**3*csc(b*x+a)**3*sec(b*x+a)**3,x)
 

output

Timed out
 

3.4.23.7 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. 5646 vs. \(2 (274) = 548\).

Time = 2.42 (sec) , antiderivative size = 5646, normalized size of antiderivative = 17.75 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^3,x, algorithm="maxima")
 

output

-1/2*(c^3*((2*sin(b*x + a)^2 - 1)/(sin(b*x + a)^4 - sin(b*x + a)^2) + 2*lo 
g(sin(b*x + a)^2 - 1) - 2*log(sin(b*x + a)^2)) - 3*a*c^2*d*((2*sin(b*x + a 
)^2 - 1)/(sin(b*x + a)^4 - sin(b*x + a)^2) + 2*log(sin(b*x + a)^2 - 1) - 2 
*log(sin(b*x + a)^2))/b + 3*a^2*c*d^2*((2*sin(b*x + a)^2 - 1)/(sin(b*x + a 
)^4 - sin(b*x + a)^2) + 2*log(sin(b*x + a)^2 - 1) - 2*log(sin(b*x + a)^2)) 
/b^2 - a^3*d^3*((2*sin(b*x + a)^2 - 1)/(sin(b*x + a)^4 - sin(b*x + a)^2) + 
 2*log(sin(b*x + a)^2 - 1) - 2*log(sin(b*x + a)^2))/b^3 + 2*(2*(8*(b*x + a 
)^3*d^3 + 9*b*c*d^2 - 9*a*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(2*b^ 
2*c^2*d - 4*a*b*c*d^2 + (2*a^2 + 1)*d^3)*(b*x + a) + (8*(b*x + a)^3*d^3 + 
9*b*c*d^2 - 9*a*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(2*b^2*c^2*d - 
4*a*b*c*d^2 + (2*a^2 + 1)*d^3)*(b*x + a))*cos(8*b*x + 8*a) - 2*(8*(b*x + a 
)^3*d^3 + 9*b*c*d^2 - 9*a*d^3 + 18*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(2*b^ 
2*c^2*d - 4*a*b*c*d^2 + (2*a^2 + 1)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - (-8 
*I*(b*x + a)^3*d^3 - 9*I*b*c*d^2 + 9*I*a*d^3 + 18*(-I*b*c*d^2 + I*a*d^3)*( 
b*x + a)^2 + 9*(-2*I*b^2*c^2*d + 4*I*a*b*c*d^2 + (-2*I*a^2 - I)*d^3)*(b*x 
+ a))*sin(8*b*x + 8*a) - 2*(8*I*(b*x + a)^3*d^3 + 9*I*b*c*d^2 - 9*I*a*d^3 
+ 18*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 9*(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 
+ (2*I*a^2 + I)*d^3)*(b*x + a))*sin(4*b*x + 4*a))*arctan2(sin(2*b*x + 2*a) 
, cos(2*b*x + 2*a) + 1) - 6*(2*(b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*d^3 + 6*( 
b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(2*b^2*c^2*d - 4*a*b*c*d^2 + (2*a^2 + ...
 

3.4.23.8 Giac [F]

\[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{3} \,d x } \]

input

integrate((d*x+c)^3*csc(b*x+a)^3*sec(b*x+a)^3,x, algorithm="giac")
 

output

integrate((d*x + c)^3*csc(b*x + a)^3*sec(b*x + a)^3, x)
 

3.4.23.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \]

input

int((c + d*x)^3/(cos(a + b*x)^3*sin(a + b*x)^3),x)
 

output

\text{Hanged}