3.4.17 \(\int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx\) [317]

3.4.17.1 Optimal result

Integrand size = 24, antiderivative size = 486 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {6 i d^2 (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^3}-\frac {3 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {6 d (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^4}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^4}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{2 b^2}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}-\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}-\frac {9 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {9 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}-\frac {3 d (c+d x)^2 \sec (a+b x)}{2 b^2}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b} \]

output

3*I*d^3*polylog(2,-I*exp(I*(b*x+a)))/b^4-3*I*(d*x+c)^3*arctan(exp(I*(b*x+a 
)))/b-6*d*(d*x+c)^2*arctanh(exp(I*(b*x+a)))/b^2-3/2*(d*x+c)^3*csc(b*x+a)/b 
-9*I*d^3*polylog(4,-I*exp(I*(b*x+a)))/b^4+9*I*d^3*polylog(4,I*exp(I*(b*x+a 
)))/b^4-6*I*d^2*(d*x+c)*arctan(exp(I*(b*x+a)))/b^3-3*I*d^3*polylog(2,I*exp 
(I*(b*x+a)))/b^4-9/2*I*d*(d*x+c)^2*polylog(2,I*exp(I*(b*x+a)))/b^2+9/2*I*d 
*(d*x+c)^2*polylog(2,-I*exp(I*(b*x+a)))/b^2-6*d^3*polylog(3,-exp(I*(b*x+a) 
))/b^4-9*d^2*(d*x+c)*polylog(3,-I*exp(I*(b*x+a)))/b^3+9*d^2*(d*x+c)*polylo 
g(3,I*exp(I*(b*x+a)))/b^3+6*d^3*polylog(3,exp(I*(b*x+a)))/b^4+6*I*d^2*(d*x 
+c)*polylog(2,-exp(I*(b*x+a)))/b^3-6*I*d^2*(d*x+c)*polylog(2,exp(I*(b*x+a) 
))/b^3-3/2*d*(d*x+c)^2*sec(b*x+a)/b^2+1/2*(d*x+c)^3*csc(b*x+a)*sec(b*x+a)^ 
2/b
 

3.4.17.2 Mathematica [A] (verified)

Time = 7.51 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.69 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {3 d \left ((c+d x)^2 \log \left (1-e^{i (a+b x)}\right )-(c+d x)^2 \log \left (1+e^{i (a+b x)}\right )+\frac {2 i d \left (b (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+i d \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )\right )}{b^2}+\frac {2 d \left (-i b (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+d \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )\right )}{b^2}\right )}{b^2}-\frac {3 \left (2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+4 i b c d^2 \arctan \left (e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )-2 b d^3 x \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )+2 b d^3 x \log \left (1+i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )-i d \left (2 d^2+3 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+i d \left (2 d^2+3 b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )\right )}{2 b^4}-\frac {\csc (a+b x) \sec ^2(a+b x) \left (b c^3+3 b c^2 d x+3 b c d^2 x^2+b d^3 x^3+3 b c^3 \cos (2 a+2 b x)+9 b c^2 d x \cos (2 a+2 b x)+9 b c d^2 x^2 \cos (2 a+2 b x)+3 b d^3 x^3 \cos (2 a+2 b x)+3 c^2 d \sin (2 a+2 b x)+6 c d^2 x \sin (2 a+2 b x)+3 d^3 x^2 \sin (2 a+2 b x)\right )}{4 b^2} \]

input

Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^3,x]
 

output

(3*d*((c + d*x)^2*Log[1 - E^(I*(a + b*x))] - (c + d*x)^2*Log[1 + E^(I*(a + 
 b*x))] + ((2*I)*d*(b*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog 
[3, -E^(I*(a + b*x))]))/b^2 + (2*d*((-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + 
b*x))] + d*PolyLog[3, E^(I*(a + b*x))]))/b^2))/b^2 - (3*((2*I)*b^3*c^3*Arc 
Tan[E^(I*(a + b*x))] + (4*I)*b*c*d^2*ArcTan[E^(I*(a + b*x))] - 3*b^3*c^2*d 
*x*Log[1 - I*E^(I*(a + b*x))] - 2*b*d^3*x*Log[1 - I*E^(I*(a + b*x))] - 3*b 
^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - I*E^(I*(a + 
b*x))] + 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] + 2*b*d^3*x*Log[1 + I*E^ 
(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + b^3*d^3*x^3* 
Log[1 + I*E^(I*(a + b*x))] - I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, (- 
I)*E^(I*(a + b*x))] + I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, I*E^(I*(a 
 + b*x))] + 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog 
[3, (-I)*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] - 6*b* 
d^3*x*PolyLog[3, I*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + 
b*x))] - (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))]))/(2*b^4) - (Csc[a + b*x] 
*Sec[a + b*x]^2*(b*c^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2 + b*d^3*x^3 + 3*b*c^3 
*Cos[2*a + 2*b*x] + 9*b*c^2*d*x*Cos[2*a + 2*b*x] + 9*b*c*d^2*x^2*Cos[2*a + 
 2*b*x] + 3*b*d^3*x^3*Cos[2*a + 2*b*x] + 3*c^2*d*Sin[2*a + 2*b*x] + 6*c*d^ 
2*x*Sin[2*a + 2*b*x] + 3*d^3*x^2*Sin[2*a + 2*b*x]))/(4*b^2)
 

3.4.17.3 Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.06, 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.

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

\(\Big \downarrow \) 4920

\(\displaystyle -3 d \int \frac {1}{2} (c+d x)^2 \left (\frac {\csc (a+b x) \sec ^2(a+b x)}{b}+\frac {3 \text {arctanh}(\sin (a+b x))}{b}-\frac {3 \csc (a+b x)}{b}\right )dx+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} d \int (c+d x)^2 \left (\frac {\csc (a+b x) \sec ^2(a+b x)}{b}+\frac {3 \text {arctanh}(\sin (a+b x))}{b}-\frac {3 \csc (a+b x)}{b}\right )dx+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {3}{2} d \int \frac {(c+d x)^2 \left (\csc (a+b x) \sec ^2(a+b x)+3 \text {arctanh}(\sin (a+b x))-3 \csc (a+b x)\right )}{b}dx+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 d \int (c+d x)^2 \left (\csc (a+b x) \sec ^2(a+b x)+3 \text {arctanh}(\sin (a+b x))-3 \csc (a+b x)\right )dx}{2 b}+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3 d \int \left (\csc (a+b x) \sec ^2(a+b x) (c+d x)^2+3 (\text {arctanh}(\sin (a+b x))-\csc (a+b x)) (c+d x)^2\right )dx}{2 b}+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 d \left (\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{d}+\frac {4 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^3 \text {arctanh}(\sin (a+b x))}{d}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {4 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {4 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^2 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^3}-\frac {4 i d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {4 i d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {6 d (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^2}-\frac {3 i (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}+\frac {3 i (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}+\frac {(c+d x)^2 \sec (a+b x)}{b}\right )}{2 b}+\frac {3 (c+d x)^3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac {(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}\)

input

Int[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^3,x]
 

output

(3*(c + d*x)^3*ArcTanh[Sin[a + b*x]])/(2*b) - (3*(c + d*x)^3*Csc[a + b*x]) 
/(2*b) + ((c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b) - (3*d*(((4*I)*d* 
(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2 + ((2*I)*(c + d*x)^3*ArcTan[E^(I*(a 
 + b*x))])/d + (4*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b + ((c + d*x)^3*A 
rcTanh[Sin[a + b*x]])/d - ((4*I)*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))]) 
/b^2 - ((2*I)*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - ((3*I)*(c + d*x) 
^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b + ((2*I)*d^2*PolyLog[2, I*E^(I*(a + 
 b*x))])/b^3 + ((3*I)*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b + ((4*I 
)*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 + (4*d^2*PolyLog[3, -E^(I*( 
a + b*x))])/b^3 + (6*d*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^2 - ( 
6*d*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^2 - (4*d^2*PolyLog[3, E^(I* 
(a + b*x))])/b^3 + ((6*I)*d^2*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^3 - ((6* 
I)*d^2*PolyLog[4, I*E^(I*(a + b*x))])/b^3 + ((c + d*x)^2*Sec[a + b*x])/b)) 
/(2*b)
 

3.4.17.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4920

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]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.4.17.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1628 vs. \(2 (428 ) = 856\).

Time = 1.36 (sec) , antiderivative size = 1629, normalized size of antiderivative = 3.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(1629\)

input

int((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
 

output

-I/b^2/(exp(2*I*(b*x+a))+1)^2/(exp(2*I*(b*x+a))-1)*(3*d^3*x^3*b*exp(5*I*(b 
*x+a))+9*c*d^2*x^2*b*exp(5*I*(b*x+a))+9*c^2*d*x*b*exp(5*I*(b*x+a))+2*d^3*x 
^3*b*exp(3*I*(b*x+a))+3*c^3*b*exp(5*I*(b*x+a))+6*c*d^2*x^2*b*exp(3*I*(b*x+ 
a))-3*I*d^3*x^2*exp(5*I*(b*x+a))+6*c^2*d*x*b*exp(3*I*(b*x+a))+3*d^3*x^3*b* 
exp(I*(b*x+a))+3*I*c^2*d*exp(I*(b*x+a))+2*c^3*b*exp(3*I*(b*x+a))+9*c*d^2*x 
^2*b*exp(I*(b*x+a))+3*I*d^3*x^2*exp(I*(b*x+a))+9*c^2*d*x*b*exp(I*(b*x+a))+ 
3*c^3*b*exp(I*(b*x+a))-6*I*c*d^2*x*exp(5*I*(b*x+a))+6*I*c*d^2*x*exp(I*(b*x 
+a))-3*I*c^2*d*exp(5*I*(b*x+a)))-9*I/b^3*a^2*c*d^2*arctan(exp(I*(b*x+a)))- 
3*I*d^3*polylog(2,I*exp(I*(b*x+a)))/b^4-9*I*d^3*polylog(4,-I*exp(I*(b*x+a) 
))/b^4+9/b^3*c*d^2*polylog(3,I*exp(I*(b*x+a)))+3/b^3*d^3*ln(1-I*exp(I*(b*x 
+a)))*x+3/b^4*d^3*ln(1-I*exp(I*(b*x+a)))*a+3/b^2*d^3*ln(1-exp(I*(b*x+a)))* 
x^2+3/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^2-3/2/b*d^3*ln(1+I*exp(I*(b*x+a)))*x^ 
3-3/b^2*d^3*ln(exp(I*(b*x+a))+1)*x^2+3/2/b*d^3*ln(1-I*exp(I*(b*x+a)))*x^3+ 
9/b^3*d^3*polylog(3,I*exp(I*(b*x+a)))*x+3/b^4*d^3*a^2*ln(exp(I*(b*x+a))-1) 
-3/b^2*c^2*d*ln(exp(I*(b*x+a))+1)+3/b^2*c^2*d*ln(exp(I*(b*x+a))-1)+3*I*d^3 
*polylog(2,-I*exp(I*(b*x+a)))/b^4+9*I*d^3*polylog(4,I*exp(I*(b*x+a)))/b^4- 
9*I/b^2*c*d^2*polylog(2,I*exp(I*(b*x+a)))*x+9*I/b^2*c*d^2*polylog(2,-I*exp 
(I*(b*x+a)))*x+9*I/b^2*a*c^2*d*arctan(exp(I*(b*x+a)))-3/b^3*d^3*ln(1+I*exp 
(I*(b*x+a)))*x-3/b^4*d^3*ln(1+I*exp(I*(b*x+a)))*a-3/2/b^4*a^3*d^3*ln(1+I*e 
xp(I*(b*x+a)))-9/b^3*c*d^2*polylog(3,-I*exp(I*(b*x+a)))+3/2/b^4*a^3*d^3...
 

3.4.17.5 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. 2226 vs. \(2 (404) = 808\).

Time = 0.43 (sec) , antiderivative size = 2226, normalized size of antiderivative = 4.58 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^3,x, algorithm="fricas")
 

output

1/4*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 18*I*d^3*cos(b*x + 
a)^2*polylog(4, I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + 18*I*d^3*cos 
(b*x + a)^2*polylog(4, I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) - 18*I* 
d^3*cos(b*x + a)^2*polylog(4, -I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) 
 - 18*I*d^3*cos(b*x + a)^2*polylog(4, -I*cos(b*x + a) - sin(b*x + a))*sin( 
b*x + a) + 12*d^3*cos(b*x + a)^2*polylog(3, cos(b*x + a) + I*sin(b*x + a)) 
*sin(b*x + a) + 12*d^3*cos(b*x + a)^2*polylog(3, cos(b*x + a) - I*sin(b*x 
+ a))*sin(b*x + a) - 12*d^3*cos(b*x + a)^2*polylog(3, -cos(b*x + a) + I*si 
n(b*x + a))*sin(b*x + a) - 12*d^3*cos(b*x + a)^2*polylog(3, -cos(b*x + a) 
- I*sin(b*x + a))*sin(b*x + a) + 2*b^3*c^3 - 12*(I*b*d^3*x + I*b*c*d^2)*co 
s(b*x + a)^2*dilog(cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - 12*(-I*b* 
d^3*x - I*b*c*d^2)*cos(b*x + a)^2*dilog(cos(b*x + a) - I*sin(b*x + a))*sin 
(b*x + a) - 3*(3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 2*I*d^3 
)*cos(b*x + a)^2*dilog(I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) - 3*(3* 
I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d + 2*I*d^3)*cos(b*x + a)^2* 
dilog(I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) - 3*(-3*I*b^2*d^3*x^2 - 
6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^2*dilog(-I*cos(b*x 
 + a) + sin(b*x + a))*sin(b*x + a) - 3*(-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x 
 - 3*I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^2*dilog(-I*cos(b*x + a) - sin(b*x 
 + a))*sin(b*x + a) - 12*(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)^2*dilog(-...
 

3.4.17.6 Sympy [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Timed out} \]

input

integrate((d*x+c)**3*csc(b*x+a)**2*sec(b*x+a)**3,x)
 

output

Timed out
 

3.4.17.7 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. 8046 vs. \(2 (404) = 808\).

Time = 3.83 (sec) , antiderivative size = 8046, normalized size of antiderivative = 16.56 \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^3,x, algorithm="maxima")
 

output

-1/4*(c^3*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 3*lo 
g(sin(b*x + a) + 1) + 3*log(sin(b*x + a) - 1)) - 3*a*c^2*d*(2*(3*sin(b*x + 
 a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 3*log(sin(b*x + a) + 1) + 3*l 
og(sin(b*x + a) - 1))/b + 3*a^2*c*d^2*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + 
 a)^3 - sin(b*x + a)) - 3*log(sin(b*x + a) + 1) + 3*log(sin(b*x + a) - 1)) 
/b^2 - a^3*d^3*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 
 3*log(sin(b*x + a) + 1) + 3*log(sin(b*x + a) - 1))/b^3 - 4*(6*((b*x + a)^ 
3*d^3 + 2*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2 
*d - 6*a*b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a) - ((b*x + a)^3*d^3 + 2*b*c*d 
^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c*d^ 
2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(6*b*x + 6*a) - ((b*x + a)^3*d^3 + 2*b* 
c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a*b*c 
*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + ((b*x + a)^3*d^3 + 2 
*b*c*d^2 - 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + (3*b^2*c^2*d - 6*a* 
b*c*d^2 + (3*a^2 + 2)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (I*(b*x + a)^3*d^ 
3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + (3*I*b 
^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x + a))*sin(6*b*x + 6*a 
) - (I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 - 2*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3) 
*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 + 2*I)*d^3)*(b*x 
+ a))*sin(4*b*x + 4*a) - (-I*(b*x + a)^3*d^3 - 2*I*b*c*d^2 + 2*I*a*d^3 ...
 

3.4.17.8 Giac [F]

\[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{3} \,d x } \]

input

integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^3,x, algorithm="giac")
 

output

integrate((d*x + c)^3*csc(b*x + a)^2*sec(b*x + a)^3, x)
 

3.4.17.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \]

input

int((c + d*x)^3/(cos(a + b*x)^3*sin(a + b*x)^2),x)
 

output

\text{Hanged}