3.3.67 \(\int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx\) [267]

3.3.67.1 Optimal result

Integrand size = 22, antiderivative size = 343 \[ \int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^3 \sec (a+b x)}{b} \]

output

6*I*d*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b^2-2*(d*x+c)^3*arctanh(exp(I*(b*x+ 
a)))/b+3*I*d*(d*x+c)^2*polylog(2,-exp(I*(b*x+a)))/b^2-6*I*d^2*(d*x+c)*poly 
log(2,-I*exp(I*(b*x+a)))/b^3+6*I*d^2*(d*x+c)*polylog(2,I*exp(I*(b*x+a)))/b 
^3-3*I*d*(d*x+c)^2*polylog(2,exp(I*(b*x+a)))/b^2-6*d^2*(d*x+c)*polylog(3,- 
exp(I*(b*x+a)))/b^3+6*d^3*polylog(3,-I*exp(I*(b*x+a)))/b^4-6*d^3*polylog(3 
,I*exp(I*(b*x+a)))/b^4+6*d^2*(d*x+c)*polylog(3,exp(I*(b*x+a)))/b^3-6*I*d^3 
*polylog(4,-exp(I*(b*x+a)))/b^4+6*I*d^3*polylog(4,exp(I*(b*x+a)))/b^4+(d*x 
+c)^3*sec(b*x+a)/b
 

3.3.67.2 Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {-2 b^3 (c+d x)^3 \text {arctanh}(\cos (a+b x)+i \sin (a+b x))-3 d \left (-2 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+2 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )+b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )-2 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )-b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )\right )+3 i d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,-\cos (a+b x)-i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,-\cos (a+b x)-i \sin (a+b x))\right )-3 i d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,\cos (a+b x)+i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,\cos (a+b x)+i \sin (a+b x))\right )+b^3 (c+d x)^3 \sec (a+b x)}{b^4} \]

input

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

output

(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] - 3*d*((-2*I)*b 
^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + 
b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b 
*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyL 
og[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b 
*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*( 
a + b*x))]) + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a 
+ b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 
 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x 
)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLo 
g[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Si 
n[a + b*x]]) + b^3*(c + d*x)^3*Sec[a + b*x])/b^4
 

3.3.67.3 Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4920, 25, 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.

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

\(\Big \downarrow \) 4920

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

-(((c + d*x)^3*ArcTanh[Cos[a + b*x]])/b) + (3*d*(((2*I)*(c + d*x)^2*ArcTan 
[E^(I*(a + b*x))])/b - (2*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/(3*d) + (( 
c + d*x)^3*ArcTanh[Cos[a + b*x]])/(3*d) + (I*(c + d*x)^2*PolyLog[2, -E^(I* 
(a + b*x))])/b - ((2*I)*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 
+ ((2*I)*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (I*(c + d*x)^2*P 
olyLog[2, E^(I*(a + b*x))])/b - (2*d*(c + d*x)*PolyLog[3, -E^(I*(a + b*x)) 
])/b^2 + (2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 - (2*d^2*PolyLog[3, 
I*E^(I*(a + b*x))])/b^3 + (2*d*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^2 
- ((2*I)*d^2*PolyLog[4, -E^(I*(a + b*x))])/b^3 + ((2*I)*d^2*PolyLog[4, E^( 
I*(a + b*x))])/b^3))/b + ((c + d*x)^3*Sec[a + b*x])/b
 

3.3.67.3.1 Defintions of rubi rules used

rule 25

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

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] 
]
 

3.3.67.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (308 ) = 616\).

Time = 1.86 (sec) , antiderivative size = 1152, normalized size of antiderivative = 3.36

method	result	size
risch	\(\text {Expression too large to display}\)	\(1152\)

input

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

output

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

3.3.67.5 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. 1705 vs. \(2 (293) = 586\).

Time = 0.37 (sec) , antiderivative size = 1705, normalized size of antiderivative = 4.97 \[ \int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

input

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

output

1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 + 6*I*d^3 
*cos(b*x + a)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - 6*I*d^3*cos(b*x 
+ a)*polylog(4, cos(b*x + a) - I*sin(b*x + a)) + 6*I*d^3*cos(b*x + a)*poly 
log(4, -cos(b*x + a) + I*sin(b*x + a)) - 6*I*d^3*cos(b*x + a)*polylog(4, - 
cos(b*x + a) - I*sin(b*x + a)) + 6*d^3*cos(b*x + a)*polylog(3, I*cos(b*x + 
 a) + sin(b*x + a)) - 6*d^3*cos(b*x + a)*polylog(3, I*cos(b*x + a) - sin(b 
*x + a)) + 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 
 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) - 3*(I*b^2* 
d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*cos(b*x + a)*dilog(cos(b*x + a) + 
 I*sin(b*x + a)) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*cos( 
b*x + a)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 6*(-I*b*d^3*x - I*b*c*d^2) 
*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a)) - 6*(-I*b*d^3*x - I*b*c 
*d^2)*cos(b*x + a)*dilog(I*cos(b*x + a) - sin(b*x + a)) - 6*(I*b*d^3*x + I 
*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - 6*(I*b*d^3* 
x + I*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) - sin(b*x + a)) - 3*(I*b 
^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*cos(b*x + a)*dilog(-cos(b*x + 
a) + I*sin(b*x + a)) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)* 
cos(b*x + a)*dilog(-cos(b*x + a) - I*sin(b*x + a)) - (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)*log(cos(b*x + a) + I*sin 
(b*x + a) + 1) - 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log...
 

3.3.67.6 Sympy [F]

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

input

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

output

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

3.3.67.7 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. 3202 vs. \(2 (293) = 586\).

Time = 0.85 (sec) , antiderivative size = 3202, normalized size of antiderivative = 9.34 \[ \int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

input

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

output

1/2*(c^3*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1)) 
- 3*a*c^2*d*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1 
))/b + 3*a^2*c*d^2*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos(b*x + 
 a) - 1))/b^2 - a^3*d^3*(2/cos(b*x + a) - log(cos(b*x + a) + 1) + log(cos( 
b*x + a) - 1))/b^3 + 2*(6*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2 
*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a 
)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*b 
^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - 
I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 
 1) + 6*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 
- a*d^3)*(b*x + a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 
+ 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c^2*d - 2*I*a*b 
*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a) 
)*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) - 2*((b*x + a 
)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a 
^2*d^3)*(b*x + a) + ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3 
*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b* 
x + a)^3*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2* 
I*a*b*c*d^2 - I*a^2*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a) 
, cos(b*x + a) + 1) - 2*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a...
 

3.3.67.8 Giac [F]

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

input

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

output

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

3.3.67.9 Mupad [F(-1)]

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

input

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

output

\text{Hanged}