\(\int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx\) [312]

Optimal result

Integrand size = 22, antiderivative size = 195 \[ \int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx=\frac {(c+d x)^2}{2 b}-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {d^2 \log (\cos (a+b x))}{b^3}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {d (c+d x) \tan (a+b x)}{b^2}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b} \] Output:

1/2*(d*x+c)^2/b-2*(d*x+c)^2*arctanh(exp(2*I*(b*x+a)))/b-d^2*ln(cos(b*x+a)) 
/b^3+I*d*(d*x+c)*polylog(2,-exp(2*I*(b*x+a)))/b^2-I*d*(d*x+c)*polylog(2,ex 
p(2*I*(b*x+a)))/b^2-1/2*d^2*polylog(3,-exp(2*I*(b*x+a)))/b^3+1/2*d^2*polyl 
og(3,exp(2*I*(b*x+a)))/b^3-d*(d*x+c)*tan(b*x+a)/b^2+1/2*(d*x+c)^2*tan(b*x+ 
a)^2/b
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(883\) vs. \(2(195)=390\).

Time = 6.54 (sec) , antiderivative size = 883, normalized size of antiderivative = 4.53 \[ \int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2* 
I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2* 
Log[1 + E^((-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, -E^((-I) 
*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, E^((-I)*(a + b*x))] + ( 
6*I)*(1 - E^((-2*I)*a))*PolyLog[3, -E^((-I)*(a + b*x))] + (6*I)*(1 - E^((- 
2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))]))/b^3 + (x*(3*c^2 + 3*c*d*x + d^2* 
x^2)*Csc[a]*Sec[a])/3 - ((I/12)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I 
)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, 
-E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a 
+ b*x))])*Sec[a])/(b^3*E^(I*a)) + ((c + d*x)^2*Sec[a + b*x]^2)/(2*b) - (c^ 
2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b* 
(Cos[a]^2 + Sin[a]^2)) - (d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]* 
Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^2*Csc[a]*(-(b*x* 
Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + S 
in[a]^2)) - (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*( 
-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot 
[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*Arc 
Tan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - 
ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a] 
^2 + Sin[a]^2)]) + (Sec[a]*Sec[a + b*x]*(-(c*d*Sin[b*x]) - d^2*x*Sin[b*...
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.

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

\(\Big \downarrow \) 4920

\(\displaystyle -2 d \int \frac {1}{2} (c+d x) \left (\frac {\tan ^2(a+b x)}{b}+\frac {2 \log (\tan (a+b x))}{b}\right )dx+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {(c+d x)^2 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -d \int (c+d x) \left (\frac {\tan ^2(a+b x)}{b}+\frac {2 \log (\tan (a+b x))}{b}\right )dx+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {(c+d x)^2 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 7292

\(\displaystyle -d \int \frac {(c+d x) \left (\tan ^2(a+b x)+2 \log (\tan (a+b x))\right )}{b}dx+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {(c+d x)^2 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \int (c+d x) \left (\tan ^2(a+b x)+2 \log (\tan (a+b x))\right )dx}{b}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {(c+d x)^2 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {d \int \left ((c+d x) \tan ^2(a+b x)+2 (c+d x) \log (\tan (a+b x))\right )dx}{b}+\frac {(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac {(c+d x)^2 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((c + d*x)^2*Log[Tan[a + b*x]])/b + ((c + d*x)^2*Tan[a + b*x]^2)/(2*b) - ( 
d*(-1/2*(c + d*x)^2/d + (2*(c + d*x)^2*ArcTanh[E^((2*I)*(a + b*x))])/d + ( 
d*Log[Cos[a + b*x]])/b^2 + ((c + d*x)^2*Log[Tan[a + b*x]])/d - (I*(c + d*x 
)*PolyLog[2, -E^((2*I)*(a + b*x))])/b + (I*(c + d*x)*PolyLog[2, E^((2*I)*( 
a + b*x))])/b + (d*PolyLog[3, -E^((2*I)*(a + b*x))])/(2*b^2) - (d*PolyLog[ 
3, E^((2*I)*(a + b*x))])/(2*b^2) + ((c + d*x)*Tan[a + b*x])/b))/b
 

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] 
]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (175 ) = 350\).

Time = 0.47 (sec) , antiderivative size = 614, normalized size of antiderivative = 3.15

method	result	size
risch	\(-\frac {2 i c d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {2 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {d^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x^{2}}{b}+\frac {d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{3}}+\frac {2 b \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+4 b c d x \,{\mathrm e}^{2 i \left (b x +a \right )}+2 b \,c^{2} {\mathrm e}^{2 i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{2 i \left (b x +a \right )}-2 i d^{2} x -2 i d c}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {2 i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {2 i c d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}+\frac {2 c d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}-\frac {2 d c \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x}{b}+\frac {2 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}+\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {c^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b}+\frac {c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}+\frac {2 d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {d^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{3}}+\frac {i d c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{2}}\)	\(614\)

Input:

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

Output:

-1/b*c^2*ln(exp(2*I*(b*x+a))+1)-2*I/b^2*d^2*polylog(2,-exp(I*(b*x+a)))*x-2 
*I/b^2*d^2*polylog(2,exp(I*(b*x+a)))*x-2*I/b^2*c*d*polylog(2,-exp(I*(b*x+a 
)))-2*I/b^2*c*d*polylog(2,exp(I*(b*x+a)))+2/b^2*c*d*ln(1-exp(I*(b*x+a)))*a 
+2*(b*d^2*x^2*exp(2*I*(b*x+a))+2*b*c*d*x*exp(2*I*(b*x+a))+b*c^2*exp(2*I*(b 
*x+a))-I*d^2*x*exp(2*I*(b*x+a))-I*c*d*exp(2*I*(b*x+a))-I*d^2*x-I*d*c)/b^2/ 
(exp(2*I*(b*x+a))+1)^2-2/b*d*c*ln(exp(2*I*(b*x+a))+1)*x-1/b*d^2*ln(exp(2*I 
*(b*x+a))+1)*x^2+I/b^2*d^2*polylog(2,-exp(2*I*(b*x+a)))*x+I/b^2*d*c*polylo 
g(2,-exp(2*I*(b*x+a)))+1/b^3*d^2*a^2*ln(exp(I*(b*x+a))-1)+1/b*d^2*ln(exp(I 
*(b*x+a))+1)*x^2+1/b*d^2*ln(1-exp(I*(b*x+a)))*x^2-1/b^3*d^2*ln(1-exp(I*(b* 
x+a)))*a^2+2/b*c*d*ln(exp(I*(b*x+a))+1)*x-2/b^2*c*d*a*ln(exp(I*(b*x+a))-1) 
+2/b*c*d*ln(1-exp(I*(b*x+a)))*x-d^2/b^3*ln(exp(2*I*(b*x+a))+1)+1/b*c^2*ln( 
exp(I*(b*x+a))-1)+1/b*c^2*ln(exp(I*(b*x+a))+1)+2/b^3*d^2*ln(exp(I*(b*x+a)) 
)+2*d^2*polylog(3,-exp(I*(b*x+a)))/b^3+2*d^2*polylog(3,exp(I*(b*x+a)))/b^3 
-1/2*d^2*polylog(3,-exp(2*I*(b*x+a)))/b^3
 

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. 1404 vs. \(2 (171) = 342\).

Time = 0.18 (sec) , antiderivative size = 1404, normalized size of antiderivative = 7.20 \[ \int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(b^2*d^2*x^2 + 2*b^2*c*d*x + 2*d^2*cos(b*x + a)^2*polylog(3, cos(b*x + 
 a) + I*sin(b*x + a)) + 2*d^2*cos(b*x + a)^2*polylog(3, cos(b*x + a) - I*s 
in(b*x + a)) - 2*d^2*cos(b*x + a)^2*polylog(3, I*cos(b*x + a) + sin(b*x + 
a)) - 2*d^2*cos(b*x + a)^2*polylog(3, I*cos(b*x + a) - sin(b*x + a)) - 2*d 
^2*cos(b*x + a)^2*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 2*d^2*cos(b 
*x + a)^2*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) + 2*d^2*cos(b*x + a)^ 
2*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) + 2*d^2*cos(b*x + a)^2*polylo 
g(3, -cos(b*x + a) - I*sin(b*x + a)) + b^2*c^2 - 2*(I*b*d^2*x + I*b*c*d)*c 
os(b*x + a)^2*dilog(cos(b*x + a) + I*sin(b*x + a)) - 2*(-I*b*d^2*x - I*b*c 
*d)*cos(b*x + a)^2*dilog(cos(b*x + a) - I*sin(b*x + a)) - 2*(I*b*d^2*x + I 
*b*c*d)*cos(b*x + a)^2*dilog(I*cos(b*x + a) + sin(b*x + a)) - 2*(-I*b*d^2* 
x - I*b*c*d)*cos(b*x + a)^2*dilog(I*cos(b*x + a) - sin(b*x + a)) - 2*(-I*b 
*d^2*x - I*b*c*d)*cos(b*x + a)^2*dilog(-I*cos(b*x + a) + sin(b*x + a)) - 2 
*(I*b*d^2*x + I*b*c*d)*cos(b*x + a)^2*dilog(-I*cos(b*x + a) - sin(b*x + a) 
) - 2*(-I*b*d^2*x - I*b*c*d)*cos(b*x + a)^2*dilog(-cos(b*x + a) + I*sin(b* 
x + a)) - 2*(I*b*d^2*x + I*b*c*d)*cos(b*x + a)^2*dilog(-cos(b*x + a) - I*s 
in(b*x + a)) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(b*x + a)^2*log(co 
s(b*x + a) + I*sin(b*x + a) + 1) - (b^2*c^2 - 2*a*b*c*d + (a^2 + 1)*d^2)*c 
os(b*x + a)^2*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b^2*d^2*x^2 + 2*b^ 
2*c*d*x + b^2*c^2)*cos(b*x + a)^2*log(cos(b*x + a) - I*sin(b*x + a) + 1...
 

Sympy [F]

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

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

Output:

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

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. 2447 vs. \(2 (171) = 342\).

Time = 0.39 (sec) , antiderivative size = 2447, normalized size of antiderivative = 12.55 \[ \int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(c^2*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x 
+ a)^2)) - 2*a*c*d*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log 
(sin(b*x + a)^2))/b + a^2*d^2*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 
 - 1) - log(sin(b*x + a)^2))/b^2 - 2*(4*(b*x + a)*d^2*cos(4*b*x + 4*a) + 4 
*I*(b*x + a)*d^2*sin(4*b*x + 4*a) - 4*b*c*d + 4*a*d^2 - 2*((b*x + a)^2*d^2 
 + 2*(b*c*d - a*d^2)*(b*x + a) + d^2 + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2 
)*(b*x + a) + d^2)*cos(4*b*x + 4*a) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^ 
2)*(b*x + a) + d^2)*cos(2*b*x + 2*a) - (-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + 
 I*a*d^2)*(b*x + a) - I*d^2)*sin(4*b*x + 4*a) - 2*(-I*(b*x + a)^2*d^2 + 2* 
(-I*b*c*d + I*a*d^2)*(b*x + a) - I*d^2)*sin(2*b*x + 2*a))*arctan2(sin(2*b* 
x + 2*a), cos(2*b*x + 2*a) + 1) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*( 
b*x + a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(4*b*x + 4*a 
) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(2*b*x + 2*a) + ( 
I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(4*b*x + 4*a) + 2* 
(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(2*b*x + 2*a))*ar 
ctan2(sin(b*x + a), cos(b*x + a) + 1) - 2*((b*x + a)^2*d^2 + 2*(b*c*d - a* 
d^2)*(b*x + a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(4*b*x 
 + 4*a) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(2*b*x + 2* 
a) - (-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b*x + a))*sin(4*b*x + 4 
*a) - 2*(-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b*x + a))*sin(2*b...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx=\frac {2 \left (\int \csc \left (b x +a \right ) \sec \left (b x +a \right )^{3} x^{2}d x \right ) \sin \left (b x +a \right )^{2} b \,d^{2}-2 \left (\int \csc \left (b x +a \right ) \sec \left (b x +a \right )^{3} x^{2}d x \right ) b \,d^{2}+4 \left (\int \csc \left (b x +a \right ) \sec \left (b x +a \right )^{3} x d x \right ) \sin \left (b x +a \right )^{2} b c d -4 \left (\int \csc \left (b x +a \right ) \sec \left (b x +a \right )^{3} x d x \right ) b c d -2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{2} c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) c^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{2} c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} c^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c^{2}-\sin \left (b x +a \right )^{2} c^{2}}{2 b \left (\sin \left (b x +a \right )^{2}-1\right )} \] Input:

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

Output:

(2*int(csc(a + b*x)*sec(a + b*x)**3*x**2,x)*sin(a + b*x)**2*b*d**2 - 2*int 
(csc(a + b*x)*sec(a + b*x)**3*x**2,x)*b*d**2 + 4*int(csc(a + b*x)*sec(a + 
b*x)**3*x,x)*sin(a + b*x)**2*b*c*d - 4*int(csc(a + b*x)*sec(a + b*x)**3*x, 
x)*b*c*d - 2*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**2*c**2 + 2*log(tan((a 
 + b*x)/2) - 1)*c**2 - 2*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**2*c**2 + 
2*log(tan((a + b*x)/2) + 1)*c**2 + 2*log(tan((a + b*x)/2))*sin(a + b*x)**2 
*c**2 - 2*log(tan((a + b*x)/2))*c**2 - sin(a + b*x)**2*c**2)/(2*b*(sin(a + 
 b*x)**2 - 1))