\(\int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx\) [268]

Optimal result

Integrand size = 22, antiderivative size = 219 \[ \int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {2 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^2 \sec (a+b x)}{b} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.66 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.45 \[ \int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx=\frac {-4 b c d \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )-4 d^2 \arctan (\cot (a)) \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )+b^2 (c+d x)^2 \log \left (1-e^{i (a+b x)}\right )-b^2 (c+d x)^2 \log \left (1+e^{i (a+b x)}\right )+\frac {2 d^2 \csc (a) \left ((b x-\arctan (\cot (a))) \left (\log \left (1-e^{i (b x-\arctan (\cot (a)))}\right )-\log \left (1+e^{i (b x-\arctan (\cot (a)))}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i (b x-\arctan (\cot (a)))}\right )-i \operatorname {PolyLog}\left (2,e^{i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {\csc ^2(a)}}+2 i d \left (b (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+i d \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )\right )+2 d \left (-i b (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+d \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )\right )+b^2 (c+d x)^2 \sec (a+b x)}{b^3} \] Input:

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

Output:

(-4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - 4*d^2*ArcTan[Cot[a]]*Arc 
Tanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + b^2*(c + d*x)^2*Log[1 - E^(I*(a + b*x 
))] - b^2*(c + d*x)^2*Log[1 + E^(I*(a + b*x))] + (2*d^2*Csc[a]*((b*x - Arc 
Tan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - 
ArcTan[Cot[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyL 
og[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2] + (2*I)*d*(b*(c + d*x 
)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog[3, -E^(I*(a + b*x))]) + 2*d*( 
(-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + d*PolyLog[3, E^(I*(a + b*x) 
)]) + b^2*(c + d*x)^2*Sec[a + b*x])/b^3
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 254, 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, 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.

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

\(\Big \downarrow \) 4920

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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] 
]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (196 ) = 392\).

Time = 0.52 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.59

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 {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {d^{2} \ln \left ({\mathrm e}^{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 \,{\mathrm e}^{i \left (b x +a \right )} \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}+\frac {2 d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {2 i d^{2} \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\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^{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 {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {2 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {2 i c d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{3}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}-\frac {2 i d^{2} \operatorname {dilog}\left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}+\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 {4 i d c \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {4 i d^{2} a \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\)	\(568\)

Input:

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

Output:

-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/b^2*c*d*ln(1-exp(I*(b*x+a)))*a+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^3*d^2*dilog(I*exp(I*(b*x 
+a))+1)+2*I/b^3*d^2*dilog(1-I*exp(I*(b*x+a)))+2/b^2*d^2*ln(I*exp(I*(b*x+a) 
)+1)*x+2/b^3*d^2*ln(I*exp(I*(b*x+a))+1)*a-2/b^2*d^2*ln(1-I*exp(I*(b*x+a))) 
*x-2/b^3*d^2*ln(1-I*exp(I*(b*x+a)))*a+4*I/b^2*d*c*arctan(exp(I*(b*x+a)))-4 
*I/b^3*d^2*a*arctan(exp(I*(b*x+a)))+2*exp(I*(b*x+a))*(d^2*x^2+2*c*d*x+c^2) 
/b/(exp(2*I*(b*x+a))+1)+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+1/b*c^2*ln(exp(I*(b*x+a))-1)-1/b*c^2*ln(e 
xp(I*(b*x+a))+1)-2*d^2*polylog(3,-exp(I*(b*x+a)))/b^3+2*d^2*polylog(3,exp( 
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. 1035 vs. \(2 (185) = 370\).

Time = 0.14 (sec) , antiderivative size = 1035, normalized size of antiderivative = 4.73 \[ \int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c + d*x)**2*csc(a + b*x)*sec(a + b*x)**2, 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. 1590 vs. \(2 (185) = 370\).

Time = 0.53 (sec) , antiderivative size = 1590, normalized size of antiderivative = 7.26 \[ \int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx=\text {Exception raised: AttributeError} \] Input:

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

Output:

Exception raised: AttributeError >> type
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

(cos(a + b*x)*int(csc(a + b*x)*sec(a + b*x)**2*x**2,x)*b*d**2 + 2*cos(a + 
b*x)*int(csc(a + b*x)*sec(a + b*x)**2*x,x)*b*c*d + cos(a + b*x)*log(tan((a 
 + b*x)/2))*c**2 - cos(a + b*x)*c**2 + c**2)/(cos(a + b*x)*b)