\(\int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx\) [318]

Optimal result

Integrand size = 24, antiderivative size = 341 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b} \] Output:

-3*I*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b+2*d^2*x*arctanh(exp(I*(b*x+a)))/b^ 
2-6*d*(d*x+c)*arctanh(exp(I*(b*x+a)))/b^2-d^2*x*arctanh(cos(b*x+a))/b^2+d* 
(d*x+c)*arctanh(cos(b*x+a))/b^2+d^2*arctanh(sin(b*x+a))/b^3-3/2*(d*x+c)^2* 
csc(b*x+a)/b+2*I*d^2*polylog(2,-exp(I*(b*x+a)))/b^3+3*I*d*(d*x+c)*polylog( 
2,-I*exp(I*(b*x+a)))/b^2-3*I*d*(d*x+c)*polylog(2,I*exp(I*(b*x+a)))/b^2-2*I 
*d^2*polylog(2,exp(I*(b*x+a)))/b^3-3*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+ 
3*d^2*polylog(3,I*exp(I*(b*x+a)))/b^3-d*(d*x+c)*sec(b*x+a)/b^2+1/2*(d*x+c) 
^2*csc(b*x+a)*sec(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. \(889\) vs. \(2(341)=682\).

Time = 7.08 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.61 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*((6*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + (4*I)*d^2*ArcTan[E^(I*(a + b 
*x))] - 6*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] - 3*b^2*d^2*x^2*Log[1 - I*E 
^(I*(a + b*x))] + 6*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] + 3*b^2*d^2*x^2*L 
og[1 + I*E^(I*(a + b*x))] - (6*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + 
b*x))] + (6*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + 6*d^2*PolyLog 
[3, (-I)*E^(I*(a + b*x))] - 6*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - ((c 
 + d*x)*Csc[a]*Sec[a]*(b*c*Cos[a] + b*d*x*Cos[a] + d*Sin[a]))/b^2 + ((4*I) 
*c*d*ArcTan[(I*Cos[a] - I*Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]]) 
/(b^2*Sqrt[Cos[a]^2 + Sin[a]^2]) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c^2*Sin 
[(b*x)/2]) - 2*c*d*x*Sin[(b*x)/2] - d^2*x^2*Sin[(b*x)/2]))/(2*b) + (Csc[a/ 
2]*Csc[a/2 + (b*x)/2]*(c^2*Sin[(b*x)/2] + 2*c*d*x*Sin[(b*x)/2] + d^2*x^2*S 
in[(b*x)/2]))/(2*b) + (c^2 + 2*c*d*x + d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] - 
 Sin[a/2 + (b*x)/2])^2) + (-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2])/(b^2* 
(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-c^2 - 
 2*c*d*x - d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])^2) + (c 
*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2] + Sin[a/2])*(Cos[a/2 
+ (b*x)/2] + Sin[a/2 + (b*x)/2])) + (2*d^2*((-2*ArcTan[Tan[a]]*ArcTanh[(-C 
os[a] + Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/Sqrt[Cos[a]^2 + S 
in[a]^2] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] 
 - Log[1 + E^(I*(b*x + ArcTan[Tan[a]]))]) + I*(PolyLog[2, -E^(I*(b*x + ...
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.09, 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.

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

\(\Big \downarrow \) 4920

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(3*(c + d*x)^2*ArcTanh[Sin[a + b*x]])/(2*b) - (3*(c + d*x)^2*Csc[a + b*x]) 
/(2*b) + ((c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b) - (d*(((3*I)*(c + 
 d*x)^2*ArcTan[E^(I*(a + b*x))])/d - (2*d*x*ArcTanh[E^(I*(a + b*x))])/b + 
(6*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b + (d*x*ArcTanh[Cos[a + b*x]])/b - 
 ((c + d*x)*ArcTanh[Cos[a + b*x]])/b - (d*ArcTanh[Sin[a + b*x]])/b^2 + (3* 
(c + d*x)^2*ArcTanh[Sin[a + b*x]])/(2*d) - ((2*I)*d*PolyLog[2, -E^(I*(a + 
b*x))])/b^2 - ((3*I)*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b + ((3*I 
)*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b + ((2*I)*d*PolyLog[2, E^(I*(a 
 + b*x))])/b^2 + (3*d*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^2 - (3*d*PolyLog 
[3, I*E^(I*(a + b*x))])/b^2 + ((c + d*x)*Sec[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. 769 vs. \(2 (310 ) = 620\).

Time = 0.71 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.26

method	result	size
risch	\(-\frac {3 i d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {i \left (3 d^{2} x^{2} b \,{\mathrm e}^{5 i \left (b x +a \right )}+6 c d x b \,{\mathrm e}^{5 i \left (b x +a \right )}+3 c^{2} b \,{\mathrm e}^{5 i \left (b x +a \right )}+2 d^{2} x^{2} b \,{\mathrm e}^{3 i \left (b x +a \right )}+4 c d x b \,{\mathrm e}^{3 i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{5 i \left (b x +a \right )}+2 c^{2} b \,{\mathrm e}^{3 i \left (b x +a \right )}+3 d^{2} x^{2} b \,{\mathrm e}^{i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{5 i \left (b x +a \right )}+6 c d x b \,{\mathrm e}^{i \left (b x +a \right )}+3 c^{2} b \,{\mathrm e}^{i \left (b x +a \right )}+2 i d^{2} x \,{\mathrm e}^{i \left (b x +a \right )}+2 i c d \,{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 c d \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {3 a^{2} d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )}{2 b^{3}}+\frac {3 d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{2 b}-\frac {3 d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{2 b}-\frac {2 i d^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 i c^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {2 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {3 a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{2 b^{3}}+\frac {6 i c d a \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}+\frac {3 i d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {3 c d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {3 i c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {3 i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 c d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {3 i c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {3 c d \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}-\frac {3 d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\)	\(770\)

Input:

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

Output:

-3/b^2*c*d*ln(I*exp(I*(b*x+a))+1)*a+3/b*c*d*ln(1-I*exp(I*(b*x+a)))*x+6*I/b 
^2*c*d*a*arctan(exp(I*(b*x+a)))+3/2/b^3*a^2*d^2*ln(I*exp(I*(b*x+a))+1)+3/2 
/b*d^2*ln(1-I*exp(I*(b*x+a)))*x^2-3/2/b*d^2*ln(I*exp(I*(b*x+a))+1)*x^2-3/2 
/b^3*a^2*d^2*ln(1-I*exp(I*(b*x+a)))-2*I/b^3*d^2*arctan(exp(I*(b*x+a)))+2*I 
/b^3*dilog(exp(I*(b*x+a))+1)*d^2+2*I/b^3*d^2*dilog(exp(I*(b*x+a)))-3*I/b*c 
^2*arctan(exp(I*(b*x+a)))+3/b^2*c*d*ln(1-I*exp(I*(b*x+a)))*a-3/b*c*d*ln(I* 
exp(I*(b*x+a))+1)*x+2*d/b^2*c*ln(exp(I*(b*x+a))-1)-2*d/b^2*c*ln(exp(I*(b*x 
+a))+1)-2*d^2/b^2*ln(exp(I*(b*x+a))+1)*x-2*d^2/b^3*a*ln(exp(I*(b*x+a))-1)- 
3*I/b^3*d^2*a^2*arctan(exp(I*(b*x+a)))-3*I/b^2*c*d*polylog(2,I*exp(I*(b*x+ 
a)))+3*I/b^2*c*d*polylog(2,-I*exp(I*(b*x+a)))-3*I/b^2*d^2*polylog(2,I*exp( 
I*(b*x+a)))*x+3*I/b^2*d^2*polylog(2,-I*exp(I*(b*x+a)))*x-I/b^2/(exp(2*I*(b 
*x+a))+1)^2/(exp(2*I*(b*x+a))-1)*(3*d^2*x^2*b*exp(5*I*(b*x+a))+6*c*d*x*b*e 
xp(5*I*(b*x+a))+3*c^2*b*exp(5*I*(b*x+a))+2*d^2*x^2*b*exp(3*I*(b*x+a))+4*c* 
d*x*b*exp(3*I*(b*x+a))-2*I*d^2*x*exp(5*I*(b*x+a))+2*c^2*b*exp(3*I*(b*x+a)) 
+3*d^2*x^2*b*exp(I*(b*x+a))-2*I*c*d*exp(5*I*(b*x+a))+6*c*d*x*b*exp(I*(b*x+ 
a))+3*c^2*b*exp(I*(b*x+a))+2*I*d^2*x*exp(I*(b*x+a))+2*I*c*d*exp(I*(b*x+a)) 
)-3*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+3*d^2*polylog(3,I*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. 1366 vs. \(2 (297) = 594\).

Time = 0.18 (sec) , antiderivative size = 1366, normalized size of antiderivative = 4.01 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

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. 3828 vs. \(2 (297) = 594\).

Time = 0.79 (sec) , antiderivative size = 3828, normalized size of antiderivative = 11.23 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

(4032*cos(a + b*x)*sin(a + b*x)**2*b**2*c*d*x - 207*cos(a + b*x)*sin(a + b 
*x)**2*b**2*d**2*x**2 - 207*cos(a + b*x)*sin(a + b*x)**2*d**2 + 640*cos(a 
+ b*x)*sin(a + b*x)*b*c*d + 2576*cos(a + b*x)*sin(a + b*x)*b*d**2*x - 3072 
*cos(a + b*x)*b**2*c*d*x + 1656*cos(a + b*x)*b**2*d**2*x**2 + 828*int(x**2 
/(tan((a + b*x)/2)**8 - 3*tan((a + b*x)/2)**6 + 3*tan((a + b*x)/2)**4 - ta 
n((a + b*x)/2)**2),x)*sin(a + b*x)**3*b**3*d**2 - 828*int(x**2/(tan((a + b 
*x)/2)**8 - 3*tan((a + b*x)/2)**6 + 3*tan((a + b*x)/2)**4 - tan((a + b*x)/ 
2)**2),x)*sin(a + b*x)*b**3*d**2 - 5244*int(x**2/(tan((a + b*x)/2)**6 - 3* 
tan((a + b*x)/2)**4 + 3*tan((a + b*x)/2)**2 - 1),x)*sin(a + b*x)**3*b**3*d 
**2 + 5244*int(x**2/(tan((a + b*x)/2)**6 - 3*tan((a + b*x)/2)**4 + 3*tan(( 
a + b*x)/2)**2 - 1),x)*sin(a + b*x)*b**3*d**2 + 736*int((tan((a + b*x)/2)* 
*7*x)/(tan((a + b*x)/2)**6 - 3*tan((a + b*x)/2)**4 + 3*tan((a + b*x)/2)**2 
 - 1),x)*sin(a + b*x)**3*b**2*d**2 - 736*int((tan((a + b*x)/2)**7*x)/(tan( 
(a + b*x)/2)**6 - 3*tan((a + b*x)/2)**4 + 3*tan((a + b*x)/2)**2 - 1),x)*si 
n(a + b*x)*b**2*d**2 - 1536*int(x/(tan((a + b*x)/2)**8 - 3*tan((a + b*x)/2 
)**6 + 3*tan((a + b*x)/2)**4 - tan((a + b*x)/2)**2),x)*sin(a + b*x)**3*b** 
3*c*d + 1536*int(x/(tan((a + b*x)/2)**8 - 3*tan((a + b*x)/2)**6 + 3*tan((a 
 + b*x)/2)**4 - tan((a + b*x)/2)**2),x)*sin(a + b*x)*b**3*c*d - 4048*int(x 
/(tan((a + b*x)/2)**7 - 3*tan((a + b*x)/2)**5 + 3*tan((a + b*x)/2)**3 - ta 
n((a + b*x)/2)),x)*sin(a + b*x)**3*b**2*d**2 + 4048*int(x/(tan((a + b*x...