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

3.4.18.1 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
 

3.4.18.2 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 = 7.33 (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=-\frac {6 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+4 i d^2 \arctan \left (e^{i (a+b x)}\right )-6 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )-3 b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+6 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )+3 b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-6 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+6 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+6 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{2 b^3}-\frac {(c+d x) \csc (a) \sec (a) (b c \cos (a)+b d x \cos (a)+d \sin (a))}{b^2}+\frac {4 i c d \arctan \left (\frac {i \cos (a)-i \sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{b^2 \sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c^2 \sin \left (\frac {b x}{2}\right )-2 c d x \sin \left (\frac {b x}{2}\right )-d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c^2 \sin \left (\frac {b x}{2}\right )+2 c d x \sin \left (\frac {b x}{2}\right )+d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {c^2+2 c d x+d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {-c d \sin \left (\frac {b x}{2}\right )-d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {-c^2-2 c d x-d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {c d \sin \left (\frac {b x}{2}\right )+d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {2 d^2 \left (-\frac {2 \arctan (\tan (a)) \text {arctanh}\left (\frac {-\cos (a)+\sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\left ((b x+\arctan (\tan (a))) \left (\log \left (1-e^{i (b x+\arctan (\tan (a)))}\right )-\log \left (1+e^{i (b x+\arctan (\tan (a)))}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (b x+\arctan (\tan (a)))}\right )-\operatorname {PolyLog}\left (2,e^{i (b x+\arctan (\tan (a)))}\right )\right )\right ) \sec (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^3} \]

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 + ...
 

3.4.18.3 Rubi [A] (verified)

Time = 1.02 (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
 

3.4.18.3.1 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] 
]
 

3.4.18.4 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 = 1.05 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.26

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

input

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

output

-2*d^2/b^3*a*ln(exp(I*(b*x+a))-1)-3*I/b*c^2*arctan(exp(I*(b*x+a)))+3/2/b*d 
^2*ln(1-I*exp(I*(b*x+a)))*x^2-3/2/b*d^2*ln(1+I*exp(I*(b*x+a)))*x^2-2*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-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+6*I/b^2*c*d*a*arctan(exp(I*(b*x+a)))-2*I/b^3*d^2*arctan(e 
xp(I*(b*x+a)))+2*I/b^3*d^2*dilog(exp(I*(b*x+a)))+2*I/b^3*d^2*dilog(exp(I*( 
b*x+a))+1)-3*I/b^3*d^2*a^2*arctan(exp(I*(b*x+a)))-3*I/b^2*d^2*polylog(2,I* 
exp(I*(b*x+a)))*x-3*I/b^2*c*d*polylog(2,I*exp(I*(b*x+a)))+3*I/b^2*c*d*poly 
log(2,-I*exp(I*(b*x+a)))+3*I/b^2*d^2*polylog(2,-I*exp(I*(b*x+a)))*x+3/b^2* 
c*d*ln(1-I*exp(I*(b*x+a)))*a-3/b^2*c*d*ln(1+I*exp(I*(b*x+a)))*a-3/b*c*d*ln 
(1+I*exp(I*(b*x+a)))*x+3/b*c*d*ln(1-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*x^2*d^2*b*exp(5*I*(b*x+a))+6*c*d*x*b*ex 
p(5*I*(b*x+a))+3*c^2*b*exp(5*I*(b*x+a))+2*x^2*d^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*x^2*d^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*d*c*exp(I*(b*x+a))) 
-3/2/b^3*a^2*d^2*ln(1-I*exp(I*(b*x+a)))+3/2/b^3*a^2*d^2*ln(1+I*exp(I*(b*x+ 
a)))
 

3.4.18.5 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. 1366 vs. \(2 (297) = 594\).

Time = 0.37 (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*...
 

3.4.18.6 Sympy [F]

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

input

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

output

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

3.4.18.7 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. 3828 vs. \(2 (297) = 594\).

Time = 1.12 (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...
 

3.4.18.8 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)
 

3.4.18.9 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}