\(\int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx\) [241]

Optimal result

Integrand size = 22, antiderivative size = 325 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}-\frac {2 (c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \] Output:

-3/2*I*d*(d*x+c)^2/b^2-1/2*(d*x+c)^3/b-2*(d*x+c)^3*arctanh(exp(2*I*(b*x+a) 
))/b-3/2*d*(d*x+c)^2*cot(b*x+a)/b^2-1/2*(d*x+c)^3*cot(b*x+a)^2/b+3*d^2*(d* 
x+c)*ln(1-exp(2*I*(b*x+a)))/b^3+3/2*I*d*(d*x+c)^2*polylog(2,-exp(2*I*(b*x+ 
a)))/b^2-3/2*I*d^3*polylog(2,exp(2*I*(b*x+a)))/b^4-3/2*I*d*(d*x+c)^2*polyl 
og(2,exp(2*I*(b*x+a)))/b^2-3/2*d^2*(d*x+c)*polylog(3,-exp(2*I*(b*x+a)))/b^ 
3+3/2*d^2*(d*x+c)*polylog(3,exp(2*I*(b*x+a)))/b^3-3/4*I*d^3*polylog(4,-exp 
(2*I*(b*x+a)))/b^4+3/4*I*d^3*polylog(4,exp(2*I*(b*x+a)))/b^4
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1526\) vs. \(2(325)=650\).

Time = 6.63 (sec) , antiderivative size = 1526, normalized size of antiderivative = 4.70 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*((c + d*x)^3*Csc[a + b*x]^2)/b - (c*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))] 
))/(2*b^3) - (d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E 
^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a) 
)*x^3*Log[1 + E^((-I)*(a + b*x))] - 6*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[2 
, -E^((-I)*(a + b*x))] - 6*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, E^((-I)*( 
a + b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyLog[3, -E^((-I)*(a + b*x))] 
 + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyLog[3, E^((-I)*(a + b*x))] + 12*(1 - 
E^((-2*I)*a))*PolyLog[4, -E^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*Poly 
Log[4, E^((-I)*(a + b*x))]))/(4*b^4) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 
 + d^3*x^3)*Csc[a]*Sec[a])/4 - ((I/4)*c*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*Pol 
yLog[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)) - ((I/8)*d^3*E^(I*a)*((2*b^4*x^4)/ 
E^((2*I)*a) - (4*I)*b^3*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x) 
)] + 6*b^2*(1 + E^((-2*I)*a))*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (...
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.08, 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.

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

\(\Big \downarrow \) 4920

\(\displaystyle -3 d \int -\frac {1}{2} (c+d x)^2 \left (\frac {\cot ^2(a+b x)}{b}-\frac {2 \log (\tan (a+b x))}{b}\right )dx-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} d \int (c+d x)^2 \left (\frac {\cot ^2(a+b x)}{b}-\frac {2 \log (\tan (a+b x))}{b}\right )dx-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {3}{2} d \int \frac {(c+d x)^2 \left (\cot ^2(a+b x)-2 \log (\tan (a+b x))\right )}{b}dx-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 d \int (c+d x)^2 \left (\cot ^2(a+b x)-2 \log (\tan (a+b x))\right )dx}{2 b}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {3 d \int \left ((c+d x)^2 \cot ^2(a+b x)-2 (c+d x)^2 \log (\tan (a+b x))\right )dx}{2 b}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 d \left (-\frac {4 (c+d x)^3 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{3 d}-\frac {i d^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}-\frac {i d^2 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {i d^2 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {d (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^2}+\frac {d (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^2}+\frac {2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}+\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^2 \cot (a+b x)}{b}-\frac {2 (c+d x)^3 \log (\tan (a+b x))}{3 d}-\frac {i (c+d x)^2}{b}-\frac {(c+d x)^3}{3 d}\right )}{2 b}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {(c+d x)^3 \log (\tan (a+b x))}{b}\)

Input:

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

Output:

-1/2*((c + d*x)^3*Cot[a + b*x]^2)/b + ((c + d*x)^3*Log[Tan[a + b*x]])/b + 
(3*d*(((-I)*(c + d*x)^2)/b - (c + d*x)^3/(3*d) - (4*(c + d*x)^3*ArcTanh[E^ 
((2*I)*(a + b*x))])/(3*d) - ((c + d*x)^2*Cot[a + b*x])/b + (2*d*(c + d*x)* 
Log[1 - E^((2*I)*(a + b*x))])/b^2 - (2*(c + d*x)^3*Log[Tan[a + b*x]])/(3*d 
) + (I*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + b*x))])/b - (I*d^2*PolyLog[2, 
 E^((2*I)*(a + b*x))])/b^3 - (I*(c + d*x)^2*PolyLog[2, E^((2*I)*(a + b*x)) 
])/b - (d*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))])/b^2 + (d*(c + d*x)*P 
olyLog[3, E^((2*I)*(a + b*x))])/b^2 - ((I/2)*d^2*PolyLog[4, -E^((2*I)*(a + 
 b*x))])/b^3 + ((I/2)*d^2*PolyLog[4, E^((2*I)*(a + b*x))])/b^3))/(2*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. 1222 vs. \(2 (279 ) = 558\).

Time = 0.63 (sec) , antiderivative size = 1223, normalized size of antiderivative = 3.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1223\)

Input:

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

Output:

-3/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2-6*I/b^2*c*d^2*polylog(2,-exp(I*(b*x+ 
a)))*x-6*I/b^2*c*d^2*polylog(2,exp(I*(b*x+a)))*x+3*I/b^2*c*d^2*polylog(2,- 
exp(2*I*(b*x+a)))*x+(2*b*d^3*x^3*exp(2*I*(b*x+a))-3*I*d^3*x^2*exp(2*I*(b*x 
+a))+6*b*c*d^2*x^2*exp(2*I*(b*x+a))-6*I*c*d^2*x*exp(2*I*(b*x+a))+6*b*c^2*d 
*x*exp(2*I*(b*x+a))-3*I*c^2*d*exp(2*I*(b*x+a))+3*I*d^3*x^2+2*b*c^3*exp(2*I 
*(b*x+a))+6*I*c*d^2*x+3*I*c^2*d)/b^2/(exp(2*I*(b*x+a))-1)^2-6*I*d^3/b^3*a* 
x-3/b*c^2*d*ln(exp(2*I*(b*x+a))+1)*x-3/b*c*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+ 
3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2+3/2*I/b^2*c^2*d*polylog(2,- 
exp(2*I*(b*x+a)))-6*d^2/b^3*c*ln(exp(I*(b*x+a)))+3*d^2/b^3*c*ln(exp(I*(b*x 
+a))-1)+3*d^2/b^3*c*ln(exp(I*(b*x+a))+1)+3*d^3/b^3*ln(exp(I*(b*x+a))+1)*x+ 
3*d^3/b^3*ln(1-exp(I*(b*x+a)))*x+3*d^3/b^4*ln(1-exp(I*(b*x+a)))*a+6*d^3/b^ 
4*a*ln(exp(I*(b*x+a)))-3*d^3/b^4*a*ln(exp(I*(b*x+a))-1)-3*I*d^3/b^2*x^2-3* 
I*d^3/b^4*a^2-3*I*d^3/b^4*polylog(2,-exp(I*(b*x+a)))-3/2/b^3*c*d^2*polylog 
(3,-exp(2*I*(b*x+a)))-3/2/b^3*d^3*polylog(3,-exp(2*I*(b*x+a)))*x+6*I/b^4*d 
^3*polylog(4,-exp(I*(b*x+a)))-1/b^4*d^3*a^3*ln(exp(I*(b*x+a))-1)+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)))+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*d^3*poly 
log(3,-exp(I*(b*x+a)))*x+6/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x+1/b^4*d^3*l 
n(1-exp(I*(b*x+a)))*a^3+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*c*d^2*ln(e...
 

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. 3475 vs. \(2 (270) = 540\).

Time = 0.23 (sec) , antiderivative size = 3475, normalized size of antiderivative = 10.69 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((d*x+c)**3*csc(b*x+a)**3*sec(b*x+a),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. 5165 vs. \(2 (270) = 540\).

Time = 2.30 (sec) , antiderivative size = 5165, normalized size of antiderivative = 15.89 \[ \int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(c^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2 
)) - 3*a*c^2*d*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log(sin(b*x + 
 a)^2))/b + 3*a^2*c*d^2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) - log( 
sin(b*x + a)^2))/b^2 - a^3*d^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1) 
 - log(sin(b*x + a)^2))/b^3 - 2*(18*b^2*c^2*d - 36*a*b*c*d^2 + 18*a^2*d^3 
- 2*(4*(b*x + a)^3*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2*c^2*d - 
2*a*b*c*d^2 + a^2*d^3)*(b*x + a) + (4*(b*x + a)^3*d^3 + 9*(b*c*d^2 - a*d^3 
)*(b*x + a)^2 + 9*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*cos(4*b*x 
 + 4*a) - 2*(4*(b*x + a)^3*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2* 
c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-4*I*(b*x + 
a)^3*d^3 + 9*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 9*(-I*b^2*c^2*d + 2*I*a* 
b*c*d^2 - I*a^2*d^3)*(b*x + a))*sin(4*b*x + 4*a) - 2*(4*I*(b*x + a)^3*d^3 
+ 9*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 9*(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(2*b*x + 2*a), cos(2*b*x 
 + 2*a) + 1) + 6*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*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 + 1)*d^3)*(b*x + a) + 
((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*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 + 1)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 
 2*((b*x + a)^3*d^3 + 3*b*c*d^2 - 3*a*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 + 1)*d^3)*(b*x + a))*cos(2*b*x + ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx=\frac {4 \left (\int \csc \left (b x +a \right )^{3} \sec \left (b x +a \right ) x^{3}d x \right ) \sin \left (b x +a \right )^{2} b \,d^{3}+12 \left (\int \csc \left (b x +a \right )^{3} \sec \left (b x +a \right ) x^{2}d x \right ) \sin \left (b x +a \right )^{2} b c \,d^{2}+12 \left (\int \csc \left (b x +a \right )^{3} \sec \left (b x +a \right ) x d x \right ) \sin \left (b x +a \right )^{2} b \,c^{2} d -4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{2} c^{3}-4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{2} c^{3}+4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} c^{3}+\sin \left (b x +a \right )^{2} c^{3}-2 c^{3}}{4 \sin \left (b x +a \right )^{2} b} \] Input:

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

Output:

(4*int(csc(a + b*x)**3*sec(a + b*x)*x**3,x)*sin(a + b*x)**2*b*d**3 + 12*in 
t(csc(a + b*x)**3*sec(a + b*x)*x**2,x)*sin(a + b*x)**2*b*c*d**2 + 12*int(c 
sc(a + b*x)**3*sec(a + b*x)*x,x)*sin(a + b*x)**2*b*c**2*d - 4*log(tan((a + 
 b*x)/2) - 1)*sin(a + b*x)**2*c**3 - 4*log(tan((a + b*x)/2) + 1)*sin(a + b 
*x)**2*c**3 + 4*log(tan((a + b*x)/2))*sin(a + b*x)**2*c**3 + sin(a + b*x)* 
*2*c**3 - 2*c**3)/(4*sin(a + b*x)**2*b)