\(\int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx\) [230]

Optimal result

Integrand size = 20, antiderivative size = 127 \[ \int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx=-\frac {2 (c+d x)^2 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}+\frac {d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \] Output:

-2*(d*x+c)^2*arctanh(exp(2*I*(b*x+a)))/b+I*d*(d*x+c)*polylog(2,-exp(2*I*(b 
*x+a)))/b^2-I*d*(d*x+c)*polylog(2,exp(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*polylog(3,exp(2*I*(b*x+a)))/b^3
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.68 \[ \int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx=\frac {-4 b^2 c^2 \text {arctanh}\left (e^{2 i (a+b x)}\right )+4 b^2 c d x \log \left (1-e^{2 i (a+b x)}\right )+2 b^2 d^2 x^2 \log \left (1-e^{2 i (a+b x)}\right )-4 b^2 c d x \log \left (1+e^{2 i (a+b x)}\right )-2 b^2 d^2 x^2 \log \left (1+e^{2 i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )-d^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )+d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \] Input:

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

Output:

(-4*b^2*c^2*ArcTanh[E^((2*I)*(a + b*x))] + 4*b^2*c*d*x*Log[1 - E^((2*I)*(a 
 + b*x))] + 2*b^2*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] - 4*b^2*c*d*x*Log[1 
 + E^((2*I)*(a + b*x))] - 2*b^2*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] + (2* 
I)*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))] - (2*I)*b*d*(c + d*x)*Po 
lyLog[2, E^((2*I)*(a + b*x))] - d^2*PolyLog[3, -E^((2*I)*(a + b*x))] + d^2 
*PolyLog[3, E^((2*I)*(a + b*x))])/(2*b^3)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4919, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 4919

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle 2 \left (\frac {d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}\right )}{b}-\frac {d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b}-\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}\right )}{b}-\frac {(c+d x)^2 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}\right )\)

\(\Big \downarrow \) 7143

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

Input:

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

Output:

2*(-(((c + d*x)^2*ArcTanh[E^((2*I)*(a + b*x))])/b) + (d*(((I/2)*(c + d*x)* 
PolyLog[2, -E^((2*I)*(a + b*x))])/b - (d*PolyLog[3, -E^((2*I)*(a + b*x))]) 
/(4*b^2)))/b - (d*(((I/2)*(c + d*x)*PolyLog[2, E^((2*I)*(a + b*x))])/b - ( 
d*PolyLog[3, E^((2*I)*(a + b*x))])/(4*b^2)))/b)
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 4919

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n 
, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (111 ) = 222\).

Time = 0.34 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.69

method	result	size
risch	\(\frac {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {i d c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{2}}-\frac {d^{2} \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{3}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{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 (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}+\frac {2 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\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 d c \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) x}{b}+\frac {2 c d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}+\frac {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{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 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 {c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}-\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 {i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 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 {2 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}\)	\(469\)

Input:

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

Output:

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

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. 1098 vs. \(2 (107) = 214\).

Time = 0.15 (sec) , antiderivative size = 1098, normalized size of antiderivative = 8.65 \[ \int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c + d*x)**2*csc(a + b*x)*sec(a + b*x), 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. 599 vs. \(2 (107) = 214\).

Time = 0.29 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.72 \[ \int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(c^2*(log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2)) - 2*a*c*d*(log(s 
in(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b + a^2*d^2*(log(sin(b*x + a)^2 
- 1) - log(sin(b*x + a)^2))/b^2 + (d^2*polylog(3, -e^(2*I*b*x + 2*I*a)) - 
4*d^2*polylog(3, -e^(I*b*x + I*a)) - 4*d^2*polylog(3, e^(I*b*x + I*a)) - 2 
*(-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b*x + a))*arctan2(sin(2*b*x 
 + 2*a), cos(2*b*x + 2*a) + 1) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d 
^2)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 2*(-I*(b*x + a)^2 
*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b*x + a))*arctan2(sin(b*x + a), -cos(b*x + 
a) + 1) - 2*(I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*dilog(-e^(2*I*b*x + 2*I* 
a)) - 4*(-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*dilog(-e^(I*b*x + I*a)) - 4 
*(-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*dilog(e^(I*b*x + I*a)) + ((b*x + a 
)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x 
+ 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)* 
(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - ((b 
*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x 
+ a)^2 - 2*cos(b*x + a) + 1))/b^2)/b
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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