\(\int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx\) [168]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\text {Int}\left (\frac {\cot (a+b x)}{c+d x},x\right ) \] Output:

-1/2*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d-1/2*cos(2*a-2*b*c/d)*Si(2*b*c/d+ 
2*b*x)/d+Defer(Int)(cot(b*x+a)/(d*x+c),x)
 

Mathematica [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \] Input:

Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x),x]
 

Output:

Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x), x]
 

Rubi [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {4908, 3042, 25, 4222, 4906, 27, 3042, 3784, 3042, 3780, 3783}

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.

\(\displaystyle \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx\)

\(\Big \downarrow \) 4908

\(\displaystyle \int \frac {\cot (a+b x)}{c+d x}dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan \left (a+b x+\frac {\pi }{2}\right )}{c+d x}dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x}dx\)

\(\Big \downarrow \) 4222

\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x}dx\)

\(\Big \downarrow \) 4906

\(\displaystyle -\int \frac {\sin (2 a+2 b x)}{2 (c+d x)}dx-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x}dx-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x}dx-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {1}{2} \left (-\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx-\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx-\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{2} \left (-\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {1}{2} \left (-\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )-\int \frac {\tan \left (\frac {1}{2} (2 a+\pi )+b x\right )}{c+d x}dx\)

Input:

Int[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x),x]
 

Output:

$Aborted
 

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 3042

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 4222

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> 
Simp[If[MatchQ[f, (f1_.)*(Complex[0, j_])], If[MatchQ[e, (e1_.) + Pi/2], I^ 
n*Unintegrable[(c + d*x)^m*Coth[(-I)*(e - Pi/2) - I*f*x]^n, x], I^n*Uninteg 
rable[(c + d*x)^m*Tanh[(-I)*e - I*f*x]^n, x]], If[MatchQ[e, (e1_.) + Pi/2], 
 (-1)^n*Unintegrable[(c + d*x)^m*Cot[e - Pi/2 + f*x]^n, x], Unintegrable[(c 
 + d*x)^m*Tan[e + f*x]^n, x]]], x] /; FreeQ[{c, d, e, f, m, n}, x] && Integ 
erQ[n]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 4908

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ 
(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr 
eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

Maple [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {\cos \left (b x +a \right )^{2} \cot \left (b x +a \right )}{d x +c}d x\]

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \] Input:

integrate(cos(b*x+a)^2*cot(b*x+a)/(d*x+c),x, algorithm="fricas")
 

Output:

integral(cos(b*x + a)^2*cot(b*x + a)/(d*x + c), x)
 

Sympy [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}}{c + d x}\, dx \] Input:

integrate(cos(b*x+a)**2*cot(b*x+a)/(d*x+c),x)
 

Output:

Integral(cos(a + b*x)**2*cot(a + b*x)/(c + d*x), x)
 

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 231, normalized size of antiderivative = 10.50 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \] Input:

integrate(cos(b*x+a)^2*cot(b*x+a)/(d*x+c),x, algorithm="maxima")
 

Output:

-1/4*((I*exp_integral_e(1, 2*(-I*b*d*x - I*b*c)/d) - I*exp_integral_e(1, - 
2*(-I*b*d*x - I*b*c)/d))*cos(-2*(b*c - a*d)/d) + 4*d*integrate(sin(b*x + a 
)/((d*x + c)*cos(b*x + a)^2 + (d*x + c)*sin(b*x + a)^2 + d*x + 2*(d*x + c) 
*cos(b*x + a) + c), x) - 4*d*integrate(sin(b*x + a)/((d*x + c)*cos(b*x + a 
)^2 + (d*x + c)*sin(b*x + a)^2 + d*x - 2*(d*x + c)*cos(b*x + a) + c), x) - 
 (exp_integral_e(1, 2*(-I*b*d*x - I*b*c)/d) + exp_integral_e(1, -2*(-I*b*d 
*x - I*b*c)/d))*sin(-2*(b*c - a*d)/d))/d
 

Giac [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c} \,d x } \] Input:

integrate(cos(b*x+a)^2*cot(b*x+a)/(d*x+c),x, algorithm="giac")
 

Output:

integrate(cos(b*x + a)^2*cot(b*x + a)/(d*x + c), x)
 

Mupad [N/A]

Not integrable

Time = 18.82 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )}{c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos \left (b x +a \right )^{2} \cot \left (b x +a \right )}{d x +c}d x \] Input:

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

Output:

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