\(\int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 650 \[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=-\frac {2 i b^2 (c+d x)^2}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^2}{(a-i b) (a+i b)^2 \left (i a+b-(i a-b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^3}{3 (a+i b)^2 d}-\frac {4 b (c+d x)^3}{3 (a+i b)^2 (i a+b) d}-\frac {4 b^2 (c+d x)^3}{3 \left (a^2+b^2\right )^2 d}+\frac {2 b^2 d (c+d x) \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {2 b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f}-\frac {2 i b^2 (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {2 b d (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a+i b)^2 (i a+b) f^2}-\frac {2 b^2 d (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {b d^2 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f^3}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3} \] Output:

-2*I*b^2*(d*x+c)^2/(a^2+b^2)^2/f-2*b^2*(d*x+c)^2/(a-I*b)/(a+I*b)^2/(I*a+b- 
(I*a-b)*exp(2*I*e+2*I*f*x))/f+1/3*(d*x+c)^3/(a+I*b)^2/d-4/3*b*(d*x+c)^3/(a 
+I*b)^2/(I*a+b)/d-4/3*b^2*(d*x+c)^3/(a^2+b^2)^2/d+2*b^2*d*(d*x+c)*ln(1-(a+ 
I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^2-2*b*(d*x+c)^2*ln(1-(a+I*b 
)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a-I*b)/(a+I*b)^2/f-2*I*b^2*(d*x+c)^2*ln(1-( 
a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f-I*b^2*d^2*polylog(2,(a+I* 
b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^3-2*b*d*(d*x+c)*polylog(2,(a+ 
I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a+I*b)^2/(I*a+b)/f^2-2*b^2*d*(d*x+c)*pol 
ylog(2,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^2-b*d^2*polylog(3 
,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a-I*b)/(a+I*b)^2/f^3-I*b^2*d^2*polyl 
og(3,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^3
 

Mathematica [A] (warning: unable to verify)

Time = 6.22 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.18 \[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\frac {\frac {2 b \left (\frac {12 c (-b d+a c f) x}{a+i b}+\frac {12 c \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (-b d+a c f) x}{a^2+b^2}+\frac {6 d (-b d+2 a c f) x^2}{a+i b}+\frac {4 a d^2 f x^3}{a+i b}-\frac {6 d \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (b d-2 a c f) x \log \left (1+\frac {(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right )}{(a-i b) (-i a+b) f}+\frac {6 a d^2 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) x^2 \log \left (1+\frac {(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right )}{(a-i b) (-i a+b)}+\frac {6 c \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (-b d+a c f) \log \left (a-i b-(a+i b) e^{2 i (e+f x)}\right )}{(a-i b) (-i a+b) f}+\frac {3 d \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (b d-2 a c f) \operatorname {PolyLog}\left (2,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )}{\left (a^2+b^2\right ) f^2}-\frac {3 a d^2 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) \left (2 f x \operatorname {PolyLog}\left (2,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )-i \operatorname {PolyLog}\left (3,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )\right )}{\left (a^2+b^2\right ) f^2}\right )}{-i a \left (-1+e^{2 i e}\right )+b \left (1+e^{2 i e}\right )}+\frac {\left (a^2-b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cos (f x)-\left (a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cos (2 e+f x)+2 b \left (3 b (c+d x)^2-a f x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \sin (f x)}{(b \cos (e)+a \sin (e)) (b \cos (e+f x)+a \sin (e+f x))}}{6 \left (a^2+b^2\right ) f} \] Input:

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

Output:

((2*b*((12*c*(-(b*d) + a*c*f)*x)/(a + I*b) + (12*c*(a*(-1 + E^((2*I)*e)) + 
 I*b*(1 + E^((2*I)*e)))*(-(b*d) + a*c*f)*x)/(a^2 + b^2) + (6*d*(-(b*d) + 2 
*a*c*f)*x^2)/(a + I*b) + (4*a*d^2*f*x^3)/(a + I*b) - (6*d*(a*(-1 + E^((2*I 
)*e)) + I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*x*Log[1 + (-a + I*b)/((a + 
I*b)*E^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)*f) + (6*a*d^2*(a*(-1 + 
 E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*x^2*Log[1 + (-a + I*b)/((a + I*b)*E 
^((2*I)*(e + f*x)))])/((a - I*b)*((-I)*a + b)) + (6*c*(a*(-1 + E^((2*I)*e) 
) + I*b*(1 + E^((2*I)*e)))*(-(b*d) + a*c*f)*Log[a - I*b - (a + I*b)*E^((2* 
I)*(e + f*x))])/((a - I*b)*((-I)*a + b)*f) + (3*d*(a*(-1 + E^((2*I)*e)) + 
I*b*(1 + E^((2*I)*e)))*(b*d - 2*a*c*f)*PolyLog[2, (a - I*b)/((a + I*b)*E^( 
(2*I)*(e + f*x)))])/((a^2 + b^2)*f^2) - (3*a*d^2*(a*(-1 + E^((2*I)*e)) + I 
*b*(1 + E^((2*I)*e)))*(2*f*x*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + 
 f*x)))] - I*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 
 + b^2)*f^2)))/((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e))) + ((a^2 - 
 b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[f*x] - (a^2 + b^2)*f*x*(3*c^2 + 
3*c*d*x + d^2*x^2)*Cos[2*e + f*x] + 2*b*(3*b*(c + d*x)^2 - a*f*x*(3*c^2 + 
3*c*d*x + d^2*x^2))*Sin[f*x])/((b*Cos[e] + a*Sin[e])*(b*Cos[e + f*x] + a*S 
in[e + f*x])))/(6*(a^2 + b^2)*f)
 

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

\(\displaystyle \int \left (-\frac {4 b^2 (c+d x)^2}{(-b+i a)^2 \left (i a \left (1-\frac {i b}{a}\right )-i a \left (1+\frac {i b}{a}\right ) e^{2 i e+2 i f x}\right )^2}+\frac {4 b (c+d x)^2}{(a+i b)^2 \left (i a \left (1+\frac {i b}{a}\right ) e^{2 i e+2 i f x}-i a \left (1-\frac {i b}{a}\right )\right )}+\frac {(c+d x)^2}{(a+i b)^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3885 vs. \(2 (584 ) = 1168\).

Time = 0.93 (sec) , antiderivative size = 3886, normalized size of antiderivative = 5.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(3886\)

Input:

int((d*x+c)^2/(a+b*cot(f*x+e))^2,x,method=_RETURNVERBOSE)
 

Output:

4*I/(b-I*a)^2/f^3/(I*a+b)*b*e^2*a*d^2/(I*b-a)*ln(exp(I*(f*x+e)))-I/(b-I*a) 
^2/f/(I*a+b)*b*a^2*c^2/(I*b-a)/(a+I*b)*ln(a^2*exp(4*I*(f*x+e))+exp(4*I*(f* 
x+e))*b^2-2*a^2*exp(2*I*(f*x+e))+2*b^2*exp(2*I*(f*x+e))+a^2+b^2)+2*I/(b-I* 
a)^2/f^2/(I*a+b)*b^3*c*d/(I*b-a)/(a+I*b)*arctan(1/2/a*b*exp(2*I*(f*x+e))+1 
/2/a*b+1/2/b*a*exp(2*I*(f*x+e))-1/2/b*a)+2*I/(b-I*a)^2/f^2/(I*a+b)*b^3*c*d 
/(a+I*b)/(I*b-a)*arctan(1/b*a)+2*I/(b-I*a)^2/f/(I*a+b)*b/(a-I*b)*d^2*a*ln( 
1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x^2-2*I/(b-I*a)^2/f^3/(I*a+b)*b/(a-I*b 
)*e^2*a*d^2*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))-8/3/(b-I*a)^2/f^3/(I*a+ 
b)*b/(a-I*b)*e^3*a*d^2-4/(b-I*a)^2/f^2/(I*a+b)*b^2/(a-I*b)*d^2*e*x+4/(b-I* 
a)^2/(I*a+b)*b/(a-I*b)*d*a*c*x^2+2/(b-I*a)^2/f^2/(I*a+b)*b^2*c*d/(I*b-a)/( 
a+I*b)*arctan(1/2/a*b*exp(2*I*(f*x+e))+1/2/a*b+1/2/b*a*exp(2*I*(f*x+e))-1/ 
2/b*a)*a+2/(b-I*a)^2/f^2/(I*a+b)*b^2*c*d/(a+I*b)/(I*b-a)*arctan(1/b*a)*a-2 
/(b-I*a)^2/f^3/(I*a+b)*b^2*e*d^2/(I*b-a)/(a+I*b)*arctan(1/2/a*b*exp(2*I*(f 
*x+e))+1/2/a*b+1/2/b*a*exp(2*I*(f*x+e))-1/2/b*a)*a-2/(b-I*a)^2/f^3/(I*a+b) 
*b^2*e*d^2/(a+I*b)/(I*b-a)*arctan(1/b*a)*a+1/(b-I*a)^2/f^3/(I*a+b)*b^2*e^2 
*a*d^2/(I*b-a)/(a+I*b)*ln(a^2*exp(4*I*(f*x+e))+exp(4*I*(f*x+e))*b^2-2*a^2* 
exp(2*I*(f*x+e))+2*b^2*exp(2*I*(f*x+e))+a^2+b^2)-2/(b-I*a)^2/f^3/(I*a+b)*b 
*e^2*a^2*d^2/(I*b-a)/(a+I*b)*arctan(1/2/a*b*exp(2*I*(f*x+e))+1/2/a*b+1/2/b 
*a*exp(2*I*(f*x+e))-1/2/b*a)-2/(b-I*a)^2/f^3/(I*a+b)*b*e^2*a^2*d^2/(a+I*b) 
/(I*b-a)*arctan(1/b*a)+8/(b-I*a)^2/f/(I*a+b)*b/(a-I*b)*d*a*c*e*x-2*I/(b...
 

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. 2038 vs. \(2 (531) = 1062\).

Time = 0.14 (sec) , antiderivative size = 2038, normalized size of antiderivative = 3.14 \[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*(2*(a^2*b - b^3)*d^2*f^3*x^3 - 6*a*b^2*c^2*f^2 - 6*(a*b^2*d^2*f^2 - (a 
^2*b - b^3)*c*d*f^3)*x^2 - 6*(2*a*b^2*c*d*f^2 - (a^2*b - b^3)*c^2*f^3)*x + 
 2*((a^2*b - b^3)*d^2*f^3*x^3 - 3*a*b^2*c^2*f^2 - 3*(a*b^2*d^2*f^2 - (a^2* 
b - b^3)*c*d*f^3)*x^2 - 3*(2*a*b^2*c*d*f^2 - (a^2*b - b^3)*c^2*f^3)*x)*cos 
(2*f*x + 2*e) - 3*(-2*I*a*b^2*d^2*f*x - 2*I*a*b^2*c*d*f + I*b^3*d^2 + (-2* 
I*a*b^2*d^2*f*x - 2*I*a*b^2*c*d*f + I*b^3*d^2)*cos(2*f*x + 2*e) + (-2*I*a^ 
2*b*d^2*f*x - 2*I*a^2*b*c*d*f + I*a*b^2*d^2)*sin(2*f*x + 2*e))*dilog(-(a^2 
 + b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (-I*a^2 + 2*a*b + I*b^2) 
*sin(2*f*x + 2*e))/(a^2 + b^2) + 1) - 3*(2*I*a*b^2*d^2*f*x + 2*I*a*b^2*c*d 
*f - I*b^3*d^2 + (2*I*a*b^2*d^2*f*x + 2*I*a*b^2*c*d*f - I*b^3*d^2)*cos(2*f 
*x + 2*e) + (2*I*a^2*b*d^2*f*x + 2*I*a^2*b*c*d*f - I*a*b^2*d^2)*sin(2*f*x 
+ 2*e))*dilog(-(a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^ 
2 + 2*a*b - I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2) + 1) - 6*(a*b^2*d^2*e^2 + 
 a*b^2*c^2*f^2 + b^3*d^2*e - (2*a*b^2*c*d*e + b^3*c*d)*f + (a*b^2*d^2*e^2 
+ a*b^2*c^2*f^2 + b^3*d^2*e - (2*a*b^2*c*d*e + b^3*c*d)*f)*cos(2*f*x + 2*e 
) + (a^2*b*d^2*e^2 + a^2*b*c^2*f^2 + a*b^2*d^2*e - (2*a^2*b*c*d*e + a*b^2* 
c*d)*f)*sin(2*f*x + 2*e))*log(1/2*a^2 + I*a*b - 1/2*b^2 - 1/2*(a^2 + b^2)* 
cos(2*f*x + 2*e) + 1/2*(I*a^2 + I*b^2)*sin(2*f*x + 2*e)) - 6*(a*b^2*d^2*e^ 
2 + a*b^2*c^2*f^2 + b^3*d^2*e - (2*a*b^2*c*d*e + b^3*c*d)*f + (a*b^2*d^2*e 
^2 + a*b^2*c^2*f^2 + b^3*d^2*e - (2*a*b^2*c*d*e + b^3*c*d)*f)*cos(2*f*x...
 

Sympy [F]

\[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (a + b \cot {\left (e + f x \right )}\right )^{2}}\, dx \] Input:

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

Output:

Integral((c + d*x)**2/(a + b*cot(e + f*x))**2, 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. 2552 vs. \(2 (531) = 1062\).

Time = 0.73 (sec) , antiderivative size = 2552, normalized size of antiderivative = 3.93 \[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(6*(b^2/((a^4 + a^2*b^2)*f*tan(f*x + e) + (a^3*b + a*b^3)*f) + 2*a*b*l 
og(a*tan(f*x + e) + b)/((a^4 + 2*a^2*b^2 + b^4)*f) - a*b*log(tan(f*x + e)^ 
2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*f) - (a^2 - b^2)*(f*x + e)/((a^4 + 2*a^2*b 
^2 + b^4)*f))*c*d*e - 3*(2*a*b*log(a*tan(f*x + e) + b)/(a^4 + 2*a^2*b^2 + 
b^4) - a*b*log(tan(f*x + e)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^2 - b^2)*( 
f*x + e)/(a^4 + 2*a^2*b^2 + b^4) + b^2/(a^3*b + a*b^3 + (a^4 + a^2*b^2)*ta 
n(f*x + e)))*c^2 - ((a^3 + I*a^2*b + a*b^2 + I*b^3)*(f*x + e)^3*d^2 + 3*(a 
^3 + I*a^2*b + a*b^2 + I*b^3)*(f*x + e)*d^2*e^2 + 6*(I*a*b^2 + b^3)*d^2*e^ 
2 - 3*((a^3 + I*a^2*b + a*b^2 + I*b^3)*d^2*e - (a^3 + I*a^2*b + a*b^2 + I* 
b^3)*c*d*f)*(f*x + e)^2 + 6*((-I*a^2*b - a*b^2)*d^2*e^2 + (-I*a*b^2 - b^3) 
*d^2*e + (I*a*b^2 + b^3)*c*d*f + ((I*a^2*b - a*b^2)*d^2*e^2 + (I*a*b^2 - b 
^3)*d^2*e + (-I*a*b^2 + b^3)*c*d*f)*cos(2*f*x + 2*e) - ((a^2*b + I*a*b^2)* 
d^2*e^2 + (a*b^2 + I*b^3)*d^2*e - (a*b^2 + I*b^3)*c*d*f)*sin(2*f*x + 2*e)) 
*arctan2(b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, a*cos(2*f*x + 2*e) - 
 b*sin(2*f*x + 2*e) - a) + 6*((-I*a^2*b - a*b^2)*(f*x + e)^2*d^2 + (2*(I*a 
^2*b + a*b^2)*d^2*e + 2*(-I*a^2*b - a*b^2)*c*d*f + (I*a*b^2 + b^3)*d^2)*(f 
*x + e) + ((I*a^2*b - a*b^2)*(f*x + e)^2*d^2 + (2*(-I*a^2*b + a*b^2)*d^2*e 
 + 2*(I*a^2*b - a*b^2)*c*d*f + (-I*a*b^2 + b^3)*d^2)*(f*x + e))*cos(2*f*x 
+ 2*e) - ((a^2*b + I*a*b^2)*(f*x + e)^2*d^2 - (2*(a^2*b + I*a*b^2)*d^2*e - 
 2*(a^2*b + I*a*b^2)*c*d*f + (a*b^2 + I*b^3)*d^2)*(f*x + e))*sin(2*f*x ...
 

Giac [F]

\[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b \cot \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^2}{(a+b \cot (e+f x))^2} \, dx=\text {too large to display} \] Input:

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

Output:

(72*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2)) 
*cos(e + f*x)*a**5*b*d**2*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)* 
b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a**4*b**2*c*d*f*i + 96*sqrt(a** 
2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x 
)*a**3*b**3*d**2*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i 
)/sqrt(a**2 + b**2))*cos(e + f*x)*a**2*b**4*c*d*f*i + 24*sqrt(a**2 + b**2) 
*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a*b**5* 
d**2*i + 72*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 
+ b**2))*sin(e + f*x)*a**6*d**2*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f* 
x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + f*x)*a**5*b*c*d*f*i + 96*sqrt( 
a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + 
f*x)*a**4*b**2*d**2*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - 
a*i)/sqrt(a**2 + b**2))*sin(e + f*x)*a**3*b**3*c*d*f*i + 24*sqrt(a**2 + b* 
*2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + f*x)*a**2 
*b**4*d**2*i - 24*cos(e + f*x)*int(x**2/(tan((e + f*x)/2)**4*b**2 - 4*tan( 
(e + f*x)/2)**3*a*b + 4*tan((e + f*x)/2)**2*a**2 - 2*tan((e + f*x)/2)**2*b 
**2 + 4*tan((e + f*x)/2)*a*b + b**2),x)*a**7*b**2*d**2*f**3 - 48*cos(e + f 
*x)*int(x**2/(tan((e + f*x)/2)**4*b**2 - 4*tan((e + f*x)/2)**3*a*b + 4*tan 
((e + f*x)/2)**2*a**2 - 2*tan((e + f*x)/2)**2*b**2 + 4*tan((e + f*x)/2)*a* 
b + b**2),x)*a**5*b**4*d**2*f**3 - 24*cos(e + f*x)*int(x**2/(tan((e + f...