\(\int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx\) [23]

Optimal result

Integrand size = 23, antiderivative size = 202 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=-\frac {i d^2 e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {i d^2 e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac {d e^{2 i e+2 i f x} (c+d x)}{4 a^2 f^2}+\frac {d e^{4 i e+4 i f x} (c+d x)}{32 a^2 f^2}+\frac {i e^{2 i e+2 i f x} (c+d x)^2}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^2}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d} \] Output:

-1/8*I*d^2*exp(2*I*e+2*I*f*x)/a^2/f^3+1/128*I*d^2*exp(4*I*e+4*I*f*x)/a^2/f 
^3-1/4*d*exp(2*I*e+2*I*f*x)*(d*x+c)/a^2/f^2+1/32*d*exp(4*I*e+4*I*f*x)*(d*x 
+c)/a^2/f^2+1/4*I*exp(2*I*e+2*I*f*x)*(d*x+c)^2/a^2/f-1/16*I*exp(4*I*e+4*I* 
f*x)*(d*x+c)^2/a^2/f+1/12*(d*x+c)^3/a^2/d
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\frac {32 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+48 ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) \cos (2 f x) (\cos (2 e)+i \sin (2 e))-3 ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d (i+(2+2 i) f x)) \cos (4 f x) (\cos (4 e)+i \sin (4 e))+48 i ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) (\cos (2 e)+i \sin (2 e)) \sin (2 f x)-3 (d-(2+2 i) c f-(2+2 i) d f x) (d+(2-2 i) c f+(2-2 i) d f x) (\cos (4 e)+i \sin (4 e)) \sin (4 f x)}{384 a^2 f^3} \] Input:

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

Output:

(32*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 48*((1 + I)*c*f + d*(-1 + (1 + I)* 
f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e 
]) - 3*((2 + 2*I)*c*f + d*(-1 + (2 + 2*I)*f*x))*((2 + 2*I)*c*f + d*(I + (2 
 + 2*I)*f*x))*Cos[4*f*x]*(Cos[4*e] + I*Sin[4*e]) + (48*I)*((1 + I)*c*f + d 
*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I)*f*x))*(Cos[2*e] + I*Sin 
[2*e])*Sin[2*f*x] - 3*(d - (2 + 2*I)*c*f - (2 + 2*I)*d*f*x)*(d + (2 - 2*I) 
*c*f + (2 - 2*I)*d*f*x)*(Cos[4*e] + I*Sin[4*e])*Sin[4*f*x])/(384*a^2*f^3)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 4212, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4212

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d (c+d x) e^{2 i e+2 i f x}}{4 a^2 f^2}+\frac {d (c+d x) e^{4 i e+4 i f x}}{32 a^2 f^2}+\frac {i (c+d x)^2 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^2 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {i d^2 e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {i d^2 e^{4 i e+4 i f x}}{128 a^2 f^3}\)

Input:

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

Output:

((-1/8*I)*d^2*E^((2*I)*e + (2*I)*f*x))/(a^2*f^3) + ((I/128)*d^2*E^((4*I)*e 
 + (4*I)*f*x))/(a^2*f^3) - (d*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(4*a^2*f^ 
2) + (d*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(32*a^2*f^2) + ((I/4)*E^((2*I)* 
e + (2*I)*f*x)*(c + d*x)^2)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + 
 d*x)^2)/(a^2*f) + (c + d*x)^3/(12*a^2*d)
 

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 4212

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

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {d^{2} x^{3}}{12 a^{2}}+\frac {d c \,x^{2}}{4 a^{2}}+\frac {c^{2} x}{4 a^{2}}+\frac {c^{3}}{12 a^{2} d}-\frac {i \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x +4 i d^{2} f x +8 c^{2} f^{2}+4 i c d f -d^{2}\right ) {\mathrm e}^{4 i \left (f x +e \right )}}{128 f^{3} a^{2}}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 i d^{2} f x +2 c^{2} f^{2}+2 i c d f -d^{2}\right ) {\mathrm e}^{2 i \left (f x +e \right )}}{8 f^{3} a^{2}}\)	\(173\)
parallelrisch	\(\frac {-60 \left (\left (-\frac {2}{15} d^{2} x^{2}-\frac {2}{5} c d x -\frac {2}{5} c^{2}\right ) f^{2}+i \left (\frac {d x}{2}+c \right ) d f -\frac {9 d^{2}}{20}\right ) f x \tan \left (f x +e \right )^{2}+\left (48 i x \left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) f^{3}+\left (-12 d^{2} x^{2}-24 c d x -72 c^{2}\right ) f^{2}-60 i d \left (\frac {d x}{10}+c \right ) f +27 d^{2}\right ) \tan \left (f x +e \right )+\left (-8 d^{2} x^{3}-24 x^{2} c d -24 x \,c^{2}\right ) f^{3}+\left (-18 i d^{2} x^{2}-36 i c d x -48 i c^{2}\right ) f^{2}+\left (21 d^{2} x +48 c d \right ) f +24 i d^{2}}{96 f^{3} a^{2} \left (-1+2 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}\right )}\)	\(224\)

Input:

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

Output:

1/12/a^2*d^2*x^3+1/4/a^2*d*c*x^2+1/4/a^2*c^2*x+1/12/a^2/d*c^3-1/128*I*(8*d 
^2*x^2*f^2+4*I*d^2*f*x+16*c*d*f^2*x+4*I*c*d*f+8*c^2*f^2-d^2)/f^3/a^2*exp(4 
*I*(f*x+e))+1/8*I*(2*d^2*x^2*f^2+2*I*d^2*f*x+4*c*d*f^2*x+2*I*c*d*f+2*c^2*f 
^2-d^2)/f^3/a^2*exp(2*I*(f*x+e))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\frac {32 \, d^{2} f^{3} x^{3} + 96 \, c d f^{3} x^{2} + 96 \, c^{2} f^{3} x - 3 \, {\left (8 i \, d^{2} f^{2} x^{2} + 8 i \, c^{2} f^{2} - 4 \, c d f - i \, d^{2} + 4 \, {\left (4 i \, c d f^{2} - d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 48 \, {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, c^{2} f^{2} + 2 \, c d f + i \, d^{2} + 2 \, {\left (-2 i \, c d f^{2} + d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{384 \, a^{2} f^{3}} \] Input:

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

Output:

1/384*(32*d^2*f^3*x^3 + 96*c*d*f^3*x^2 + 96*c^2*f^3*x - 3*(8*I*d^2*f^2*x^2 
 + 8*I*c^2*f^2 - 4*c*d*f - I*d^2 + 4*(4*I*c*d*f^2 - d^2*f)*x)*e^(4*I*f*x + 
 4*I*e) - 48*(-2*I*d^2*f^2*x^2 - 2*I*c^2*f^2 + 2*c*d*f + I*d^2 + 2*(-2*I*c 
*d*f^2 + d^2*f)*x)*e^(2*I*f*x + 2*I*e))/(a^2*f^3)
 

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\begin {cases} \frac {\left (256 i a^{2} c^{2} f^{5} e^{2 i e} + 512 i a^{2} c d f^{5} x e^{2 i e} - 256 a^{2} c d f^{4} e^{2 i e} + 256 i a^{2} d^{2} f^{5} x^{2} e^{2 i e} - 256 a^{2} d^{2} f^{4} x e^{2 i e} - 128 i a^{2} d^{2} f^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 64 i a^{2} c^{2} f^{5} e^{4 i e} - 128 i a^{2} c d f^{5} x e^{4 i e} + 32 a^{2} c d f^{4} e^{4 i e} - 64 i a^{2} d^{2} f^{5} x^{2} e^{4 i e} + 32 a^{2} d^{2} f^{4} x e^{4 i e} + 8 i a^{2} d^{2} f^{3} e^{4 i e}\right ) e^{4 i f x}}{1024 a^{4} f^{6}} & \text {for}\: a^{4} f^{6} \neq 0 \\\frac {x^{3} \left (d^{2} e^{4 i e} - 2 d^{2} e^{2 i e}\right )}{12 a^{2}} + \frac {x^{2} \left (c d e^{4 i e} - 2 c d e^{2 i e}\right )}{4 a^{2}} + \frac {x \left (c^{2} e^{4 i e} - 2 c^{2} e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{2} x}{4 a^{2}} + \frac {c d x^{2}}{4 a^{2}} + \frac {d^{2} x^{3}}{12 a^{2}} \] Input:

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

Output:

Piecewise((((256*I*a**2*c**2*f**5*exp(2*I*e) + 512*I*a**2*c*d*f**5*x*exp(2 
*I*e) - 256*a**2*c*d*f**4*exp(2*I*e) + 256*I*a**2*d**2*f**5*x**2*exp(2*I*e 
) - 256*a**2*d**2*f**4*x*exp(2*I*e) - 128*I*a**2*d**2*f**3*exp(2*I*e))*exp 
(2*I*f*x) + (-64*I*a**2*c**2*f**5*exp(4*I*e) - 128*I*a**2*c*d*f**5*x*exp(4 
*I*e) + 32*a**2*c*d*f**4*exp(4*I*e) - 64*I*a**2*d**2*f**5*x**2*exp(4*I*e) 
+ 32*a**2*d**2*f**4*x*exp(4*I*e) + 8*I*a**2*d**2*f**3*exp(4*I*e))*exp(4*I* 
f*x))/(1024*a**4*f**6), Ne(a**4*f**6, 0)), (x**3*(d**2*exp(4*I*e) - 2*d**2 
*exp(2*I*e))/(12*a**2) + x**2*(c*d*exp(4*I*e) - 2*c*d*exp(2*I*e))/(4*a**2) 
 + x*(c**2*exp(4*I*e) - 2*c**2*exp(2*I*e))/(4*a**2), True)) + c**2*x/(4*a* 
*2) + c*d*x**2/(4*a**2) + d**2*x**3/(12*a**2)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\frac {32 \, d^{2} f^{3} x^{3} + 96 \, c d f^{3} x^{2} - 24 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 \, c^{2} f^{3} x - 48 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 192 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} - 24 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 96 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} - 96 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 48 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{384 \, a^{2} f^{3}} \] Input:

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

Output:

1/384*(32*d^2*f^3*x^3 + 96*c*d*f^3*x^2 - 24*I*d^2*f^2*x^2*e^(4*I*f*x + 4*I 
*e) + 96*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) + 96*c^2*f^3*x - 48*I*c*d*f^2*x 
*e^(4*I*f*x + 4*I*e) + 192*I*c*d*f^2*x*e^(2*I*f*x + 2*I*e) - 24*I*c^2*f^2* 
e^(4*I*f*x + 4*I*e) + 12*d^2*f*x*e^(4*I*f*x + 4*I*e) + 96*I*c^2*f^2*e^(2*I 
*f*x + 2*I*e) - 96*d^2*f*x*e^(2*I*f*x + 2*I*e) + 12*c*d*f*e^(4*I*f*x + 4*I 
*e) - 96*c*d*f*e^(2*I*f*x + 2*I*e) + 3*I*d^2*e^(4*I*f*x + 4*I*e) - 48*I*d^ 
2*e^(2*I*f*x + 2*I*e))/(a^2*f^3)
 

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx={\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{8\,a^2\,f^3}+\frac {d^2\,x^2\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^2\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{128\,a^2\,f^3}+\frac {d^2\,x^2\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^2\,f^2}\right )+\frac {c^2\,x}{4\,a^2}+\frac {d^2\,x^3}{12\,a^2}+\frac {c\,d\,x^2}{4\,a^2} \] Input:

int((c + d*x)^2/(a + a*cot(e + f*x)*1i)^2,x)
 

Output:

exp(e*2i + f*x*2i)*(((2*c^2*f^2 - d^2 + c*d*f*2i)*1i)/(8*a^2*f^3) + (d^2*x 
^2*1i)/(4*a^2*f) + (d*x*(d*1i + 2*c*f)*1i)/(4*a^2*f^2)) - exp(e*4i + f*x*4 
i)*(((8*c^2*f^2 - d^2 + c*d*f*4i)*1i)/(128*a^2*f^3) + (d^2*x^2*1i)/(16*a^2 
*f) + (d*x*(d*1i + 4*c*f)*1i)/(32*a^2*f^2)) + (c^2*x)/(4*a^2) + (d^2*x^3)/ 
(12*a^2) + (c*d*x^2)/(4*a^2)
 

Reduce [F]

\[ \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx=\frac {-\left (\int \frac {x^{2}}{\cot \left (f x +e \right )^{2}-2 \cot \left (f x +e \right ) i -1}d x \right ) d^{2}-2 \left (\int \frac {x}{\cot \left (f x +e \right )^{2}-2 \cot \left (f x +e \right ) i -1}d x \right ) c d -\left (\int \frac {1}{\cot \left (f x +e \right )^{2}-2 \cot \left (f x +e \right ) i -1}d x \right ) c^{2}}{a^{2}} \] Input:

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

Output:

( - int(x**2/(cot(e + f*x)**2 - 2*cot(e + f*x)*i - 1),x)*d**2 - 2*int(x/(c 
ot(e + f*x)**2 - 2*cot(e + f*x)*i - 1),x)*c*d - int(1/(cot(e + f*x)**2 - 2 
*cot(e + f*x)*i - 1),x)*c**2)/a**2