\(\int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx\) [17]

Optimal result

Integrand size = 23, antiderivative size = 137 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=-\frac {d^2 x}{4 a f^2}+\frac {i (c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}+\frac {i d^2}{4 f^3 (a+i a \cot (e+f x))}+\frac {d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))} \] Output:

-1/4*d^2*x/a/f^2+1/4*I*(d*x+c)^2/a/f+1/6*(d*x+c)^3/a/d+1/4*I*d^2/f^3/(a+I* 
a*cot(f*x+e))+1/2*d*(d*x+c)/f^2/(a+I*a*cot(f*x+e))-1/2*I*(d*x+c)^2/f/(a+I* 
a*cot(f*x+e))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+3 ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) \cos (2 f x) (\cos (2 e)+i \sin (2 e))+3 i ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) (\cos (2 e)+i \sin (2 e)) \sin (2 f x)}{24 a f^3} \] Input:

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

Output:

(4*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*((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*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])/(24*a*f^3)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4206, 3042, 4206, 3042, 3960, 24}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4206

\(\displaystyle \frac {i d \int \frac {c+d x}{i \cot (e+f x) a+a}dx}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {i d \int \frac {c+d x}{a-i a \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\)

\(\Big \downarrow \) 4206

\(\displaystyle \frac {i d \left (\frac {i d \int \frac {1}{i \cot (e+f x) a+a}dx}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {i d \left (\frac {i d \int \frac {1}{a-i a \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\)

\(\Big \downarrow \) 3960

\(\displaystyle \frac {i d \left (\frac {i d \left (\frac {\int 1dx}{2 a}-\frac {i}{2 f (a+i a \cot (e+f x))}\right )}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {i d \left (-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {i d \left (\frac {x}{2 a}-\frac {i}{2 f (a+i a \cot (e+f x))}\right )}{2 f}\right )}{f}+\frac {(c+d x)^3}{6 a d}\)

Input:

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

Output:

(c + d*x)^3/(6*a*d) - ((I/2)*(c + d*x)^2)/(f*(a + I*a*Cot[e + f*x])) + (I* 
d*((c + d*x)^2/(4*a*d) - ((I/2)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])) + (( 
I/2)*d*(x/(2*a) - (I/2)/(f*(a + I*a*Cot[e + f*x]))))/f))/f
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

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

rule 3960

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + 
b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a)   Int[(a + b*Tan[c + d*x])^ 
(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
 

rule 4206

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + (Simp[a*d*(m/(2*b*f)) 
   Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x], x] - Simp[a*((c + d*x)^m 
/(2*b*f*(a + b*Tan[e + f*x]))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[ 
a^2 + b^2, 0] && GtQ[m, 0]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {d^{2} x^{3}}{6 a}+\frac {d c \,x^{2}}{2 a}+\frac {c^{2} x}{2 a}+\frac {c^{3}}{6 a d}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 i d^{2} f x +2 c^{2} f^{2}+2 i c d f -d^{2}\right ) {\mathrm e}^{2 i \left (f x +e \right )}}{8 f^{3} a}\)	\(108\)
parallelrisch	\(\frac {-6 \left (\left (-\frac {1}{3} d^{2} x^{2}-c d x -c^{2}\right ) f^{2}+i \left (\frac {d x}{2}+c \right ) d f -\frac {d^{2}}{2}\right ) f x \tan \left (f x +e \right )+6 i x \left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) f^{3}+\left (-3 d^{2} x^{2}-6 c d x -6 c^{2}\right ) f^{2}-6 i \left (\frac {d x}{2}+c \right ) d f +3 d^{2}}{12 f^{3} a \left (i+\tan \left (f x +e \right )\right )}\)	\(133\)
default	\(\frac {c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 c d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 c d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\sin \left (f x +e \right )^{2}}{4}\right )}{f}+\frac {d^{2} e^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}-\frac {2 d^{2} e \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\sin \left (f x +e \right )^{2}}{4}\right )}{f^{2}}+\frac {d^{2} \left (\left (f x +e \right )^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right ) \cos \left (f x +e \right )^{2}}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}-\frac {\left (f x +e \right )^{3}}{3}\right )}{f^{2}}+\frac {i c^{2} \cos \left (f x +e \right )^{2}}{2}-\frac {i c d e \cos \left (f x +e \right )^{2}}{f}-\frac {2 i c d \left (-\frac {\left (f x +e \right ) \cos \left (f x +e \right )^{2}}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f}+\frac {i d^{2} e^{2} \cos \left (f x +e \right )^{2}}{2 f^{2}}+\frac {2 i d^{2} e \left (-\frac {\left (f x +e \right ) \cos \left (f x +e \right )^{2}}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {i d^{2} \left (-\frac {\left (f x +e \right )^{2} \cos \left (f x +e \right )^{2}}{2}+\left (f x +e \right ) \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\sin \left (f x +e \right )^{2}}{4}\right )}{f^{2}}}{a f}\)	\(509\)

Input:

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

Output:

1/6/a*d^2*x^3+1/2/a*d*c*x^2+1/2/a*c^2*x+1/6/a/d*c^3+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*exp(2*I*(f*x+e))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 12 \, c^{2} f^{3} x - 3 \, {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, c^{2} f^{2} + 2 \, c d f + i \, d^{2} + 2 \, {\left (-2 i \, c d f^{2} + d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a f^{3}} \] Input:

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

Output:

1/24*(4*d^2*f^3*x^3 + 12*c*d*f^3*x^2 + 12*c^2*f^3*x - 3*(-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*f^3)
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\begin {cases} \frac {\left (2 i c^{2} f^{2} e^{2 i e} + 4 i c d f^{2} x e^{2 i e} - 2 c d f e^{2 i e} + 2 i d^{2} f^{2} x^{2} e^{2 i e} - 2 d^{2} f x e^{2 i e} - i d^{2} e^{2 i e}\right ) e^{2 i f x}}{8 a f^{3}} & \text {for}\: a f^{3} \neq 0 \\- \frac {c^{2} x e^{2 i e}}{2 a} - \frac {c d x^{2} e^{2 i e}}{2 a} - \frac {d^{2} x^{3} e^{2 i e}}{6 a} & \text {otherwise} \end {cases} + \frac {c^{2} x}{2 a} + \frac {c d x^{2}}{2 a} + \frac {d^{2} x^{3}}{6 a} \] Input:

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

Output:

Piecewise(((2*I*c**2*f**2*exp(2*I*e) + 4*I*c*d*f**2*x*exp(2*I*e) - 2*c*d*f 
*exp(2*I*e) + 2*I*d**2*f**2*x**2*exp(2*I*e) - 2*d**2*f*x*exp(2*I*e) - I*d* 
*2*exp(2*I*e))*exp(2*I*f*x)/(8*a*f**3), Ne(a*f**3, 0)), (-c**2*x*exp(2*I*e 
)/(2*a) - c*d*x**2*exp(2*I*e)/(2*a) - d**2*x**3*exp(2*I*e)/(6*a), True)) + 
 c**2*x/(2*a) + c*d*x**2/(2*a) + d**2*x**3/(6*a)
 

Maxima [F(-2)]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 6 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c^{2} f^{3} x + 12 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a f^{3}} \] Input:

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

Output:

1/24*(4*d^2*f^3*x^3 + 12*c*d*f^3*x^2 + 6*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) 
 + 12*c^2*f^3*x + 12*I*c*d*f^2*x*e^(2*I*f*x + 2*I*e) + 6*I*c^2*f^2*e^(2*I* 
f*x + 2*I*e) - 6*d^2*f*x*e^(2*I*f*x + 2*I*e) - 6*c*d*f*e^(2*I*f*x + 2*I*e) 
 - 3*I*d^2*e^(2*I*f*x + 2*I*e))/(a*f^3)
 

Mupad [B] (verification not implemented)

Time = 10.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=-\frac {6\,c^2\,f^2\,\sin \left (2\,e+2\,f\,x\right )-12\,c^2\,f^3\,x-3\,d^2\,\sin \left (2\,e+2\,f\,x\right )-4\,d^2\,f^3\,x^3+6\,c\,d\,f\,\cos \left (2\,e+2\,f\,x\right )+6\,d^2\,f^2\,x^2\,\sin \left (2\,e+2\,f\,x\right )-12\,c\,d\,f^3\,x^2+6\,d^2\,f\,x\,\cos \left (2\,e+2\,f\,x\right )+12\,c\,d\,f^2\,x\,\sin \left (2\,e+2\,f\,x\right )+d^2\,\cos \left (2\,e+2\,f\,x\right )\,3{}\mathrm {i}-c^2\,f^2\,\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+c\,d\,f\,\sin \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}-d^2\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+d^2\,f\,x\,\sin \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}-c\,d\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )\,12{}\mathrm {i}}{24\,a\,f^3} \] Input:

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

Output:

-(d^2*cos(2*e + 2*f*x)*3i - 3*d^2*sin(2*e + 2*f*x) - 12*c^2*f^3*x - c^2*f^ 
2*cos(2*e + 2*f*x)*6i + 6*c^2*f^2*sin(2*e + 2*f*x) - 4*d^2*f^3*x^3 + 6*c*d 
*f*cos(2*e + 2*f*x) + c*d*f*sin(2*e + 2*f*x)*6i - d^2*f^2*x^2*cos(2*e + 2* 
f*x)*6i + 6*d^2*f^2*x^2*sin(2*e + 2*f*x) - 12*c*d*f^3*x^2 + 6*d^2*f*x*cos( 
2*e + 2*f*x) + d^2*f*x*sin(2*e + 2*f*x)*6i - c*d*f^2*x*cos(2*e + 2*f*x)*12 
i + 12*c*d*f^2*x*sin(2*e + 2*f*x))/(24*a*f^3)
 

Reduce [F]

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

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

Output:

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