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

Optimal. Leaf size=202 \[ -\frac {d (c+d x) e^{2 i e+2 i f x}}{4 a^2 f^2}+\frac {d (c+d x) e^{4 i e+4 i f x}}{32 a^2 f^2}+\frac {i (c+d x)^2 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^2 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {i d^2 e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {i d^2 e^{4 i e+4 i f x}}{128 a^2 f^3} \]

[Out]

-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

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3729, 2176, 2194} \[ -\frac {d (c+d x) e^{2 i e+2 i f x}}{4 a^2 f^2}+\frac {d (c+d x) e^{4 i e+4 i f x}}{32 a^2 f^2}+\frac {i (c+d x)^2 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^2 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {i d^2 e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {i d^2 e^{4 i e+4 i f x}}{128 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/8)*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)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 3729

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*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(c+d x)^2}{(a+i a \cot (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^2}{4 a^2}-\frac {e^{2 i e+2 i f x} (c+d x)^2}{2 a^2}+\frac {e^{4 i e+4 i f x} (c+d x)^2}{4 a^2}\right ) \, dx\\ &=\frac {(c+d x)^3}{12 a^2 d}+\frac {\int e^{4 i e+4 i f x} (c+d x)^2 \, dx}{4 a^2}-\frac {\int e^{2 i e+2 i f x} (c+d x)^2 \, dx}{2 a^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}+\frac {(i d) \int e^{4 i e+4 i f x} (c+d x) \, dx}{8 a^2 f}-\frac {(i d) \int e^{2 i e+2 i f x} (c+d x) \, dx}{2 a^2 f}\\ &=-\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}-\frac {d^2 \int e^{4 i e+4 i f x} \, dx}{32 a^2 f^2}+\frac {d^2 \int e^{2 i e+2 i f x} \, dx}{4 a^2 f^2}\\ &=-\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.74, size = 255, normalized size = 1.26 \[ \frac {32 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+48 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-3 (\cos (4 e)+i \sin (4 e)) \cos (4 f x) ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d ((2+2 i) f x+i))+48 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-3 (\cos (4 e)+i \sin (4 e)) \sin (4 f x) (-(2+2 i) c f+(-2-2 i) d f x+d) ((2-2 i) c f+(2-2 i) d f x+d)}{384 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)

________________________________________________________________________________________

fricas [A]  time = 1.51, size = 151, normalized size = 0.75 \[ \frac {32 \, d^{2} f^{3} x^{3} + 96 \, c d f^{3} x^{2} + 96 \, c^{2} f^{3} x + {\left (-24 i \, d^{2} f^{2} x^{2} - 24 i \, c^{2} f^{2} + 12 \, c d f + 3 i \, d^{2} + {\left (-48 i \, c d f^{2} + 12 \, d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (96 i \, d^{2} f^{2} x^{2} + 96 i \, c^{2} f^{2} - 96 \, c d f - 48 i \, d^{2} + {\left (192 i \, c d f^{2} - 96 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{384 \, a^{2} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.50, size = 247, normalized size = 1.22 \[ \frac {32 \, d^{2} f^{3} x^{3} + 96 \, c d f^{3} x^{2} - 24 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 \, c^{2} f^{3} x - 48 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 192 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} - 24 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 96 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} - 96 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 48 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{384 \, a^{2} f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

________________________________________________________________________________________

maple [B]  time = 1.95, size = 1073, normalized size = 5.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2/f*(-I/f*c*d*e*sin(f*x+e)^4-4*I/f^2*d^2*e*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*sin(f*x+e))*c
os(f*x+e)-3/32*f*x-3/32*e)+2*I/f^2*d^2*(1/4*(f*x+e)^2*sin(f*x+e)^4-1/2*(f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x
+e))*cos(f*x+e)+3/8*f*x+3/8*e)+3/32*(f*x+e)^2-1/32*sin(f*x+e)^4-3/32*sin(f*x+e)^2)+1/2*I/f^2*d^2*e^2*sin(f*x+e
)^4+4*I/f*c*d*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-3/32*f*x-3/32*e)+1/2*I*c
^2*sin(f*x+e)^4+2/f^2*d^2*((f*x+e)^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*
sin(f*x+e)*cos(f*x+e)+7/64*f*x+7/64*e-1/12*(f*x+e)^3-(f*x+e)^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+
3/8*f*x+3/8*e)-1/8*(f*x+e)*sin(f*x+e)^4-1/32*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e))+4/f*c*d*((f*x+e)*(-1/2*
sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/16*(f*x+e)^2+1/16*sin(f*x+e)^2-(f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+
e))*cos(f*x+e)+3/8*f*x+3/8*e)-1/16*sin(f*x+e)^4)-4/f^2*d^2*e*((f*x+e)*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*
e)-1/16*(f*x+e)^2+1/16*sin(f*x+e)^2-(f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-1/16
*sin(f*x+e)^4)+2*c^2*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)-4/f*c*d*e*(-1/4*si
n(f*x+e)*cos(f*x+e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)+2/f^2*d^2*e^2*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8
*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)-1/f^2*d^2*((f*x+e)^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/2*(f*x
+e)*cos(f*x+e)^2+1/4*sin(f*x+e)*cos(f*x+e)+1/4*f*x+1/4*e-1/3*(f*x+e)^3)-2/f*c*d*((f*x+e)*(-1/2*sin(f*x+e)*cos(
f*x+e)+1/2*f*x+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)+2/f^2*d^2*e*((f*x+e)*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x
+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)-c^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+2/f*c*d*e*(-1/2*sin(f*x
+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/f^2*d^2*e^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [B]  time = 0.58, size = 186, normalized size = 0.92 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{8\,a^2\,f^3}+\frac {d^2\,x^2\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^2\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{128\,a^2\,f^3}+\frac {d^2\,x^2\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^2\,f^2}\right )+\frac {c^2\,x}{4\,a^2}+\frac {d^2\,x^3}{12\,a^2}+\frac {c\,d\,x^2}{4\,a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*4i)*(((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)

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 406, normalized size = 2.01 \[ \begin {cases} \frac {\left (256 i a^{2} c^{2} f^{5} e^{2 i e} + 512 i a^{2} c d f^{5} x e^{2 i e} - 256 a^{2} c d f^{4} e^{2 i e} + 256 i a^{2} d^{2} f^{5} x^{2} e^{2 i e} - 256 a^{2} d^{2} f^{4} x e^{2 i e} - 128 i a^{2} d^{2} f^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 64 i a^{2} c^{2} f^{5} e^{4 i e} - 128 i a^{2} c d f^{5} x e^{4 i e} + 32 a^{2} c d f^{4} e^{4 i e} - 64 i a^{2} d^{2} f^{5} x^{2} e^{4 i e} + 32 a^{2} d^{2} f^{4} x e^{4 i e} + 8 i a^{2} d^{2} f^{3} e^{4 i e}\right ) e^{4 i f x}}{1024 a^{4} f^{6}} & \text {for}\: 1024 a^{4} f^{6} \neq 0 \\\frac {x^{3} \left (d^{2} e^{4 i e} - 2 d^{2} e^{2 i e}\right )}{12 a^{2}} + \frac {x^{2} \left (c d e^{4 i e} - 2 c d e^{2 i e}\right )}{4 a^{2}} + \frac {x \left (c^{2} e^{4 i e} - 2 c^{2} e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{2} x}{4 a^{2}} + \frac {c d x^{2}}{4 a^{2}} + \frac {d^{2} x^{3}}{12 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))*e
xp(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(1024*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)

________________________________________________________________________________________