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]

((-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)

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Rubi [A]  time = 0.198422, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 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]

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]

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 = 0.674306, 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)

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Maple [B]  time = 0.138, size = 1073, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

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*(-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))*cos(f*x+e)-3/32*f*x-3/32
*e)+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*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)-I/f*c*d
*e*sin(f*x+e)^4+2/f^2*d^2*((f*x+e)^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*
cos(f*x+e)*sin(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*
cos(f*x+e)*sin(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*cos(f*x+e)*sin(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*cos(f*x+e)*sin(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*cos(f*x+e)*sin(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
*cos(f*x+e)*sin(f*x+e)+1/8*f*x+1/8*e)-1/f^2*d^2*((f*x+e)^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/2*(f*x
+e)*cos(f*x+e)^2+1/4*cos(f*x+e)*sin(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*cos(f*x+e)*sin(
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*cos(f*x+e)*sin(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*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)+2/f*c*d*e*(-1/2*cos(f*x
+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/f^2*d^2*e^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [A]  time = 1.63045, size = 383, normalized size = 1.9 \begin{align*} \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}} \end{align*}

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)

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Sympy [A]  time = 0.940646, size = 406, normalized size = 2.01 \begin{align*} \begin{cases} \frac{\left (256 i a^{10} c^{2} f^{11} e^{2 i e} + 512 i a^{10} c d f^{11} x e^{2 i e} - 256 a^{10} c d f^{10} e^{2 i e} + 256 i a^{10} d^{2} f^{11} x^{2} e^{2 i e} - 256 a^{10} d^{2} f^{10} x e^{2 i e} - 128 i a^{10} d^{2} f^{9} e^{2 i e}\right ) e^{2 i f x} + \left (- 64 i a^{10} c^{2} f^{11} e^{4 i e} - 128 i a^{10} c d f^{11} x e^{4 i e} + 32 a^{10} c d f^{10} e^{4 i e} - 64 i a^{10} d^{2} f^{11} x^{2} e^{4 i e} + 32 a^{10} d^{2} f^{10} x e^{4 i e} + 8 i a^{10} d^{2} f^{9} e^{4 i e}\right ) e^{4 i f x}}{1024 a^{12} f^{12}} & \text{for}\: 1024 a^{12} f^{12} \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}} \end{align*}

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**10*c**2*f**11*exp(2*I*e) + 512*I*a**10*c*d*f**11*x*exp(2*I*e) - 256*a**10*c*d*f**10*exp(
2*I*e) + 256*I*a**10*d**2*f**11*x**2*exp(2*I*e) - 256*a**10*d**2*f**10*x*exp(2*I*e) - 128*I*a**10*d**2*f**9*ex
p(2*I*e))*exp(2*I*f*x) + (-64*I*a**10*c**2*f**11*exp(4*I*e) - 128*I*a**10*c*d*f**11*x*exp(4*I*e) + 32*a**10*c*
d*f**10*exp(4*I*e) - 64*I*a**10*d**2*f**11*x**2*exp(4*I*e) + 32*a**10*d**2*f**10*x*exp(4*I*e) + 8*I*a**10*d**2
*f**9*exp(4*I*e))*exp(4*I*f*x))/(1024*a**12*f**12), Ne(1024*a**12*f**12, 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)

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Giac [A]  time = 1.28269, size = 333, normalized size = 1.65 \begin{align*} \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}} \end{align*}

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)