3.29 \(\int \frac{c+d x}{(a+i a \cot (e+f x))^3} \, dx\)

Optimal. Leaf size=209 \[ -\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{x (c+d x)}{8 a^3}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3} \]

[Out]

(((11*I)/96)*d*x)/(a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) + d/(36*f^2*(a + I*a*Cot[e + f*x])^3) - (
(I/6)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^3) + (5*d)/(96*a*f^2*(a + I*a*Cot[e + f*x])^2) - ((I/8)*(c + d*x))/
(a*f*(a + I*a*Cot[e + f*x])^2) + (11*d)/(96*f^2*(a^3 + I*a^3*Cot[e + f*x])) - ((I/8)*(c + d*x))/(f*(a^3 + I*a^
3*Cot[e + f*x]))

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Rubi [A]  time = 0.217833, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3479, 8, 3730} \[ -\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{x (c+d x)}{8 a^3}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(((11*I)/96)*d*x)/(a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) + d/(36*f^2*(a + I*a*Cot[e + f*x])^3) - (
(I/6)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^3) + (5*d)/(96*a*f^2*(a + I*a*Cot[e + f*x])^2) - ((I/8)*(c + d*x))/
(a*f*(a + I*a*Cot[e + f*x])^2) + (11*d)/(96*f^2*(a^3 + I*a^3*Cot[e + f*x])) - ((I/8)*(c + d*x))/(f*(a^3 + I*a^
3*Cot[e + f*x]))

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[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 8

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

Rule 3730

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{u = IntHide[(a
+ b*Tan[e + f*x])^n, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[Dist[(c + d*x)^(m - 1), u, x], x], x]] /; Fr
eeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]

Rubi steps

\begin{align*} \int \frac{c+d x}{(a+i a \cot (e+f x))^3} \, dx &=\frac{x (c+d x)}{8 a^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}-d \int \left (\frac{x}{8 a^3}-\frac{i}{6 f (a+i a \cot (e+f x))^3}-\frac{i}{8 a f (a+i a \cot (e+f x))^2}-\frac{i}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}\right ) \, dx\\ &=-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{(i d) \int \frac{1}{a^3+i a^3 \cot (e+f x)} \, dx}{8 f}+\frac{(i d) \int \frac{1}{(a+i a \cot (e+f x))^3} \, dx}{6 f}+\frac{(i d) \int \frac{1}{(a+i a \cot (e+f x))^2} \, dx}{8 a f}\\ &=-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{d}{32 a f^2 (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac{d}{16 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{(i d) \int 1 \, dx}{16 a^3 f}+\frac{(i d) \int \frac{1}{a+i a \cot (e+f x)} \, dx}{16 a^2 f}+\frac{(i d) \int \frac{1}{(a+i a \cot (e+f x))^2} \, dx}{12 a f}\\ &=\frac{i d x}{16 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac{3 d}{32 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{(i d) \int 1 \, dx}{32 a^3 f}+\frac{(i d) \int \frac{1}{a+i a \cot (e+f x)} \, dx}{24 a^2 f}\\ &=\frac{3 i d x}{32 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac{(i d) \int 1 \, dx}{48 a^3 f}\\ &=\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \cot (e+f x))^3}-\frac{i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac{i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac{i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.610507, size = 244, normalized size = 1.17 \[ \frac{108 i (2 c f+d (2 f x+i)) \cos (2 (e+f x))+27 (-4 i c f-4 i d f x+d) \cos (4 (e+f x))-216 c f \sin (2 (e+f x))+108 c f \sin (4 (e+f x))-24 c f \sin (6 (e+f x))+24 i c f \cos (6 (e+f x))+144 c e f+144 c f^2 x-72 d e^2-108 i d \sin (2 (e+f x))-216 d f x \sin (2 (e+f x))+27 i d \sin (4 (e+f x))+108 d f x \sin (4 (e+f x))-4 i d \sin (6 (e+f x))-24 d f x \sin (6 (e+f x))-4 d \cos (6 (e+f x))+24 i d f x \cos (6 (e+f x))+72 d f^2 x^2}{1152 a^3 f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(-72*d*e^2 + 144*c*e*f + 144*c*f^2*x + 72*d*f^2*x^2 + (108*I)*(2*c*f + d*(I + 2*f*x))*Cos[2*(e + f*x)] + 27*(d
- (4*I)*c*f - (4*I)*d*f*x)*Cos[4*(e + f*x)] - 4*d*Cos[6*(e + f*x)] + (24*I)*c*f*Cos[6*(e + f*x)] + (24*I)*d*f
*x*Cos[6*(e + f*x)] - (108*I)*d*Sin[2*(e + f*x)] - 216*c*f*Sin[2*(e + f*x)] - 216*d*f*x*Sin[2*(e + f*x)] + (27
*I)*d*Sin[4*(e + f*x)] + 108*c*f*Sin[4*(e + f*x)] + 108*d*f*x*Sin[4*(e + f*x)] - (4*I)*d*Sin[6*(e + f*x)] - 24
*c*f*Sin[6*(e + f*x)] - 24*d*f*x*Sin[6*(e + f*x)])/(1152*a^3*f^2)

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Maple [B]  time = 0.074, size = 653, normalized size = 3.1 \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)/(a+I*a*cot(f*x+e))^3,x)

[Out]

1/f^2/a^3*(4*I*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)-1/24*f*x-1/24*e-1/6*(
f*x+e)*sin(f*x+e)^6-1/36*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e))+4*I*c*f*(-1/6*sin(f*x+e)^
2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)-4*I*d*e*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)-4*d*((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)-1/32*(f*x+e)^2+1/96*sin(f*x+e)^4+1/32*sin(f*x+e)^2-(
f*x+e)*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)-4*
c*f*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*sin(f*x+e)*cos(f*x+e)^3+1/16*cos(f*x+e)*sin(f*x+e)+1/16*f*x+1/16*e)+4*
d*e*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*sin(f*x+e)*cos(f*x+e)^3+1/16*cos(f*x+e)*sin(f*x+e)+1/16*f*x+1/16*e)-3*
I*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)-3/4*I*c*f*sin(f*x
+e)^4+3/4*I*d*e*sin(f*x+e)^4+d*((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)-3/16*(f*
x+e)^2+1/16*sin(f*x+e)^4+3/16*sin(f*x+e)^2)+c*f*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-
d*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))

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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)/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.58715, size = 285, normalized size = 1.36 \begin{align*} \frac{72 \, d f^{2} x^{2} + 144 \, c f^{2} x +{\left (24 i \, d f x + 24 i \, c f - 4 \, d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-108 i \, d f x - 108 i \, c f + 27 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (216 i \, d f x + 216 i \, c f - 108 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

1/1152*(72*d*f^2*x^2 + 144*c*f^2*x + (24*I*d*f*x + 24*I*c*f - 4*d)*e^(6*I*f*x + 6*I*e) + (-108*I*d*f*x - 108*I
*c*f + 27*d)*e^(4*I*f*x + 4*I*e) + (216*I*d*f*x + 216*I*c*f - 108*d)*e^(2*I*f*x + 2*I*e))/(a^3*f^2)

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Sympy [A]  time = 0.98689, size = 299, normalized size = 1.43 \begin{align*} \begin{cases} \frac{\left (221184 i a^{15} c f^{8} e^{2 i e} + 221184 i a^{15} d f^{8} x e^{2 i e} - 110592 a^{15} d f^{7} e^{2 i e}\right ) e^{2 i f x} + \left (- 110592 i a^{15} c f^{8} e^{4 i e} - 110592 i a^{15} d f^{8} x e^{4 i e} + 27648 a^{15} d f^{7} e^{4 i e}\right ) e^{4 i f x} + \left (24576 i a^{15} c f^{8} e^{6 i e} + 24576 i a^{15} d f^{8} x e^{6 i e} - 4096 a^{15} d f^{7} e^{6 i e}\right ) e^{6 i f x}}{1179648 a^{18} f^{9}} & \text{for}\: 1179648 a^{18} f^{9} \neq 0 \\\frac{x^{2} \left (- d e^{6 i e} + 3 d e^{4 i e} - 3 d e^{2 i e}\right )}{16 a^{3}} + \frac{x \left (- c e^{6 i e} + 3 c e^{4 i e} - 3 c e^{2 i e}\right )}{8 a^{3}} & \text{otherwise} \end{cases} + \frac{c x}{8 a^{3}} + \frac{d x^{2}}{16 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(a+I*a*cot(f*x+e))**3,x)

[Out]

Piecewise((((221184*I*a**15*c*f**8*exp(2*I*e) + 221184*I*a**15*d*f**8*x*exp(2*I*e) - 110592*a**15*d*f**7*exp(2
*I*e))*exp(2*I*f*x) + (-110592*I*a**15*c*f**8*exp(4*I*e) - 110592*I*a**15*d*f**8*x*exp(4*I*e) + 27648*a**15*d*
f**7*exp(4*I*e))*exp(4*I*f*x) + (24576*I*a**15*c*f**8*exp(6*I*e) + 24576*I*a**15*d*f**8*x*exp(6*I*e) - 4096*a*
*15*d*f**7*exp(6*I*e))*exp(6*I*f*x))/(1179648*a**18*f**9), Ne(1179648*a**18*f**9, 0)), (x**2*(-d*exp(6*I*e) +
3*d*exp(4*I*e) - 3*d*exp(2*I*e))/(16*a**3) + x*(-c*exp(6*I*e) + 3*c*exp(4*I*e) - 3*c*exp(2*I*e))/(8*a**3), Tru
e)) + c*x/(8*a**3) + d*x**2/(16*a**3)

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Giac [A]  time = 1.3295, size = 204, normalized size = 0.98 \begin{align*} \frac{72 \, d f^{2} x^{2} + 144 \, c f^{2} x + 24 i \, d f x e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, d e^{\left (6 i \, f x + 6 i \, e\right )} + 27 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - 108 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")

[Out]

1/1152*(72*d*f^2*x^2 + 144*c*f^2*x + 24*I*d*f*x*e^(6*I*f*x + 6*I*e) - 108*I*d*f*x*e^(4*I*f*x + 4*I*e) + 216*I*
d*f*x*e^(2*I*f*x + 2*I*e) + 24*I*c*f*e^(6*I*f*x + 6*I*e) - 108*I*c*f*e^(4*I*f*x + 4*I*e) + 216*I*c*f*e^(2*I*f*
x + 2*I*e) - 4*d*e^(6*I*f*x + 6*I*e) + 27*d*e^(4*I*f*x + 4*I*e) - 108*d*e^(2*I*f*x + 2*I*e))/(a^3*f^2)