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]

11/96*I*d*x/a^3/f-1/16*d*x^2/a^3+1/8*x*(d*x+c)/a^3+1/36*d/f^2/(a+I*a*cot(f*x+e))^3-1/6*I*(d*x+c)/f/(a+I*a*cot(
f*x+e))^3+5/96*d/a/f^2/(a+I*a*cot(f*x+e))^2-1/8*I*(d*x+c)/a/f/(a+I*a*cot(f*x+e))^2+11/96*d/f^2/(a^3+I*a^3*cot(
f*x+e))-1/8*I*(d*x+c)/f/(a^3+I*a^3*cot(f*x+e))

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Rubi [A]  time = 0.22, antiderivative size = 209, normalized size of antiderivative = 1.00, 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 8

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

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Mathematica [A]  time = 0.66, 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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fricas [A]  time = 0.66, size = 94, normalized size = 0.45 \[ \frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x + {\left (24 i \, d f x + 24 i \, c f - 4 \, d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-108 i \, d f x - 108 i \, c f + 27 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (216 i \, d f x + 216 i \, c f - 108 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \]

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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giac [A]  time = 2.08, size = 151, normalized size = 0.72 \[ \frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x + 24 i \, d f x e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, d e^{\left (6 i \, f x + 6 i \, e\right )} + 27 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - 108 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \]

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)

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maple [B]  time = 1.06, size = 653, normalized size = 3.12 \[ \text {result too large to display} \]

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*sin(f*x+e)*cos(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*sin(f*x+e)*cos(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 = 0.00, 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)/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.62, size = 144, normalized size = 0.69 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{16\,a^3\,f}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (12\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{128\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{32\,a^3\,f}\right )+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{288\,a^3\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{48\,a^3\,f}\right )+\frac {d\,x^2}{16\,a^3}+\frac {c\,x}{8\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + a*cot(e + f*x)*1i)^3,x)

[Out]

exp(e*2i + f*x*2i)*(((d*3i + 6*c*f)*1i)/(32*a^3*f^2) + (d*x*3i)/(16*a^3*f)) - exp(e*4i + f*x*4i)*(((d*3i + 12*
c*f)*1i)/(128*a^3*f^2) + (d*x*3i)/(32*a^3*f)) + exp(e*6i + f*x*6i)*(((d*1i + 6*c*f)*1i)/(288*a^3*f^2) + (d*x*1
i)/(48*a^3*f)) + (d*x^2)/(16*a^3) + (c*x)/(8*a^3)

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sympy [A]  time = 0.46, size = 301, normalized size = 1.44 \[ \begin {cases} - \frac {\left (- 221184 i a^{6} c f^{5} e^{2 i e} - 221184 i a^{6} d f^{5} x e^{2 i e} + 110592 a^{6} d f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (110592 i a^{6} c f^{5} e^{4 i e} + 110592 i a^{6} d f^{5} x e^{4 i e} - 27648 a^{6} d f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (- 24576 i a^{6} c f^{5} e^{6 i e} - 24576 i a^{6} d f^{5} x e^{6 i e} + 4096 a^{6} d f^{4} e^{6 i e}\right ) e^{6 i f x}}{1179648 a^{9} f^{6}} & \text {for}\: 1179648 a^{9} f^{6} \neq 0 \\\frac {x^{2} \left (- d e^{6 i e} + 3 d e^{4 i e} - 3 d e^{2 i e}\right )}{16 a^{3}} + \frac {x \left (- c e^{6 i e} + 3 c e^{4 i e} - 3 c e^{2 i e}\right )}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c x}{8 a^{3}} + \frac {d x^{2}}{16 a^{3}} \]

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**6*c*f**5*exp(2*I*e) - 221184*I*a**6*d*f**5*x*exp(2*I*e) + 110592*a**6*d*f**4*exp(2*
I*e))*exp(2*I*f*x) + (110592*I*a**6*c*f**5*exp(4*I*e) + 110592*I*a**6*d*f**5*x*exp(4*I*e) - 27648*a**6*d*f**4*
exp(4*I*e))*exp(4*I*f*x) + (-24576*I*a**6*c*f**5*exp(6*I*e) - 24576*I*a**6*d*f**5*x*exp(6*I*e) + 4096*a**6*d*f
**4*exp(6*I*e))*exp(6*I*f*x))/(1179648*a**9*f**6), Ne(1179648*a**9*f**6, 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), True)) + c*x
/(8*a**3) + d*x**2/(16*a**3)

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