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

Optimal. Leaf size=84 \[ -\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}+\frac {i d x}{4 a f} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3723, 3479, 8} \[ -\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}+\frac {i d x}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*d*x)/(a*f) + (c + d*x)^2/(4*a*d) + d/(4*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x))/(f*(a + I*a*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 3723

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(2*
a*d*(m + 1)), x] + (Dist[(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]

Rubi steps

\begin {align*} \int \frac {c+d x}{a+i a \cot (e+f x)} \, dx &=\frac {(c+d x)^2}{4 a d}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(i d) \int \frac {1}{a+i a \cot (e+f x)} \, dx}{2 f}\\ &=\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(i d) \int 1 \, dx}{4 a f}\\ &=\frac {i d x}{4 a f}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 107, normalized size = 1.27 \[ \frac {(\cos (e+f x)+i \sin (e+f x)) \left (\left (2 c f (2 f x+i)+d \left (2 f^2 x^2+2 i f x-1\right )\right ) \cos (e+f x)-i \left (2 c f (2 f x-i)+d \left (2 f^2 x^2-2 i f x+1\right )\right ) \sin (e+f x)\right )}{8 a f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[e + f*x] + I*Sin[e + f*x])*((2*c*f*(I + 2*f*x) + d*(-1 + (2*I)*f*x + 2*f^2*x^2))*Cos[e + f*x] - I*(2*c*f
*(-I + 2*f*x) + d*(1 - (2*I)*f*x + 2*f^2*x^2))*Sin[e + f*x]))/(8*a*f^2)

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fricas [A]  time = 0.48, size = 48, normalized size = 0.57 \[ \frac {2 \, d f^{2} x^{2} + 4 \, c f^{2} x + {\left (2 i \, d f x + 2 i \, c f - d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, a f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*d*f^2*x^2 + 4*c*f^2*x + (2*I*d*f*x + 2*I*c*f - d)*e^(2*I*f*x + 2*I*e))/(a*f^2)

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giac [A]  time = 0.65, size = 67, normalized size = 0.80 \[ \frac {2 \, d f^{2} x^{2} + 4 \, c f^{2} x + 2 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - d e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, a f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*d*f^2*x^2 + 4*c*f^2*x + 2*I*d*f*x*e^(2*I*f*x + 2*I*e) + 2*I*c*f*e^(2*I*f*x + 2*I*e) - d*e^(2*I*f*x + 2*
I*e))/(a*f^2)

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maple [B]  time = 0.62, size = 177, normalized size = 2.11 \[ \frac {d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )+c f \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-i d \left (-\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )+\frac {i c f \left (\cos ^{2}\left (f x +e \right )\right )}{2}-\frac {i d e \left (\cos ^{2}\left (f x +e \right )\right )}{2}}{f^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f^2/a*(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)+c*f*(-1/2*sin(f
*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-d*e*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-I*d*(-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)+1/2*I*c*f*cos(f*x+e)^2-1/2*I*d*e*cos(f*x+e)^2)

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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)),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 = 0.38, size = 105, normalized size = 1.25 \[ -\frac {d\,\cos \left (2\,e+2\,f\,x\right )-2\,d\,f^2\,x^2+2\,c\,f\,\sin \left (2\,e+2\,f\,x\right )-4\,c\,f^2\,x+2\,d\,f\,x\,\sin \left (2\,e+2\,f\,x\right )+d\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-c\,f\,\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}-d\,f\,x\,\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}}{8\,a\,f^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*cos(2*e + 2*f*x) + d*sin(2*e + 2*f*x)*1i - 2*d*f^2*x^2 - c*f*cos(2*e + 2*f*x)*2i + 2*c*f*sin(2*e + 2*f*x)
- 4*c*f^2*x - d*f*x*cos(2*e + 2*f*x)*2i + 2*d*f*x*sin(2*e + 2*f*x))/(8*a*f^2)

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sympy [A]  time = 0.23, size = 104, normalized size = 1.24 \[ \begin {cases} - \frac {\left (- 2 i c f e^{2 i e} - 2 i d f x e^{2 i e} + d e^{2 i e}\right ) e^{2 i f x}}{8 a f^{2}} & \text {for}\: 8 a f^{2} \neq 0 \\- \frac {c x e^{2 i e}}{2 a} - \frac {d x^{2} e^{2 i e}}{4 a} & \text {otherwise} \end {cases} + \frac {c x}{2 a} + \frac {d x^{2}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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