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]

((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]))

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Rubi [A]  time = 0.0536668, antiderivative size = 84, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.22995, 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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Maple [A]  time = 0.143, size = 139, normalized size = 1.7 \begin{align*}{\frac{1}{1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2}} \left ({\frac{d{x}^{2}}{4\,a}}-{\frac{-2\,icf+d}{4\,a{f}^{2}}}+{\frac{d{x}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{4\,a}}-{\frac{ \left ( id+2\,cf \right ) \tan \left ( fx+e \right ) }{4\,a{f}^{2}}}+{\frac{ \left ( id+2\,cf \right ) x}{4\,af}}-{\frac{dx\tan \left ( fx+e \right ) }{2\,af}}+{\frac{ \left ( -id+2\,cf \right ) x \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{4\,af}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.60447, size = 117, normalized size = 1.39 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 0.328362, size = 116, normalized size = 1.38 \begin{align*} \begin{cases} \frac{\left (2 i a c f^{2} e^{2 i e} + 2 i a d f^{2} x e^{2 i e} - a d f e^{2 i e}\right ) e^{2 i f x}}{8 a^{2} f^{3}} & \text{for}\: 8 a^{2} f^{3} \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} \end{align*}

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*a*c*f**2*exp(2*I*e) + 2*I*a*d*f**2*x*exp(2*I*e) - a*d*f*exp(2*I*e))*exp(2*I*f*x)/(8*a**2*f**3)
, Ne(8*a**2*f**3, 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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Giac [A]  time = 1.25435, size = 90, normalized size = 1.07 \begin{align*} \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}} \end{align*}

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)