3.17 \(\int \frac{(c+d x)^2}{a+i a \cot (e+f x)} \, dx\)

Optimal. Leaf size=137 \[ \frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{i d^2}{4 f^3 (a+i a \cot (e+f x))}-\frac{d^2 x}{4 a f^2} \]

[Out]

-(d^2*x)/(4*a*f^2) + ((I/4)*(c + d*x)^2)/(a*f) + (c + d*x)^3/(6*a*d) + ((I/4)*d^2)/(f^3*(a + I*a*Cot[e + f*x])
) + (d*(c + d*x))/(2*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^2)/(f*(a + I*a*Cot[e + f*x]))

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Rubi [A]  time = 0.122501, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3723, 3479, 8} \[ \frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{i d^2}{4 f^3 (a+i a \cot (e+f x))}-\frac{d^2 x}{4 a f^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + I*a*Cot[e + f*x]),x]

[Out]

-(d^2*x)/(4*a*f^2) + ((I/4)*(c + d*x)^2)/(a*f) + (c + d*x)^3/(6*a*d) + ((I/4)*d^2)/(f^3*(a + I*a*Cot[e + f*x])
) + (d*(c + d*x))/(2*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^2)/(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)^2}{a+i a \cot (e+f x)} \, dx &=\frac{(c+d x)^3}{6 a d}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac{(i d) \int \frac{c+d x}{a+i a \cot (e+f x)} \, dx}{f}\\ &=\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}-\frac{d^2 \int \frac{1}{a+i a \cot (e+f x)} \, dx}{2 f^2}\\ &=\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{i d^2}{4 f^3 (a+i a \cot (e+f x))}+\frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}-\frac{d^2 \int 1 \, dx}{4 a f^2}\\ &=-\frac{d^2 x}{4 a f^2}+\frac{i (c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}+\frac{i d^2}{4 f^3 (a+i a \cot (e+f x))}+\frac{d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^2}{2 f (a+i a \cot (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.330341, size = 149, normalized size = 1.09 \[ \frac{4 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+3 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))+3 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))}{24 a f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + I*a*Cot[e + f*x]),x]

[Out]

(4*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*((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*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])/(24*a*f^3)

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Maple [A]  time = 0.178, size = 108, normalized size = 0.8 \begin{align*}{\frac{{d}^{2}{x}^{3}}{6\,a}}+{\frac{cd{x}^{2}}{2\,a}}+{\frac{{c}^{2}x}{2\,a}}+{\frac{{c}^{3}}{6\,ad}}+{\frac{{\frac{i}{8}} \left ( 2\,{d}^{2}{x}^{2}{f}^{2}+2\,i{d}^{2}fx+4\,cd{f}^{2}x+2\,icdf+2\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{2\,i \left ( fx+e \right ) }}}{a{f}^{3}}} \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)),x)

[Out]

1/6/a*d^2*x^3+1/2/a*d*c*x^2+1/2/a*c^2*x+1/6/a/d*c^3+1/8*I*(2*d^2*x^2*f^2+2*I*d^2*f*x+4*c*d*f^2*x+2*I*c*d*f+2*c
^2*f^2-d^2)/a/f^3*exp(2*I*(f*x+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)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.64781, size = 223, normalized size = 1.63 \begin{align*} \frac{4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 12 \, c^{2} f^{3} x +{\left (6 i \, d^{2} f^{2} x^{2} + 6 i \, c^{2} f^{2} - 6 \, c d f - 3 i \, d^{2} +{\left (12 i \, c d f^{2} - 6 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a 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)),x, algorithm="fricas")

[Out]

1/24*(4*d^2*f^3*x^3 + 12*c*d*f^3*x^2 + 12*c^2*f^3*x + (6*I*d^2*f^2*x^2 + 6*I*c^2*f^2 - 6*c*d*f - 3*I*d^2 + (12
*I*c*d*f^2 - 6*d^2*f)*x)*e^(2*I*f*x + 2*I*e))/(a*f^3)

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Sympy [A]  time = 0.50811, size = 226, normalized size = 1.65 \begin{align*} \begin{cases} \frac{\left (2 i a^{2} c^{2} f^{5} e^{2 i e} + 4 i a^{2} c d f^{5} x e^{2 i e} - 2 a^{2} c d f^{4} e^{2 i e} + 2 i a^{2} d^{2} f^{5} x^{2} e^{2 i e} - 2 a^{2} d^{2} f^{4} x e^{2 i e} - i a^{2} d^{2} f^{3} e^{2 i e}\right ) e^{2 i f x}}{8 a^{3} f^{6}} & \text{for}\: 8 a^{3} f^{6} \neq 0 \\- \frac{c^{2} x e^{2 i e}}{2 a} - \frac{c d x^{2} e^{2 i e}}{2 a} - \frac{d^{2} x^{3} e^{2 i e}}{6 a} & \text{otherwise} \end{cases} + \frac{c^{2} x}{2 a} + \frac{c d x^{2}}{2 a} + \frac{d^{2} x^{3}}{6 a} \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)),x)

[Out]

Piecewise(((2*I*a**2*c**2*f**5*exp(2*I*e) + 4*I*a**2*c*d*f**5*x*exp(2*I*e) - 2*a**2*c*d*f**4*exp(2*I*e) + 2*I*
a**2*d**2*f**5*x**2*exp(2*I*e) - 2*a**2*d**2*f**4*x*exp(2*I*e) - I*a**2*d**2*f**3*exp(2*I*e))*exp(2*I*f*x)/(8*
a**3*f**6), Ne(8*a**3*f**6, 0)), (-c**2*x*exp(2*I*e)/(2*a) - c*d*x**2*exp(2*I*e)/(2*a) - d**2*x**3*exp(2*I*e)/
(6*a), True)) + c**2*x/(2*a) + c*d*x**2/(2*a) + d**2*x**3/(6*a)

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Giac [A]  time = 1.22021, size = 193, normalized size = 1.41 \begin{align*} \frac{4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 6 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c^{2} f^{3} x + 12 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a 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)),x, algorithm="giac")

[Out]

1/24*(4*d^2*f^3*x^3 + 12*c*d*f^3*x^2 + 6*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) + 12*c^2*f^3*x + 12*I*c*d*f^2*x*e^(
2*I*f*x + 2*I*e) + 6*I*c^2*f^2*e^(2*I*f*x + 2*I*e) - 6*d^2*f*x*e^(2*I*f*x + 2*I*e) - 6*c*d*f*e^(2*I*f*x + 2*I*
e) - 3*I*d^2*e^(2*I*f*x + 2*I*e))/(a*f^3)