Integrand size = 21, antiderivative size = 151 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\frac {3 i d x}{16 a^2 f}-\frac {d x^2}{8 a^2}+\frac {x (c+d x)}{4 a^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {3 d}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )} \] Output:
3/16*I*d*x/a^2/f-1/8*d*x^2/a^2+1/4*x*(d*x+c)/a^2+1/16*d/f^2/(a+I*a*cot(f*x +e))^2-1/4*I*(d*x+c)/f/(a+I*a*cot(f*x+e))^2+3/16*d/f^2/(a^2+I*a^2*cot(f*x+ e))-1/4*I*(d*x+c)/f/(a^2+I*a^2*cot(f*x+e))
Time = 0.80 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\frac {-8 d e^2+16 c e f+16 c f^2 x+8 d f^2 x^2+8 i (2 c f+d (i+2 f x)) \cos (2 (e+f x))+(d-4 i c f-4 i d f x) \cos (4 (e+f x))-8 i d \sin (2 (e+f x))-16 c f \sin (2 (e+f x))-16 d f x \sin (2 (e+f x))+i d \sin (4 (e+f x))+4 c f \sin (4 (e+f x))+4 d f x \sin (4 (e+f x))}{64 a^2 f^2} \] Input:
Integrate[(c + d*x)/(a + I*a*Cot[e + f*x])^2,x]
Output:
(-8*d*e^2 + 16*c*e*f + 16*c*f^2*x + 8*d*f^2*x^2 + (8*I)*(2*c*f + d*(I + 2* f*x))*Cos[2*(e + f*x)] + (d - (4*I)*c*f - (4*I)*d*f*x)*Cos[4*(e + f*x)] - (8*I)*d*Sin[2*(e + f*x)] - 16*c*f*Sin[2*(e + f*x)] - 16*d*f*x*Sin[2*(e + f *x)] + I*d*Sin[4*(e + f*x)] + 4*c*f*Sin[4*(e + f*x)] + 4*d*f*x*Sin[4*(e + f*x)])/(64*a^2*f^2)
Time = 0.39 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4213, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {c+d x}{\left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 4213 |
\(\displaystyle -d \int \left (\frac {x}{4 a^2}-\frac {i}{4 f \left (i \cot (e+f x) a^2+a^2\right )}-\frac {i}{4 f (i \cot (e+f x) a+a)^2}\right )dx-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {x (c+d x)}{4 a^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {x (c+d x)}{4 a^2}-d \left (-\frac {3}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}-\frac {3 i x}{16 a^2 f}+\frac {x^2}{8 a^2}-\frac {1}{16 f^2 (a+i a \cot (e+f x))^2}\right )-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}\) |
Input:
Int[(c + d*x)/(a + I*a*Cot[e + f*x])^2,x]
Output:
(x*(c + d*x))/(4*a^2) - ((I/4)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^2) - ( (I/4)*(c + d*x))/(f*(a^2 + I*a^2*Cot[e + f*x])) - d*((((-3*I)/16)*x)/(a^2* f) + x^2/(8*a^2) - 1/(16*f^2*(a + I*a*Cot[e + f*x])^2) - 3/(16*f^2*(a^2 + I*a^2*Cot[e + f*x])))
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]}, Simp[(c + d*x) ^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1) u, x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]
Time = 0.81 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {d \,x^{2}}{8 a^{2}}+\frac {c x}{4 a^{2}}-\frac {i \left (4 d f x +4 c f +i d \right ) {\mathrm e}^{4 i \left (f x +e \right )}}{64 f^{2} a^{2}}+\frac {i \left (2 d f x +2 c f +i d \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{8 f^{2} a^{2}}\) | \(82\) |
parallelrisch | \(\frac {\left (\left (2 d \,x^{2}+4 c x \right ) f^{2}-5 i d f x \right ) \tan \left (f x +e \right )^{2}+\left (8 i \left (\frac {d x}{2}+c \right ) x \,f^{2}+\left (-2 d x -12 c \right ) f -5 i d \right ) \tan \left (f x +e \right )+\left (-2 d \,x^{2}-4 c x \right ) f^{2}+\left (-3 i d x -8 i c \right ) f +4 d}{16 f^{2} a^{2} \left (-1+2 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}\right )}\) | \(127\) |
Input:
int((d*x+c)/(a+I*a*cot(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
1/8*d*x^2/a^2+1/4/a^2*c*x-1/64*I*(4*d*f*x+I*d+4*c*f)/f^2/a^2*exp(4*I*(f*x+ e))+1/8*I*(2*d*f*x+I*d+2*c*f)/f^2/a^2*exp(2*I*(f*x+e))
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.45 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\frac {8 \, d f^{2} x^{2} + 16 \, c f^{2} x + {\left (-4 i \, d f x - 4 i \, c f + d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 8 \, {\left (-2 i \, d f x - 2 i \, c f + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, a^{2} f^{2}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="fricas")
Output:
1/64*(8*d*f^2*x^2 + 16*c*f^2*x + (-4*I*d*f*x - 4*I*c*f + d)*e^(4*I*f*x + 4 *I*e) - 8*(-2*I*d*f*x - 2*I*c*f + d)*e^(2*I*f*x + 2*I*e))/(a^2*f^2)
Time = 0.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.40 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\begin {cases} \frac {\left (128 i a^{2} c f^{3} e^{2 i e} + 128 i a^{2} d f^{3} x e^{2 i e} - 64 a^{2} d f^{2} e^{2 i e}\right ) e^{2 i f x} + \left (- 32 i a^{2} c f^{3} e^{4 i e} - 32 i a^{2} d f^{3} x e^{4 i e} + 8 a^{2} d f^{2} e^{4 i e}\right ) e^{4 i f x}}{512 a^{4} f^{4}} & \text {for}\: a^{4} f^{4} \neq 0 \\\frac {x^{2} \left (d e^{4 i e} - 2 d e^{2 i e}\right )}{8 a^{2}} + \frac {x \left (c e^{4 i e} - 2 c e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c x}{4 a^{2}} + \frac {d x^{2}}{8 a^{2}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))**2,x)
Output:
Piecewise((((128*I*a**2*c*f**3*exp(2*I*e) + 128*I*a**2*d*f**3*x*exp(2*I*e) - 64*a**2*d*f**2*exp(2*I*e))*exp(2*I*f*x) + (-32*I*a**2*c*f**3*exp(4*I*e) - 32*I*a**2*d*f**3*x*exp(4*I*e) + 8*a**2*d*f**2*exp(4*I*e))*exp(4*I*f*x)) /(512*a**4*f**4), Ne(a**4*f**4, 0)), (x**2*(d*exp(4*I*e) - 2*d*exp(2*I*e)) /(8*a**2) + x*(c*exp(4*I*e) - 2*c*exp(2*I*e))/(4*a**2), True)) + c*x/(4*a* *2) + d*x**2/(8*a**2)
Exception generated. \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\frac {8 \, d f^{2} x^{2} + 16 \, c f^{2} x - 4 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} - 4 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + d e^{\left (4 i \, f x + 4 i \, e\right )} - 8 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, a^{2} f^{2}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="giac")
Output:
1/64*(8*d*f^2*x^2 + 16*c*f^2*x - 4*I*d*f*x*e^(4*I*f*x + 4*I*e) + 16*I*d*f* x*e^(2*I*f*x + 2*I*e) - 4*I*c*f*e^(4*I*f*x + 4*I*e) + 16*I*c*f*e^(2*I*f*x + 2*I*e) + d*e^(4*I*f*x + 4*I*e) - 8*d*e^(2*I*f*x + 2*I*e))/(a^2*f^2)
Time = 0.36 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx={\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,a^2\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{4\,a^2\,f}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^2\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{16\,a^2\,f}\right )+\frac {d\,x^2}{8\,a^2}+\frac {c\,x}{4\,a^2} \] Input:
int((c + d*x)/(a + a*cot(e + f*x)*1i)^2,x)
Output:
exp(e*2i + f*x*2i)*(((d*1i + 2*c*f)*1i)/(8*a^2*f^2) + (d*x*1i)/(4*a^2*f)) - exp(e*4i + f*x*4i)*(((d*1i + 4*c*f)*1i)/(64*a^2*f^2) + (d*x*1i)/(16*a^2* f)) + (d*x^2)/(8*a^2) + (c*x)/(4*a^2)
\[ \int \frac {c+d x}{(a+i a \cot (e+f x))^2} \, dx=\frac {-\left (\int \frac {x}{\cot \left (f x +e \right )^{2}-2 \cot \left (f x +e \right ) i -1}d x \right ) d -\left (\int \frac {1}{\cot \left (f x +e \right )^{2}-2 \cot \left (f x +e \right ) i -1}d x \right ) c}{a^{2}} \] Input:
int((d*x+c)/(a+I*a*cot(f*x+e))^2,x)
Output:
( - (int(x/(cot(e + f*x)**2 - 2*cot(e + f*x)*i - 1),x)*d + int(1/(cot(e + f*x)**2 - 2*cot(e + f*x)*i - 1),x)*c))/a**2