Integrand size = 21, antiderivative size = 209 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}+\frac {d}{36 f^2 (a+i a \cot (e+f x))^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )} \] Output:
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))
Time = 0.84 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\frac {-72 d e^2+144 c e f+144 c f^2 x+72 d f^2 x^2+108 i (2 c f+d (i+2 f x)) \cos (2 (e+f x))+27 (d-4 i c f-4 i d f x) \cos (4 (e+f x))-4 d \cos (6 (e+f x))+24 i c f \cos (6 (e+f x))+24 i d f x \cos (6 (e+f x))-108 i d \sin (2 (e+f x))-216 c f \sin (2 (e+f x))-216 d f x \sin (2 (e+f x))+27 i d \sin (4 (e+f x))+108 c f \sin (4 (e+f x))+108 d f x \sin (4 (e+f x))-4 i d \sin (6 (e+f x))-24 c f \sin (6 (e+f x))-24 d f x \sin (6 (e+f x))}{1152 a^3 f^2} \] Input:
Integrate[(c + d*x)/(a + I*a*Cot[e + f*x])^3,x]
Output:
(-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)
Time = 0.51 (sec) , antiderivative size = 208, 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.
| \(\displaystyle \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {c+d x}{\left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4213 |
| \(\displaystyle -d \int \left (\frac {x}{8 a^3}-\frac {i}{8 f \left (i \cot (e+f x) a^3+a^3\right )}-\frac {i}{8 a f (i \cot (e+f x) a+a)^2}-\frac {i}{6 f (i \cot (e+f x) a+a)^3}\right )dx-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}-d \left (-\frac {11}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {11 i x}{96 a^3 f}+\frac {x^2}{16 a^3}-\frac {5}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac {1}{36 f^2 (a+i a \cot (e+f x))^3}\right )-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}\) |
Input:
Int[(c + d*x)/(a + I*a*Cot[e + f*x])^3,x]
Output:
(x*(c + d*x))/(8*a^3) - ((I/6)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^3) - ( (I/8)*(c + d*x))/(a*f*(a + I*a*Cot[e + f*x])^2) - ((I/8)*(c + d*x))/(f*(a^ 3 + I*a^3*Cot[e + f*x])) - d*((((-11*I)/96)*x)/(a^3*f) + x^2/(16*a^3) - 1/ (36*f^2*(a + I*a*Cot[e + f*x])^3) - 5/(96*a*f^2*(a + I*a*Cot[e + f*x])^2) - 11/(96*f^2*(a^3 + I*a^3*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 = 1.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {d \,x^{2}}{16 a^{3}}+\frac {c x}{8 a^{3}}+\frac {i \left (6 d f x +6 c f +i d \right ) {\mathrm e}^{6 i \left (f x +e \right )}}{288 a^{3} f^{2}}-\frac {3 i \left (4 d f x +4 c f +i d \right ) {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{3} f^{2}}+\frac {3 i \left (2 d f x +2 c f +i d \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{32 a^{3} f^{2}}\) | \(114\) |
| parallelrisch | \(\frac {\left (\left (18 d \,x^{2}+36 c x \right ) f^{2}-87 i d f x \right ) \tan \left (f x +e \right )^{3}+\left (108 i \left (\frac {d x}{2}+c \right ) x \,f^{2}+\left (9 d x -252 c \right ) f -87 i d \right ) \tan \left (f x +e \right )^{2}+\left (\left (-54 d \,x^{2}-108 c x \right ) f^{2}+\left (-63 i d x -324 i c \right ) f +135 d \right ) \tan \left (f x +e \right )-36 i \left (\frac {d x}{2}+c \right ) x \,f^{2}+\left (33 d x +120 c \right ) f +56 i d}{288 f^{2} a^{3} \left (-i-3 \tan \left (f x +e \right )+3 i \tan \left (f x +e \right )^{2}+\tan \left (f x +e \right )^{3}\right )}\) | \(175\) |
Input:
int((d*x+c)/(a+I*a*cot(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
1/16*d*x^2/a^3+1/8/a^3*c*x+1/288*I*(6*d*f*x+I*d+6*c*f)/a^3/f^2*exp(6*I*(f* x+e))-3/128*I*(4*d*f*x+I*d+4*c*f)/a^3/f^2*exp(4*I*(f*x+e))+3/32*I*(2*d*f*x +I*d+2*c*f)/a^3/f^2*exp(2*I*(f*x+e))
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.44 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x - 4 \, {\left (-6 i \, d f x - 6 i \, c f + d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 27 \, {\left (4 i \, d f x + 4 i \, c f - d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 108 \, {\left (-2 i \, d f x - 2 i \, c f + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")
Output:
1/1152*(72*d*f^2*x^2 + 144*c*f^2*x - 4*(-6*I*d*f*x - 6*I*c*f + d)*e^(6*I*f *x + 6*I*e) - 27*(4*I*d*f*x + 4*I*c*f - d)*e^(4*I*f*x + 4*I*e) - 108*(-2*I *d*f*x - 2*I*c*f + d)*e^(2*I*f*x + 2*I*e))/(a^3*f^2)
Time = 0.26 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\begin {cases} \frac {\left (221184 i a^{6} c f^{5} e^{2 i e} + 221184 i a^{6} d f^{5} x e^{2 i e} - 110592 a^{6} d f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (- 110592 i a^{6} c f^{5} e^{4 i e} - 110592 i a^{6} d f^{5} x e^{4 i e} + 27648 a^{6} d f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (24576 i a^{6} c f^{5} e^{6 i e} + 24576 i a^{6} d f^{5} x e^{6 i e} - 4096 a^{6} d f^{4} e^{6 i e}\right ) e^{6 i f x}}{1179648 a^{9} f^{6}} & \text {for}\: a^{9} f^{6} \neq 0 \\\frac {x^{2} \left (- d e^{6 i e} + 3 d e^{4 i e} - 3 d e^{2 i e}\right )}{16 a^{3}} + \frac {x \left (- c e^{6 i e} + 3 c e^{4 i e} - 3 c e^{2 i e}\right )}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c x}{8 a^{3}} + \frac {d x^{2}}{16 a^{3}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))**3,x)
Output:
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(a**9*f**6, 0)), (x**2*(-d*exp(6*I*e) + 3*d*exp(4*I*e) - 3*d*ex p(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)
Exception generated. \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x + 24 i \, d f x e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, d e^{\left (6 i \, f x + 6 i \, e\right )} + 27 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - 108 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \] Input:
integrate((d*x+c)/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")
Output:
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)
Time = 0.65 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.69 \[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx={\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{16\,a^3\,f}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (12\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{128\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{32\,a^3\,f}\right )+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{288\,a^3\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{48\,a^3\,f}\right )+\frac {d\,x^2}{16\,a^3}+\frac {c\,x}{8\,a^3} \] Input:
int((c + d*x)/(a + a*cot(e + f*x)*1i)^3,x)
Output:
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*1i) /(48*a^3*f)) + (d*x^2)/(16*a^3) + (c*x)/(8*a^3)
\[ \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx=\frac {3 \left (\int \frac {\sin \left (f x +e \right )^{3}}{4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} i -\cos \left (f x +e \right ) i +4 \sin \left (f x +e \right )^{3}-3 \sin \left (f x +e \right )}d x \right ) c f -3 \left (\int \frac {x}{\cot \left (f x +e \right )^{3} i +3 \cot \left (f x +e \right )^{2}-3 \cot \left (f x +e \right ) i -1}d x \right ) d f +3 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) c i -\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} i -6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-15 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} i +20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+15 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} i -6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-i \right ) c i +3 c f x}{3 a^{3} f} \] Input:
int((d*x+c)/(a+I*a*cot(f*x+e))^3,x)
Output:
(3*int(sin(e + f*x)**3/(4*cos(e + f*x)*sin(e + f*x)**2*i - cos(e + f*x)*i + 4*sin(e + f*x)**3 - 3*sin(e + f*x)),x)*c*f - 3*int(x/(cot(e + f*x)**3*i + 3*cot(e + f*x)**2 - 3*cot(e + f*x)*i - 1),x)*d*f + 3*log(tan((e + f*x)/2 )**2 + 1)*c*i - log(tan((e + f*x)/2)**6*i - 6*tan((e + f*x)/2)**5 - 15*tan ((e + f*x)/2)**4*i + 20*tan((e + f*x)/2)**3 + 15*tan((e + f*x)/2)**2*i - 6 *tan((e + f*x)/2) - i)*c*i + 3*c*f*x)/(3*a**3*f)