Integrand size = 23, antiderivative size = 137 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=-\frac {d^2 x}{4 a f^2}+\frac {i (c+d x)^2}{4 a f}+\frac {(c+d x)^3}{6 a d}+\frac {i d^2}{4 f^3 (a+i a \cot (e+f x))}+\frac {d (c+d x)}{2 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))} \]
-1/4*d^2*x/a/f^2+1/4*I*(d*x+c)^2/a/f+1/6*(d*x+c)^3/a/d+1/4*I*d^2/f^3/(a+I* a*cot(f*x+e))+1/2*d*(d*x+c)/f^2/(a+I*a*cot(f*x+e))-1/2*I*(d*x+c)^2/f/(a+I* a*cot(f*x+e))
Time = 0.50 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+3 ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) \cos (2 f x) (\cos (2 e)+i \sin (2 e))+3 i ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d (i+(1+i) f x)) (\cos (2 e)+i \sin (2 e)) \sin (2 f x)}{24 a f^3} \]
(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)
Time = 0.48 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4206, 3042, 4206, 3042, 3960, 24}
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.
\(\displaystyle \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d x)^2}{a-i a \tan \left (e+f x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4206 |
\(\displaystyle \frac {i d \int \frac {c+d x}{i \cot (e+f x) a+a}dx}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i d \int \frac {c+d x}{a-i a \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\) |
\(\Big \downarrow \) 4206 |
\(\displaystyle \frac {i d \left (\frac {i d \int \frac {1}{i \cot (e+f x) a+a}dx}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i d \left (\frac {i d \int \frac {1}{a-i a \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle \frac {i d \left (\frac {i d \left (\frac {\int 1dx}{2 a}-\frac {i}{2 f (a+i a \cot (e+f x))}\right )}{2 f}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^3}{6 a d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {i (c+d x)^2}{2 f (a+i a \cot (e+f x))}+\frac {i d \left (-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {i d \left (\frac {x}{2 a}-\frac {i}{2 f (a+i a \cot (e+f x))}\right )}{2 f}\right )}{f}+\frac {(c+d x)^3}{6 a d}\) |
(c + d*x)^3/(6*a*d) - ((I/2)*(c + d*x)^2)/(f*(a + I*a*Cot[e + f*x])) + (I* d*((c + d*x)^2/(4*a*d) - ((I/2)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])) + (( I/2)*d*(x/(2*a) - (I/2)/(f*(a + I*a*Cot[e + f*x]))))/f))/f
3.1.17.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy mbol] :> Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + (Simp[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]
Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {d^{2} x^{3}}{6 a}+\frac {d c \,x^{2}}{2 a}+\frac {c^{2} x}{2 a}+\frac {c^{3}}{6 a d}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 i d^{2} f x +2 c^{2} f^{2}+2 i c d f -d^{2}\right ) {\mathrm e}^{2 i \left (f x +e \right )}}{8 a \,f^{3}}\) | \(108\) |
parallelrisch | \(\frac {-2 \left (\left (-\frac {1}{3} d^{2} x^{2}-c d x -c^{2}\right ) f^{2}+i \left (\frac {d x}{2}+c \right ) d f -\frac {d^{2}}{2}\right ) f x \tan \left (f x +e \right )+2 i \left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) x \,f^{3}+\left (-d^{2} x^{2}-2 c d x -2 c^{2}\right ) f^{2}-2 i \left (\frac {d x}{2}+c \right ) d f +d^{2}}{4 f^{3} a \left (i+\tan \left (f x +e \right )\right )}\) | \(131\) |
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))
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 12 \, c^{2} f^{3} x - 3 \, {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, c^{2} f^{2} + 2 \, c d f + i \, d^{2} + 2 \, {\left (-2 i \, c d f^{2} + d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a f^{3}} \]
1/24*(4*d^2*f^3*x^3 + 12*c*d*f^3*x^2 + 12*c^2*f^3*x - 3*(-2*I*d^2*f^2*x^2 - 2*I*c^2*f^2 + 2*c*d*f + I*d^2 + 2*(-2*I*c*d*f^2 + d^2*f)*x)*e^(2*I*f*x + 2*I*e))/(a*f^3)
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\begin {cases} \frac {\left (2 i c^{2} f^{2} e^{2 i e} + 4 i c d f^{2} x e^{2 i e} - 2 c d f e^{2 i e} + 2 i d^{2} f^{2} x^{2} e^{2 i e} - 2 d^{2} f x e^{2 i e} - i d^{2} e^{2 i e}\right ) e^{2 i f x}}{8 a f^{3}} & \text {for}\: a f^{3} \neq 0 \\- \frac {c^{2} x e^{2 i e}}{2 a} - \frac {c d x^{2} e^{2 i e}}{2 a} - \frac {d^{2} x^{3} e^{2 i e}}{6 a} & \text {otherwise} \end {cases} + \frac {c^{2} x}{2 a} + \frac {c d x^{2}}{2 a} + \frac {d^{2} x^{3}}{6 a} \]
Piecewise(((2*I*c**2*f**2*exp(2*I*e) + 4*I*c*d*f**2*x*exp(2*I*e) - 2*c*d*f *exp(2*I*e) + 2*I*d**2*f**2*x**2*exp(2*I*e) - 2*d**2*f*x*exp(2*I*e) - I*d* *2*exp(2*I*e))*exp(2*I*f*x)/(8*a*f**3), Ne(a*f**3, 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)
Exception generated. \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=\frac {4 \, d^{2} f^{3} x^{3} + 12 \, c d f^{3} x^{2} + 6 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c^{2} f^{3} x + 12 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{24 \, a f^{3}} \]
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)
Time = 12.78 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^2}{a+i a \cot (e+f x)} \, dx=-\frac {6\,c^2\,f^2\,\sin \left (2\,e+2\,f\,x\right )-12\,c^2\,f^3\,x-3\,d^2\,\sin \left (2\,e+2\,f\,x\right )-4\,d^2\,f^3\,x^3+6\,c\,d\,f\,\cos \left (2\,e+2\,f\,x\right )+6\,d^2\,f^2\,x^2\,\sin \left (2\,e+2\,f\,x\right )-12\,c\,d\,f^3\,x^2+6\,d^2\,f\,x\,\cos \left (2\,e+2\,f\,x\right )+12\,c\,d\,f^2\,x\,\sin \left (2\,e+2\,f\,x\right )+d^2\,\cos \left (2\,e+2\,f\,x\right )\,3{}\mathrm {i}-c^2\,f^2\,\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+c\,d\,f\,\sin \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}-d^2\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+d^2\,f\,x\,\sin \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}-c\,d\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )\,12{}\mathrm {i}}{24\,a\,f^3} \]
-(d^2*cos(2*e + 2*f*x)*3i - 3*d^2*sin(2*e + 2*f*x) - 12*c^2*f^3*x - c^2*f^ 2*cos(2*e + 2*f*x)*6i + 6*c^2*f^2*sin(2*e + 2*f*x) - 4*d^2*f^3*x^3 + 6*c*d *f*cos(2*e + 2*f*x) + c*d*f*sin(2*e + 2*f*x)*6i - d^2*f^2*x^2*cos(2*e + 2* f*x)*6i + 6*d^2*f^2*x^2*sin(2*e + 2*f*x) - 12*c*d*f^3*x^2 + 6*d^2*f*x*cos( 2*e + 2*f*x) + d^2*f*x*sin(2*e + 2*f*x)*6i - c*d*f^2*x*cos(2*e + 2*f*x)*12 i + 12*c*d*f^2*x*sin(2*e + 2*f*x))/(24*a*f^3)