Integrand size = 24, antiderivative size = 70 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {3 x}{2 a}-\frac {3 \cot (c+d x)}{2 a d}-\frac {i \log (\sin (c+d x))}{a d}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))} \] Output:
-3/2*x/a-3/2*cot(d*x+c)/a/d-I*ln(sin(d*x+c))/a/d+1/2*cot(d*x+c)/d/(a+I*a*t an(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\frac {-3 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )-2 i (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a}+\frac {\cot (c+d x)}{a+i a \tan (c+d x)}}{2 d} \] Input:
Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]
Output:
((-3*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] - (2*I) *(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/a + Cot[c + d*x]/(a + I*a*Tan[c + d*x]))/(2*d)
Time = 0.49 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 4035, 25, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}
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.
\(\displaystyle \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^2 (a+i a \tan (c+d x))}dx\) |
\(\Big \downarrow \) 4035 |
\(\displaystyle \frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac {\int -\cot ^2(c+d x) (3 a-2 i a \tan (c+d x))dx}{2 a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot ^2(c+d x) (3 a-2 i a \tan (c+d x))dx}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {3 a-2 i a \tan (c+d x)}{\tan (c+d x)^2}dx}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {-\frac {3 a \cot (c+d x)}{d}+\int -\cot (c+d x) (3 \tan (c+d x) a+2 i a)dx}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {3 a \cot (c+d x)}{d}-\int \cot (c+d x) (3 \tan (c+d x) a+2 i a)dx}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {3 a \cot (c+d x)}{d}-\int \frac {3 \tan (c+d x) a+2 i a}{\tan (c+d x)}dx}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle \frac {-2 i a \int \cot (c+d x)dx-\frac {3 a \cot (c+d x)}{d}-3 a x}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-2 i a \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {3 a \cot (c+d x)}{d}-3 a x}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 i a \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {3 a \cot (c+d x)}{d}-3 a x}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {-\frac {3 a \cot (c+d x)}{d}-\frac {2 i a \log (-\sin (c+d x))}{d}-3 a x}{2 a^2}+\frac {\cot (c+d x)}{2 d (a+i a \tan (c+d x))}\) |
Input:
Int[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]
Output:
(-3*a*x - (3*a*Cot[c + d*x])/d - ((2*I)*a*Log[-Sin[c + d*x]])/d)/(2*a^2) + Cot[c + d*x]/(2*d*(a + I*a*Tan[c + d*x]))
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d)) Int[(c + d *Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]
Time = 0.84 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-\frac {5 x}{2 a}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a d}-\frac {2 c}{a d}-\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\) | \(78\) |
derivativedivides | \(\frac {i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d a}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}-\frac {1}{a d \tan \left (d x +c \right )}-\frac {i \ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {1}{2 d a \left (-i+\tan \left (d x +c \right )\right )}\) | \(88\) |
default | \(\frac {i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d a}-\frac {3 \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}-\frac {1}{a d \tan \left (d x +c \right )}-\frac {i \ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {1}{2 d a \left (-i+\tan \left (d x +c \right )\right )}\) | \(88\) |
norman | \(\frac {-\frac {1}{a d}-\frac {3 x \tan \left (d x +c \right )}{2 a}-\frac {3 x \tan \left (d x +c \right )^{3}}{2 a}-\frac {3 \tan \left (d x +c \right )^{2}}{2 a d}-\frac {i \tan \left (d x +c \right )}{2 d a}}{\tan \left (d x +c \right ) \left (1+\tan \left (d x +c \right )^{2}\right )}-\frac {i \ln \left (\tan \left (d x +c \right )\right )}{a d}+\frac {i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d a}\) | \(125\) |
Input:
int(cot(d*x+c)^2/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-5/2*x/a-1/4*I/a/d*exp(-2*I*(d*x+c))-2/a/d*c-2*I/d/a/(exp(2*I*(d*x+c))-1)- I/a/d*ln(exp(2*I*(d*x+c))-1)
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.41 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {10 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (10 \, d x - 9 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - i}{4 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
-1/4*(10*d*x*e^(4*I*d*x + 4*I*c) - (10*d*x - 9*I)*e^(2*I*d*x + 2*I*c) + 4* (I*e^(4*I*d*x + 4*I*c) - I*e^(2*I*d*x + 2*I*c))*log(e^(2*I*d*x + 2*I*c) - 1) - I)/(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\begin {cases} - \frac {i e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: a d e^{2 i c} \neq 0 \\x \left (\frac {\left (- 5 e^{2 i c} - 1\right ) e^{- 2 i c}}{2 a} + \frac {5}{2 a}\right ) & \text {otherwise} \end {cases} - \frac {2 i}{a d e^{2 i c} e^{2 i d x} - a d} - \frac {5 x}{2 a} - \frac {i \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a d} \] Input:
integrate(cot(d*x+c)**2/(a+I*a*tan(d*x+c)),x)
Output:
Piecewise((-I*exp(-2*I*c)*exp(-2*I*d*x)/(4*a*d), Ne(a*d*exp(2*I*c), 0)), ( x*((-5*exp(2*I*c) - 1)*exp(-2*I*c)/(2*a) + 5/(2*a)), True)) - 2*I/(a*d*exp (2*I*c)*exp(2*I*d*x) - a*d) - 5*x/(2*a) - I*log(exp(2*I*d*x) - exp(-2*I*c) )/(a*d)
Exception generated. \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i \, \log \left (\tan \left (d x + c\right ) + i\right )}{4 \, a d} + \frac {5 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{4 \, a d} - \frac {i \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a d} + \frac {i \, {\left (3 i \, \tan \left (d x + c\right ) + 2\right )}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )} \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
-1/4*I*log(tan(d*x + c) + I)/(a*d) + 5/4*I*log(tan(d*x + c) - I)/(a*d) - I *log(abs(tan(d*x + c)))/(a*d) + 1/2*I*(3*I*tan(d*x + c) + 2)/(a*d*(tan(d*x + c) - I)*tan(d*x + c))
Time = 0.99 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.37 \[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,5{}\mathrm {i}}{4\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a\,d}-\frac {\frac {1}{a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}}{2\,a}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,1{}\mathrm {i}}{a\,d} \] Input:
int(cot(c + d*x)^2/(a + a*tan(c + d*x)*1i),x)
Output:
(log(tan(c + d*x) - 1i)*5i)/(4*a*d) - (log(tan(c + d*x) + 1i)*1i)/(4*a*d) - ((tan(c + d*x)*3i)/(2*a) + 1/a)/(d*(tan(c + d*x) + tan(c + d*x)^2*1i)) - (log(tan(c + d*x))*1i)/(a*d)
\[ \int \frac {\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\int \frac {\cot \left (d x +c \right )^{2}}{\tan \left (d x +c \right ) i +1}d x}{a} \] Input:
int(cot(d*x+c)^2/(a+I*a*tan(d*x+c)),x)
Output:
int(cot(c + d*x)**2/(tan(c + d*x)*i + 1),x)/a