Integrand size = 22, antiderivative size = 100 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {i a \log (\sin (c+d x))}{d} \] Output:
-a*x-a*cot(d*x+c)/d+1/2*I*a*cot(d*x+c)^2/d+1/3*a*cot(d*x+c)^3/d-1/4*I*a*co t(d*x+c)^4/d-1/5*a*cot(d*x+c)^5/d+I*a*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {i a \csc ^2(c+d x)}{d}-\frac {i a \csc ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {i a \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]
Output:
(I*a*Csc[c + d*x]^2)/d - ((I/4)*a*Csc[c + d*x]^4)/d - (a*Cot[c + d*x]^5*Hy pergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + (I*a*Log[Sin[c + d*x]])/d
Time = 0.79 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4012, 3042, 4012, 25, 3042, 4012, 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.
\(\displaystyle \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\tan (c+d x)^6}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (i a-a \tan (c+d x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \cot ^5(c+d x)}{5 d}+\int \frac {i a-a \tan (c+d x)}{\tan (c+d x)^5}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int -\cot ^4(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot ^4(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {i \tan (c+d x) a+a}{\tan (c+d x)^4}dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int \cot ^3(c+d x) (i a-a \tan (c+d x))dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {i a-a \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int -\cot ^2(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cot ^2(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan (c+d x) a+a}{\tan (c+d x)^2}dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int \cot (c+d x) (i a-a \tan (c+d x))dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i a-a \tan (c+d x)}{\tan (c+d x)}dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle i a \int \cot (c+d x)dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}-a x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i a \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}-a x\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -i a \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}-a x\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {i a \log (-\sin (c+d x))}{d}-a x\) |
Input:
Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]
Output:
-(a*x) - (a*Cot[c + d*x])/d + ((I/2)*a*Cot[c + d*x]^2)/d + (a*Cot[c + d*x] ^3)/(3*d) - ((I/4)*a*Cot[c + d*x]^4)/d - (a*Cot[c + d*x]^5)/(5*d) + (I*a*L og[-Sin[c + d*x]])/d
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]
Time = 1.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {\left (i \cot \left (d x +c \right )^{4}+\frac {4 \cot \left (d x +c \right )^{5}}{5}-2 i \cot \left (d x +c \right )^{2}-\frac {4 \cot \left (d x +c \right )^{3}}{3}-4 i \ln \left (\tan \left (d x +c \right )\right )+2 i \ln \left (\sec \left (d x +c \right )^{2}\right )+4 d x +4 \cot \left (d x +c \right )\right ) a}{4 d}\) | \(84\) |
derivativedivides | \(\frac {a \left (-\frac {1}{5 \tan \left (d x +c \right )^{5}}-\frac {1}{\tan \left (d x +c \right )}+\frac {i}{2 \tan \left (d x +c \right )^{2}}-\frac {i}{4 \tan \left (d x +c \right )^{4}}+i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
default | \(\frac {a \left (-\frac {1}{5 \tan \left (d x +c \right )^{5}}-\frac {1}{\tan \left (d x +c \right )}+\frac {i}{2 \tan \left (d x +c \right )^{2}}-\frac {i}{4 \tan \left (d x +c \right )^{4}}+i \ln \left (\tan \left (d x +c \right )\right )+\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
risch | \(\frac {2 a c}{d}-\frac {2 i a \left (75 \,{\mathrm e}^{8 i \left (d x +c \right )}-150 \,{\mathrm e}^{6 i \left (d x +c \right )}+200 \,{\mathrm e}^{4 i \left (d x +c \right )}-100 \,{\mathrm e}^{2 i \left (d x +c \right )}+23\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(94\) |
norman | \(\frac {-\frac {a}{5 d}-a x \tan \left (d x +c \right )^{5}+\frac {a \tan \left (d x +c \right )^{2}}{3 d}-\frac {a \tan \left (d x +c \right )^{4}}{d}-\frac {i a \tan \left (d x +c \right )}{4 d}+\frac {i a \tan \left (d x +c \right )^{3}}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {i a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(118\) |
Input:
int(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/4*(I*cot(d*x+c)^4+4/5*cot(d*x+c)^5-2*I*cot(d*x+c)^2-4/3*cot(d*x+c)^3-4* I*ln(tan(d*x+c))+2*I*ln(sec(d*x+c)^2)+4*d*x+4*cot(d*x+c))*a/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (86) = 172\).
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.97 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {-150 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 300 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 400 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 200 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 15 \, {\left (-i \, a e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 46 i \, a}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="fricas")
Output:
1/15*(-150*I*a*e^(8*I*d*x + 8*I*c) + 300*I*a*e^(6*I*d*x + 6*I*c) - 400*I*a *e^(4*I*d*x + 4*I*c) + 200*I*a*e^(2*I*d*x + 2*I*c) - 15*(-I*a*e^(10*I*d*x + 10*I*c) + 5*I*a*e^(8*I*d*x + 8*I*c) - 10*I*a*e^(6*I*d*x + 6*I*c) + 10*I* a*e^(4*I*d*x + 4*I*c) - 5*I*a*e^(2*I*d*x + 2*I*c) + I*a)*log(e^(2*I*d*x + 2*I*c) - 1) - 46*I*a)/(d*e^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2* I*c) - d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {i a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 150 i a e^{8 i c} e^{8 i d x} + 300 i a e^{6 i c} e^{6 i d x} - 400 i a e^{4 i c} e^{4 i d x} + 200 i a e^{2 i c} e^{2 i d x} - 46 i a}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \] Input:
integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c)),x)
Output:
I*a*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-150*I*a*exp(8*I*c)*exp(8*I*d*x) + 300*I*a*exp(6*I*c)*exp(6*I*d*x) - 400*I*a*exp(4*I*c)*exp(4*I*d*x) + 200* I*a*exp(2*I*c)*exp(2*I*d*x) - 46*I*a)/(15*d*exp(10*I*c)*exp(10*I*d*x) - 75 *d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d*x) - 150*d*exp(4*I *c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {60 \, {\left (d x + c\right )} a + 30 i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a \tan \left (d x + c\right )^{4} - 30 i \, a \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 i \, a \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="maxima")
Output:
-1/60*(60*(d*x + c)*a + 30*I*a*log(tan(d*x + c)^2 + 1) - 60*I*a*log(tan(d* x + c)) + (60*a*tan(d*x + c)^4 - 30*I*a*tan(d*x + c)^3 - 20*a*tan(d*x + c) ^2 + 15*I*a*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.83 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {1}{60} i \, a {\left (\frac {60 \, \log \left (\tan \left (d x + c\right ) + i\right )}{d} - \frac {60 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {-60 i \, \tan \left (d x + c\right )^{4} - 30 \, \tan \left (d x + c\right )^{3} + 20 i \, \tan \left (d x + c\right )^{2} + 15 \, \tan \left (d x + c\right ) - 12 i}{d \tan \left (d x + c\right )^{5}}\right )} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="giac")
Output:
-1/60*I*a*(60*log(tan(d*x + c) + I)/d - 60*log(abs(tan(d*x + c)))/d + (-60 *I*tan(d*x + c)^4 - 30*tan(d*x + c)^3 + 20*I*tan(d*x + c)^2 + 15*tan(d*x + c) - 12*I)/(d*tan(d*x + c)^5))
Time = 1.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2\,a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {1{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{4}+\frac {a}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \] Input:
int(cot(c + d*x)^6*(a + a*tan(c + d*x)*1i),x)
Output:
- (2*a*atan(2*tan(c + d*x) + 1i))/d - (a/5 + (a*tan(c + d*x)*1i)/4 - (a*ta n(c + d*x)^2)/3 - (a*tan(c + d*x)^3*1i)/2 + a*tan(c + d*x)^4)/(d*tan(c + d *x)^5)
Time = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.44 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \left (-736 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+352 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-96 \cos \left (d x +c \right )-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{5} i +480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} i -480 \sin \left (d x +c \right )^{5} d x -195 \sin \left (d x +c \right )^{5} i +480 \sin \left (d x +c \right )^{3} i -120 \sin \left (d x +c \right ) i \right )}{480 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+I*a*tan(d*x+c)),x)
Output:
(a*( - 736*cos(c + d*x)*sin(c + d*x)**4 + 352*cos(c + d*x)*sin(c + d*x)**2 - 96*cos(c + d*x) - 480*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**5*i + 480*log(tan((c + d*x)/2))*sin(c + d*x)**5*i - 480*sin(c + d*x)**5*d*x - 19 5*sin(c + d*x)**5*i + 480*sin(c + d*x)**3*i - 120*sin(c + d*x)*i))/(480*si n(c + d*x)**5*d)