Integrand size = 22, antiderivative size = 83 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=i a x+\frac {i a \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d} \]
I*a*x+I*a*cot(d*x+c)/d+1/2*a*cot(d*x+c)^2/d-1/3*I*a*cot(d*x+c)^3/d-1/4*a*c ot(d*x+c)^4/d+a*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {a \log (\tan (c+d x))}{d} \]
(a*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^4)/(4*d) - ((I/3)*a*Cot[c + d*x ]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d + (a*Log[Cos[c + d*x]])/d + (a*Log[Tan[c + d*x]])/d
Time = 0.64 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4012, 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.
\(\displaystyle \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\tan (c+d x)^5}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (i a-a \tan (c+d x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \cot ^4(c+d x)}{4 d}+\int \frac {i a-a \tan (c+d x)}{\tan (c+d x)^4}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int -\cot ^3(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot ^3(c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {i \tan (c+d x) a+a}{\tan (c+d x)^3}dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int \cot ^2(c+d x) (i a-a \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {i a-a \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int -\cot (c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cot (c+d x) (i \tan (c+d x) a+a)dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan (c+d x) a+a}{\tan (c+d x)}dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle a \int \cot (c+d x)dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}+i a x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}+i a x\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}+i a x\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {a \cot ^4(c+d x)}{4 d}-\frac {i a \cot ^3(c+d x)}{3 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {i a \cot (c+d x)}{d}+\frac {a \log (-\sin (c+d x))}{d}+i a x\) |
I*a*x + (I*a*Cot[c + d*x])/d + (a*Cot[c + d*x]^2)/(2*d) - ((I/3)*a*Cot[c + d*x]^3)/d - (a*Cot[c + d*x]^4)/(4*d) + (a*Log[-Sin[c + d*x]])/d
3.1.11.3.1 Defintions of rubi rules used
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 = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(-\frac {a \left (3 \left (\cot ^{4}\left (d x +c \right )\right )+4 i \left (\cot ^{3}\left (d x +c \right )\right )-12 i d x -6 \left (\cot ^{2}\left (d x +c \right )\right )-12 i \cot \left (d x +c \right )-12 \ln \left (\tan \left (d x +c \right )\right )+6 \ln \left (\sec ^{2}\left (d x +c \right )\right )\right )}{12 d}\) | \(73\) |
derivativedivides | \(\frac {a \left (-\frac {1}{4 \tan \left (d x +c \right )^{4}}+\ln \left (\tan \left (d x +c \right )\right )+\frac {i}{\tan \left (d x +c \right )}-\frac {i}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{2 \tan \left (d x +c \right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(79\) |
default | \(\frac {a \left (-\frac {1}{4 \tan \left (d x +c \right )^{4}}+\ln \left (\tan \left (d x +c \right )\right )+\frac {i}{\tan \left (d x +c \right )}-\frac {i}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{2 \tan \left (d x +c \right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(79\) |
risch | \(-\frac {2 i a c}{d}-\frac {4 a \left (6 \,{\mathrm e}^{6 i \left (d x +c \right )}-9 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}-2\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(81\) |
norman | \(\frac {i a x \left (\tan ^{4}\left (d x +c \right )\right )+\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a}{4 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i a \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(102\) |
-1/12*a*(3*cot(d*x+c)^4+4*I*cot(d*x+c)^3-12*I*d*x-6*cot(d*x+c)^2-12*I*cot( d*x+c)-12*ln(tan(d*x+c))+6*ln(sec(d*x+c)^2))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (71) = 142\).
Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.88 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {24 \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 32 \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 3 \, {\left (a e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 8 \, a}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1/3*(24*a*e^(6*I*d*x + 6*I*c) - 36*a*e^(4*I*d*x + 4*I*c) + 32*a*e^(2*I*d* x + 2*I*c) - 3*(a*e^(8*I*d*x + 8*I*c) - 4*a*e^(6*I*d*x + 6*I*c) + 6*a*e^(4 *I*d*x + 4*I*c) - 4*a*e^(2*I*d*x + 2*I*c) + a)*log(e^(2*I*d*x + 2*I*c) - 1 ) - 8*a)/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*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. 158 vs. \(2 (70) = 140\).
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.90 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 24 a e^{6 i c} e^{6 i d x} + 36 a e^{4 i c} e^{4 i d x} - 32 a e^{2 i c} e^{2 i d x} + 8 a}{3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} + 3 d} \]
a*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-24*a*exp(6*I*c)*exp(6*I*d*x) + 36* a*exp(4*I*c)*exp(4*I*d*x) - 32*a*exp(2*I*c)*exp(2*I*d*x) + 8*a)/(3*d*exp(8 *I*c)*exp(8*I*d*x) - 12*d*exp(6*I*c)*exp(6*I*d*x) + 18*d*exp(4*I*c)*exp(4* I*d*x) - 12*d*exp(2*I*c)*exp(2*I*d*x) + 3*d)
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {-12 i \, {\left (d x + c\right )} a + 6 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \log \left (\tan \left (d x + c\right )\right ) - \frac {12 i \, a \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 i \, a \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
-1/12*(-12*I*(d*x + c)*a + 6*a*log(tan(d*x + c)^2 + 1) - 12*a*log(tan(d*x + c)) - (12*I*a*tan(d*x + c)^3 + 6*a*tan(d*x + c)^2 - 4*I*a*tan(d*x + c) - 3*a)/tan(d*x + c)^4)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (71) = 142\).
Time = 0.54 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.90 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 384 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 120 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {400 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
-1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 - 8*I*a*tan(1/2*d*x + 1/2*c)^3 - 36*a*t an(1/2*d*x + 1/2*c)^2 + 384*a*log(tan(1/2*d*x + 1/2*c) + I) - 192*a*log(ta n(1/2*d*x + 1/2*c)) + 120*I*a*tan(1/2*d*x + 1/2*c) + (400*a*tan(1/2*d*x + 1/2*c)^4 - 120*I*a*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/2*d*x + 1/2*c)^2 + 8*I*a*tan(1/2*d*x + 1/2*c) + 3*a)/tan(1/2*d*x + 1/2*c)^4)/d
Time = 4.83 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d}-\frac {-1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {1{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {a}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \]