Integrand size = 21, antiderivative size = 120 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=-\left (\left (a^2-b^2\right ) x\right )-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a b \log (\sin (c+d x))}{d} \]
-(a^2-b^2)*x-(a^2-b^2)*cot(d*x+c)/d+a*b*cot(d*x+c)^2/d+1/3*(a^2-b^2)*cot(d *x+c)^3/d-1/2*a*b*cot(d*x+c)^4/d-1/5*a^2*cot(d*x+c)^5/d+2*a*b*ln(sin(d*x+c ))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a b \cot ^2(c+d x)}{d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}-\frac {b^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {2 a b \log (\cos (c+d x))}{d}+\frac {2 a b \log (\tan (c+d x))}{d} \]
(a*b*Cot[c + d*x]^2)/d - (a*b*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5* Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) - (b^2*Cot[c + d* x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (2*a*b*Log [Cos[c + d*x]])/d + (2*a*b*Log[Tan[c + d*x]])/d
Time = 0.91 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4025, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^2}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \int \cot ^5(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx-\frac {a^2 \cot ^5(c+d x)}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^5}dx-\frac {a^2 \cot ^5(c+d x)}{5 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int -\cot ^4(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot ^4(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)^4}dx-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int \cot ^3(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^3}dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\int -\cot ^2(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \cot ^2(c+d x) \left (a^2+2 b \tan (c+d x) a-b^2\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+2 b \tan (c+d x) a-b^2}{\tan (c+d x)^2}dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \int \cot (c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)}dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle 2 a b \int \cot (c+d x)dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a b \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 a b \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {2 a b \log (-\sin (c+d x))}{d}\) |
-((a^2 - b^2)*x) - ((a^2 - b^2)*Cot[c + d*x])/d + (a*b*Cot[c + d*x]^2)/d + ((a^2 - b^2)*Cot[c + d*x]^3)/(3*d) - (a*b*Cot[c + d*x]^4)/(2*d) - (a^2*Co t[c + d*x]^5)/(5*d) + (2*a*b*Log[-Sin[c + d*x]])/d
3.5.34.3.1 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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*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^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.87 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(103\) |
| default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(103\) |
| parallelrisch | \(\frac {-30 a b \ln \left (\sec ^{2}\left (d x +c \right )\right )+60 a b \ln \left (\tan \left (d x +c \right )\right )-6 \left (\cot ^{5}\left (d x +c \right )\right ) a^{2}-15 \left (\cot ^{4}\left (d x +c \right )\right ) a b +10 \left (a^{2}-b^{2}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+30 \left (\cot ^{2}\left (d x +c \right )\right ) a b +30 \left (-a^{2}+b^{2}\right ) \cot \left (d x +c \right )-30 d x \left (a -b \right ) \left (a +b \right )}{30 d}\) | \(116\) |
| norman | \(\frac {\left (-a^{2}+b^{2}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a b \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{5 d}-\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a b \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(142\) |
| risch | \(-2 i a b x -a^{2} x +b^{2} x -\frac {4 i a b c}{d}-\frac {2 i \left (-60 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-30 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-90 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+90 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+140 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-110 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}-70 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+70 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+23 a^{2}-20 b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(245\) |
1/d*(a^2*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+2*a*b*(-1/4 *cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+b^2*(-1/3*cot(d*x+c)^3+cot( d*x+c)+d*x+c))
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {30 \, a b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} - 15 \, {\left (2 \, {\left (a^{2} - b^{2}\right )} d x - 3 \, a b\right )} \tan \left (d x + c\right )^{5} + 30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{30 \, d \tan \left (d x + c\right )^{5}} \]
1/30*(30*a*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 - 15* (2*(a^2 - b^2)*d*x - 3*a*b)*tan(d*x + c)^5 + 30*a*b*tan(d*x + c)^3 - 30*(a ^2 - b^2)*tan(d*x + c)^4 - 15*a*b*tan(d*x + c) + 10*(a^2 - b^2)*tan(d*x + c)^2 - 6*a^2)/(d*tan(d*x + c)^5)
Time = 1.80 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\- a^{2} x - \frac {a^{2}}{d \tan {\left (c + d x \right )}} + \frac {a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a^{2}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {a b}{2 d \tan ^{4}{\left (c + d x \right )}} + b^{2} x + \frac {b^{2}}{d \tan {\left (c + d x \right )}} - \frac {b^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*a**2*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**2*cot(c)** 6, Eq(d, 0)), (zoo*a**2*x, Eq(c, -d*x)), (-a**2*x - a**2/(d*tan(c + d*x)) + a**2/(3*d*tan(c + d*x)**3) - a**2/(5*d*tan(c + d*x)**5) - a*b*log(tan(c + d*x)**2 + 1)/d + 2*a*b*log(tan(c + d*x))/d + a*b/(d*tan(c + d*x)**2) - a *b/(2*d*tan(c + d*x)**4) + b**2*x + b**2/(d*tan(c + d*x)) - b**2/(3*d*tan( c + d*x)**3), True))
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {30 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, a b \log \left (\tan \left (d x + c\right )\right ) + 30 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \]
-1/30*(30*a*b*log(tan(d*x + c)^2 + 1) - 60*a*b*log(tan(d*x + c)) + 30*(a^2 - b^2)*(d*x + c) - (30*a*b*tan(d*x + c)^3 - 30*(a^2 - b^2)*tan(d*x + c)^4 - 15*a*b*tan(d*x + c) + 10*(a^2 - b^2)*tan(d*x + c)^2 - 6*a^2)/tan(d*x + c)^5)/d
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (114) = 228\).
Time = 1.10 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.39 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 480 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {2192 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
1/480*(3*a^2*tan(1/2*d*x + 1/2*c)^5 - 15*a*b*tan(1/2*d*x + 1/2*c)^4 - 35*a ^2*tan(1/2*d*x + 1/2*c)^3 + 20*b^2*tan(1/2*d*x + 1/2*c)^3 + 180*a*b*tan(1/ 2*d*x + 1/2*c)^2 - 960*a*b*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 960*a*b*log(a bs(tan(1/2*d*x + 1/2*c))) + 330*a^2*tan(1/2*d*x + 1/2*c) - 300*b^2*tan(1/2 *d*x + 1/2*c) - 480*(a^2 - b^2)*(d*x + c) - (2192*a*b*tan(1/2*d*x + 1/2*c) ^5 + 330*a^2*tan(1/2*d*x + 1/2*c)^4 - 300*b^2*tan(1/2*d*x + 1/2*c)^4 - 180 *a*b*tan(1/2*d*x + 1/2*c)^3 - 35*a^2*tan(1/2*d*x + 1/2*c)^2 + 20*b^2*tan(1 /2*d*x + 1/2*c)^2 + 15*a*b*tan(1/2*d*x + 1/2*c) + 3*a^2)/tan(1/2*d*x + 1/2 *c)^5)/d
Time = 5.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2-b^2\right )+\frac {a^2}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{3}\right )+\frac {a\,b\,\mathrm {tan}\left (c+d\,x\right )}{2}-a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d} \]