Integrand size = 21, antiderivative size = 120 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=-\left (\left (a^2-b^2\right ) x\right )-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a b \log (\sin (c+d x))}{d} \] Output:
-(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 = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2 a b \csc ^2(c+d x)}{d}-\frac {a b \csc ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}-\frac {b^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {2 a b \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]
Output:
(2*a*b*Csc[c + d*x]^2)/d - (a*b*Csc[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*L og[Sin[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 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {2 a b \log (-\sin (c+d x))}{d}\) |
Input:
Int[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]
Output:
-((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
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 = 1.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {b^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(103\) |
| default | \(\frac {b^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(103\) |
| parallelrisch | \(\frac {-30 a b \ln \left (\sec \left (d x +c \right )^{2}\right )+60 a b \ln \left (\tan \left (d x +c \right )\right )-6 \cot \left (d x +c \right )^{5} a^{2}-15 \cot \left (d x +c \right )^{4} a b +10 \cot \left (d x +c \right )^{3} \left (a^{2}-b^{2}\right )+30 \cot \left (d x +c \right )^{2} a b +30 \left (-a^{2}+b^{2}\right ) \cot \left (d x +c \right )-30 d x \left (a -b \right ) \left (a +b \right )}{30 d}\) | \(116\) |
| norman | \(\frac {\left (-a^{2}+b^{2}\right ) x \tan \left (d x +c \right )^{5}+\frac {a b \tan \left (d x +c \right )^{3}}{d}-\frac {a^{2}}{5 d}-\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{4}}{d}+\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2}}{3 d}-\frac {a b \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(142\) |
| risch | \(-2 i a b x -a^{2} x +b^{2} x -\frac {4 i a b c}{d}-\frac {2 i \left (-60 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-30 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-90 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+90 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+140 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-110 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}-70 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+70 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+23 a^{2}-20 b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(245\) |
Input:
int(cot(d*x+c)^6*(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(b^2*(-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)))+a^2*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot( d*x+c)-d*x-c))
Time = 0.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {30 \, a b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} - 15 \, {\left (2 \, {\left (a^{2} - b^{2}\right )} d x - 3 \, a b\right )} \tan \left (d x + c\right )^{5} + 30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{30 \, d \tan \left (d x + c\right )^{5}} \] Input:
integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
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 = 2.02 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.42 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\- a^{2} x - \frac {a^{2}}{d \tan {\left (c + d x \right )}} + \frac {a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a^{2}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {a b}{2 d \tan ^{4}{\left (c + d x \right )}} + b^{2} x + \frac {b^{2}}{d \tan {\left (c + d x \right )}} - \frac {b^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cot(d*x+c)**6*(a+b*tan(d*x+c))**2,x)
Output:
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 = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {30 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, a b \log \left (\tan \left (d x + c\right )\right ) + 30 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-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
Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d} + \frac {2 \, a b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{d} + \frac {30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{30 \, d \tan \left (d x + c\right )^{5}} \] Input:
integrate(cot(d*x+c)^6*(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
-a*b*log(tan(d*x + c)^2 + 1)/d + 2*a*b*log(abs(tan(d*x + c)))/d - (a^2 - b ^2)*(d*x + c)/d + 1/30*(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 = 1.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2-b^2\right )+\frac {a^2}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{3}\right )+\frac {a\,b\,\mathrm {tan}\left (c+d\,x\right )}{2}-a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d} \] Input:
int(cot(c + d*x)^6*(a + b*tan(c + d*x))^2,x)
Output:
(2*a*b*log(tan(c + d*x)))/d - (log(tan(c + d*x) + 1i)*(a - b*1i)^2*1i)/(2* d) - (log(tan(c + d*x) - 1i)*(a*1i - b)^2*1i)/(2*d) - (cot(c + d*x)^5*(tan (c + d*x)^4*(a^2 - b^2) + a^2/5 - tan(c + d*x)^2*(a^2/3 - b^2/3) + (a*b*ta n(c + d*x))/2 - a*b*tan(c + d*x)^3))/d
Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {-368 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-48 \cos \left (d x +c \right ) a^{2}-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{5} a b +480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} a b -240 \sin \left (d x +c \right )^{5} a^{2} d x -195 \sin \left (d x +c \right )^{5} a b +240 \sin \left (d x +c \right )^{5} b^{2} d x +480 \sin \left (d x +c \right )^{3} a b -120 \sin \left (d x +c \right ) a b}{240 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+b*tan(d*x+c))^2,x)
Output:
( - 368*cos(c + d*x)*sin(c + d*x)**4*a**2 + 320*cos(c + d*x)*sin(c + d*x)* *4*b**2 + 176*cos(c + d*x)*sin(c + d*x)**2*a**2 - 80*cos(c + d*x)*sin(c + d*x)**2*b**2 - 48*cos(c + d*x)*a**2 - 480*log(tan((c + d*x)/2)**2 + 1)*sin (c + d*x)**5*a*b + 480*log(tan((c + d*x)/2))*sin(c + d*x)**5*a*b - 240*sin (c + d*x)**5*a**2*d*x - 195*sin(c + d*x)**5*a*b + 240*sin(c + d*x)**5*b**2 *d*x + 480*sin(c + d*x)**3*a*b - 120*sin(c + d*x)*a*b)/(240*sin(c + d*x)** 5*d)