Integrand size = 21, antiderivative size = 170 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=-\left (\left (a^4-6 a^2 b^2+b^4\right ) x\right )-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}+\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
-(a^4-6*a^2*b^2+b^4)*x-(a^4-6*a^2*b^2+b^4)*cot(d*x+c)/d+2*a*b*(a^2-b^2)*co t(d*x+c)^2/d+1/15*a^2*(5*a^2-27*b^2)*cot(d*x+c)^3/d-3/5*a^3*b*cot(d*x+c)^4 /d+4*a*b*(a^2-b^2)*ln(sin(d*x+c))/d-1/5*a^2*cot(d*x+c)^5*(a+b*tan(d*x+c))^ 2/d
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)-2 a (a-b) b (a+b) \cot ^2(c+d x)-\frac {1}{3} a^2 \left (a^2-6 b^2\right ) \cot ^3(c+d x)+a^3 b \cot ^4(c+d x)+\frac {1}{5} a^4 \cot ^5(c+d x)+\frac {1}{2} i (a-i b)^4 \log (i-\cot (c+d x))-\frac {1}{2} i (a+i b)^4 \log (i+\cot (c+d x))}{d} \]
-(((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x] - 2*a*(a - b)*b*(a + b)*Cot[c + d* x]^2 - (a^2*(a^2 - 6*b^2)*Cot[c + d*x]^3)/3 + a^3*b*Cot[c + d*x]^4 + (a^4* Cot[c + d*x]^5)/5 + (I/2)*(a - I*b)^4*Log[I - Cot[c + d*x]] - (I/2)*(a + I *b)^4*Log[I + Cot[c + d*x]])/d)
Time = 1.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4048, 3042, 4118, 25, 3042, 4111, 27, 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.
| \(\displaystyle \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^4}{\tan (c+d x)^6}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (12 b a^2-5 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right )dx-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {(a+b \tan (c+d x)) \left (12 b a^2-5 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (3 a^2-5 b^2\right ) \tan (c+d x)^2\right )}{\tan (c+d x)^5}dx-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 4118 |
| \(\displaystyle \frac {1}{5} \left (\int -\cot ^4(c+d x) \left (\left (5 a^2-27 b^2\right ) a^2+20 b \left (a^2-b^2\right ) \tan (c+d x) a+b^2 \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right )dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (-\int \cot ^4(c+d x) \left (\left (5 a^2-27 b^2\right ) a^2+20 b \left (a^2-b^2\right ) \tan (c+d x) a+b^2 \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right )dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {\left (5 a^2-27 b^2\right ) a^2+20 b \left (a^2-b^2\right ) \tan (c+d x) a+b^2 \left (3 a^2-5 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^4}dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {1}{5} \left (-\int 5 \cot ^3(c+d x) \left (4 a b \left (a^2-b^2\right )-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)\right )dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (-5 \int \cot ^3(c+d x) \left (4 a b \left (a^2-b^2\right )-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)\right )dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-5 \int \frac {4 a b \left (a^2-b^2\right )-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (\int -\cot ^2(c+d x) \left (a^4-6 b^2 a^2+4 b \left (a^2-b^2\right ) \tan (c+d x) a+b^4\right )dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-\int \cot ^2(c+d x) \left (a^4-6 b^2 a^2+4 b \left (a^2-b^2\right ) \tan (c+d x) a+b^4\right )dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-\int \frac {a^4-6 b^2 a^2+4 b \left (a^2-b^2\right ) \tan (c+d x) a+b^4}{\tan (c+d x)^2}dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-\int \cot (c+d x) \left (4 a b \left (a^2-b^2\right )-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)\right )dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-\int \frac {4 a b \left (a^2-b^2\right )-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)}{\tan (c+d x)}dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-4 a b \left (a^2-b^2\right ) \int \cot (c+d x)dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+x \left (a^4-6 a^2 b^2+b^4\right )\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (-4 a b \left (a^2-b^2\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+x \left (a^4-6 a^2 b^2+b^4\right )\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (-5 \left (4 a b \left (a^2-b^2\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+x \left (a^4-6 a^2 b^2+b^4\right )\right )-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{5} \left (-\frac {3 a^3 b \cot ^4(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{3 d}-5 \left (-\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}-\frac {4 a b \left (a^2-b^2\right ) \log (-\sin (c+d x))}{d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+x \left (a^4-6 a^2 b^2+b^4\right )\right )\right )-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\) |
((a^2*(5*a^2 - 27*b^2)*Cot[c + d*x]^3)/(3*d) - (3*a^3*b*Cot[c + d*x]^4)/d - 5*((a^4 - 6*a^2*b^2 + b^4)*x + ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x])/d - (2*a*b*(a^2 - b^2)*Cot[c + d*x]^2)/d - (4*a*b*(a^2 - b^2)*Log[-Sin[c + d *x]])/d))/5 - (a^2*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*d)
3.5.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((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[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_. )*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(c^2*C - B*c*d + A*d^2)*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Simp[1/(d*(c^2 + d^2)) Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b* (c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d) *Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n , -1]
Time = 0.68 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02
| method | result | size |
| parallelrisch | \(\frac {30 \left (-a^{3} b +a \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+60 \left (a^{3} b -a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-3 \left (\cot ^{5}\left (d x +c \right )\right ) a^{4}-15 \left (\cot ^{4}\left (d x +c \right )\right ) a^{3} b +5 \left (a^{4}-6 a^{2} b^{2}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+30 \left (a^{3} b -a \,b^{3}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+15 \left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \cot \left (d x +c \right )-15 a^{4} d x +90 a^{2} b^{2} d x -15 b^{4} d x}{15 d}\) | \(173\) |
| derivativedivides | \(\frac {-\frac {a^{4}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{4}-6 a^{2} b^{2}+b^{4}}{\tan \left (d x +c \right )}+\frac {a^{2} \left (a^{2}-6 b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} b}{\tan \left (d x +c \right )^{4}}+\frac {2 a b \left (a^{2}-b^{2}\right )}{\tan \left (d x +c \right )^{2}}+4 a b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(174\) |
| default | \(\frac {-\frac {a^{4}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{4}-6 a^{2} b^{2}+b^{4}}{\tan \left (d x +c \right )}+\frac {a^{2} \left (a^{2}-6 b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} b}{\tan \left (d x +c \right )^{4}}+\frac {2 a b \left (a^{2}-b^{2}\right )}{\tan \left (d x +c \right )^{2}}+4 a b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(174\) |
| norman | \(\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {a^{4}}{5 d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {a^{2} \left (a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a^{3} b \tan \left (d x +c \right )}{d}+\frac {2 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d}}{\tan \left (d x +c \right )^{5}}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(191\) |
| risch | \(-4 i a^{3} b x +4 i a \,b^{3} x -a^{4} x +6 a^{2} b^{2} x -b^{4} x -\frac {8 i a^{3} b c}{d}+\frac {8 i a \,b^{3} c}{d}-\frac {2 i \left (240 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+120 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+45 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-180 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+180 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-240 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-90 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+540 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+60 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-120 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+140 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-660 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+90 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-180 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-60 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-70 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+420 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+23 a^{4}-120 a^{2} b^{2}+15 b^{4}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(449\) |
1/15*(30*(-a^3*b+a*b^3)*ln(sec(d*x+c)^2)+60*(a^3*b-a*b^3)*ln(tan(d*x+c))-3 *cot(d*x+c)^5*a^4-15*cot(d*x+c)^4*a^3*b+5*(a^4-6*a^2*b^2)*cot(d*x+c)^3+30* (a^3*b-a*b^3)*cot(d*x+c)^2+15*(-a^4+6*a^2*b^2-b^4)*cot(d*x+c)-15*a^4*d*x+9 0*a^2*b^2*d*x-15*b^4*d*x)/d
Time = 0.25 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {30 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, a^{3} b - 2 \, a b^{3} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 15 \, a^{3} b \tan \left (d x + c\right ) - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 30 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 5 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{15 \, d \tan \left (d x + c\right )^{5}} \]
1/15*(30*(a^3*b - a*b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 + 15*(3*a^3*b - 2*a*b^3 - (a^4 - 6*a^2*b^2 + b^4)*d*x)*tan(d*x + c) ^5 - 15*a^3*b*tan(d*x + c) - 15*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^4 - 3 *a^4 + 30*(a^3*b - a*b^3)*tan(d*x + c)^3 + 5*(a^4 - 6*a^2*b^2)*tan(d*x + c )^2)/(d*tan(d*x + c)^5)
Time = 4.00 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=\begin {cases} \tilde {\infty } a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{4} x & \text {for}\: c = - d x \\- a^{4} x - \frac {a^{4}}{d \tan {\left (c + d x \right )}} + \frac {a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a^{4}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {2 a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{3} b}{d \tan ^{4}{\left (c + d x \right )}} + 6 a^{2} b^{2} x + \frac {6 a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 a^{2} b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 a b^{3}}{d \tan ^{2}{\left (c + d x \right )}} - b^{4} x - \frac {b^{4}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**4*cot(c)** 6, Eq(d, 0)), (zoo*a**4*x, Eq(c, -d*x)), (-a**4*x - a**4/(d*tan(c + d*x)) + a**4/(3*d*tan(c + d*x)**3) - a**4/(5*d*tan(c + d*x)**5) - 2*a**3*b*log(t an(c + d*x)**2 + 1)/d + 4*a**3*b*log(tan(c + d*x))/d + 2*a**3*b/(d*tan(c + d*x)**2) - a**3*b/(d*tan(c + d*x)**4) + 6*a**2*b**2*x + 6*a**2*b**2/(d*ta n(c + d*x)) - 2*a**2*b**2/(d*tan(c + d*x)**3) + 2*a*b**3*log(tan(c + d*x)* *2 + 1)/d - 4*a*b**3*log(tan(c + d*x))/d - 2*a*b**3/(d*tan(c + d*x)**2) - b**4*x - b**4/(d*tan(c + d*x)), True))
Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} + 30 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {15 \, a^{3} b \tan \left (d x + c\right ) + 15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 3 \, a^{4} - 30 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 5 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{5}}}{15 \, d} \]
-1/15*(15*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c) + 30*(a^3*b - a*b^3)*log(tan(d *x + c)^2 + 1) - 60*(a^3*b - a*b^3)*log(tan(d*x + c)) + (15*a^3*b*tan(d*x + c) + 15*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^4 + 3*a^4 - 30*(a^3*b - a*b ^3)*tan(d*x + c)^3 - 5*(a^4 - 6*a^2*b^2)*tan(d*x + c)^2)/tan(d*x + c)^5)/d
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (164) = 328\).
Time = 1.20 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.45 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 330 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 480 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} - 1920 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 1920 \, {\left (a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4384 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 30 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
1/480*(3*a^4*tan(1/2*d*x + 1/2*c)^5 - 30*a^3*b*tan(1/2*d*x + 1/2*c)^4 - 35 *a^4*tan(1/2*d*x + 1/2*c)^3 + 120*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 360*a^3 *b*tan(1/2*d*x + 1/2*c)^2 - 240*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 330*a^4*tan (1/2*d*x + 1/2*c) - 1800*a^2*b^2*tan(1/2*d*x + 1/2*c) + 240*b^4*tan(1/2*d* x + 1/2*c) - 480*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c) - 1920*(a^3*b - a*b^3)* log(tan(1/2*d*x + 1/2*c)^2 + 1) + 1920*(a^3*b - a*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (4384*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 4384*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 330*a^4*tan(1/2*d*x + 1/2*c)^4 - 1800*a^2*b^2*tan(1/2*d*x + 1 /2*c)^4 + 240*b^4*tan(1/2*d*x + 1/2*c)^4 - 360*a^3*b*tan(1/2*d*x + 1/2*c)^ 3 + 240*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 35*a^4*tan(1/2*d*x + 1/2*c)^2 + 120 *a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 30*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/t an(1/2*d*x + 1/2*c)^5)/d
Time = 5.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {4\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a\,b^3-2\,a^3\,b\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{3}-2\,a^2\,b^2\right )+\frac {a^4}{5}+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4-6\,a^2\,b^2+b^4\right )+a^3\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d} \]
(log(tan(c + d*x) - 1i)*(a*1i - b)^4*1i)/(2*d) - (cot(c + d*x)^5*(tan(c + d*x)^3*(2*a*b^3 - 2*a^3*b) - tan(c + d*x)^2*(a^4/3 - 2*a^2*b^2) + a^4/5 + tan(c + d*x)^4*(a^4 + b^4 - 6*a^2*b^2) + a^3*b*tan(c + d*x)))/d - (log(tan (c + d*x) + 1i)*(a - b*1i)^4*1i)/(2*d) + (4*a*b*log(tan(c + d*x))*(a^2 - b ^2))/d