Integrand size = 31, antiderivative size = 323 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\left (\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x\right )-\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)}{d}-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac {a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\sin (c+d x))}{d}-\frac {a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d} \]
-(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*x-(4*A*a^3*b-4*A*a*b^3+B*a^ 4-6*B*a^2*b^2+B*b^4)*cot(d*x+c)/d-1/2*(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4 *B*a*b^3)*cot(d*x+c)^2/d+1/15*a*(20*A*a^2*b-13*A*b^3+5*B*a^3-27*B*a*b^2)*c ot(d*x+c)^3/d+1/20*a^2*(5*A*a^2-8*A*b^2-12*B*a*b)*cot(d*x+c)^4/d-(A*a^4-6* A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^3)*ln(sin(d*x+c))/d-1/10*a*(3*A*b+2*B*a) *cot(d*x+c)^5*(a+b*tan(d*x+c))^2/d-1/6*a*A*cot(d*x+c)^6*(a+b*tan(d*x+c))^3 /d
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.93 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {-60 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)-30 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \cot ^2(c+d x)+20 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \cot ^3(c+d x)+15 a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \cot ^4(c+d x)-12 a^3 (4 A b+a B) \cot ^5(c+d x)-10 a^4 A \cot ^6(c+d x)+30 (a+i b)^4 (A+i B) \log (i-\tan (c+d x))-60 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\tan (c+d x))+30 (a-i b)^4 (A-i B) \log (i+\tan (c+d x))}{60 d} \]
(-60*(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Cot[c + d*x] - 30*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2 + 20*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x]^3 + 15*a^2*(a^ 2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^4 - 12*a^3*(4*A*b + a*B)*Cot[c + d*x ]^5 - 10*a^4*A*Cot[c + d*x]^6 + 30*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d *x]] - 60*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Tan[c + d*x]] + 30*(a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]])/(60*d)
Time = 2.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.05, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.677, Rules used = {3042, 4088, 27, 3042, 4128, 27, 3042, 4118, 3042, 4111, 27, 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.
| \(\displaystyle \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^4 (A+B \tan (c+d x))}{\tan (c+d x)^7}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {1}{6} \int 3 \cot ^6(c+d x) (a+b \tan (c+d x))^2 \left (-b (a A-2 b B) \tan ^2(c+d x)-2 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (3 A b+2 a B)\right )dx-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \left (-b (a A-2 b B) \tan ^2(c+d x)-2 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (3 A b+2 a B)\right )dx-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {(a+b \tan (c+d x))^2 \left (-b (a A-2 b B) \tan (c+d x)^2-2 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (3 A b+2 a B)\right )}{\tan (c+d x)^6}dx-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int -2 \cot ^5(c+d x) (a+b \tan (c+d x)) \left (b \left (3 B a^2+7 A b a-5 b^2 B\right ) \tan ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 A a^2-12 b B a-8 A b^2\right )\right )dx-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (b \left (3 B a^2+7 A b a-5 b^2 B\right ) \tan ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 A a^2-12 b B a-8 A b^2\right )\right )dx-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \int \frac {(a+b \tan (c+d x)) \left (b \left (3 B a^2+7 A b a-5 b^2 B\right ) \tan (c+d x)^2+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 A a^2-12 b B a-8 A b^2\right )\right )}{\tan (c+d x)^5}dx-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4118 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\int \cot ^4(c+d x) \left (b^2 \left (3 B a^2+7 A b a-5 b^2 B\right ) \tan ^2(c+d x)-5 \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)+a \left (5 B a^3+20 A b a^2-27 b^2 B a-13 A b^3\right )\right )dx-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\int \frac {b^2 \left (3 B a^2+7 A b a-5 b^2 B\right ) \tan (c+d x)^2-5 \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)+a \left (5 B a^3+20 A b a^2-27 b^2 B a-13 A b^3\right )}{\tan (c+d x)^4}dx-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (\int -5 \cot ^3(c+d x) \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)\right )dx-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \int \cot ^3(c+d x) \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)\right )dx-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \int \frac {A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (\int \cot ^2(c+d x) \left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B-\left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)\right )dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (\int \frac {B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B-\left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (\int -\cot (c+d x) \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)\right )dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (-\int \cot (c+d x) \left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)\right )dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (-\int \frac {A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4+\left (B a^4+4 A b a^3-6 b^2 B a^2-4 A b^3 a+b^4 B\right ) \tan (c+d x)}{\tan (c+d x)}dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (-\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \int \cot (c+d x)dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (-\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-5 \left (\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )\right )-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2}{5} \left (-\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{4 d}-\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{3 d}-5 \left (-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \log (-\sin (c+d x))}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )\right )\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\right )-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\) |
-1/6*(a*A*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3)/d + ((-2*(-1/3*(a*(20*a^2 *A*b - 13*A*b^3 + 5*a^3*B - 27*a*b^2*B)*Cot[c + d*x]^3)/d - (a^2*(5*a^2*A - 8*A*b^2 - 12*a*b*B)*Cot[c + d*x]^4)/(4*d) - 5*(-((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x) - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^ 2*b^2*B + b^4*B)*Cot[c + d*x])/d - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b *B + 4*a*b^3*B)*Cot[c + d*x]^2)/(2*d) - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4* a^3*b*B + 4*a*b^3*B)*Log[-Sin[c + d*x]])/d)))/5 - (a*(3*A*b + 2*a*B)*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*d))/2
3.3.66.3.1 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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((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 - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d* (b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[ e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] & & LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*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 [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Time = 0.37 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02
| method | result | size |
| parallelrisch | \(\frac {\left (30 A \,a^{4}-180 A \,a^{2} b^{2}+30 A \,b^{4}-120 B \,a^{3} b +120 B a \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-60 A \,a^{4}+360 A \,a^{2} b^{2}-60 A \,b^{4}+240 B \,a^{3} b -240 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-10 A \left (\cot ^{6}\left (d x +c \right )\right ) a^{4}+\left (-48 A \,a^{3} b -12 B \,a^{4}\right ) \left (\cot ^{5}\left (d x +c \right )\right )+15 a^{2} \left (\cot ^{4}\left (d x +c \right )\right ) \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )+\left (80 A \,a^{3} b -80 A a \,b^{3}+20 B \,a^{4}-120 B \,a^{2} b^{2}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+\left (-30 A \,a^{4}+180 A \,a^{2} b^{2}-30 A \,b^{4}+120 B \,a^{3} b -120 B a \,b^{3}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+\left (-240 A \,a^{3} b +240 A a \,b^{3}-60 B \,a^{4}+360 B \,a^{2} b^{2}-60 B \,b^{4}\right ) \cot \left (d x +c \right )-240 d \left (A \,a^{3} b -A a \,b^{3}+\frac {1}{4} B \,a^{4}-\frac {3}{2} B \,a^{2} b^{2}+\frac {1}{4} B \,b^{4}\right ) x}{60 d}\) | \(330\) |
| derivativedivides | \(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}}{\tan \left (d x +c \right )}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {A \,a^{4}}{6 \tan \left (d x +c \right )^{6}}-\frac {a^{3} \left (4 A b +B a \right )}{5 \tan \left (d x +c \right )^{5}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{4 \tan \left (d x +c \right )^{4}}}{d}\) | \(331\) |
| default | \(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}}{\tan \left (d x +c \right )}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {A \,a^{4}}{6 \tan \left (d x +c \right )^{6}}-\frac {a^{3} \left (4 A b +B a \right )}{5 \tan \left (d x +c \right )^{5}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{4 \tan \left (d x +c \right )^{4}}}{d}\) | \(331\) |
| norman | \(\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) x \left (\tan ^{6}\left (d x +c \right )\right )-\frac {A \,a^{4}}{6 d}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{d}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} \left (4 A b +B a \right ) \tan \left (d x +c \right )}{5 d}}{\tan \left (d x +c \right )^{6}}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(352\) |
| risch | \(\text {Expression too large to display}\) | \(1137\) |
1/60*((30*A*a^4-180*A*a^2*b^2+30*A*b^4-120*B*a^3*b+120*B*a*b^3)*ln(sec(d*x +c)^2)+(-60*A*a^4+360*A*a^2*b^2-60*A*b^4+240*B*a^3*b-240*B*a*b^3)*ln(tan(d *x+c))-10*A*cot(d*x+c)^6*a^4+(-48*A*a^3*b-12*B*a^4)*cot(d*x+c)^5+15*a^2*co t(d*x+c)^4*(A*a^2-6*A*b^2-4*B*a*b)+(80*A*a^3*b-80*A*a*b^3+20*B*a^4-120*B*a ^2*b^2)*cot(d*x+c)^3+(-30*A*a^4+180*A*a^2*b^2-30*A*b^4+120*B*a^3*b-120*B*a *b^3)*cot(d*x+c)^2+(-240*A*a^3*b+240*A*a*b^3-60*B*a^4+360*B*a^2*b^2-60*B*b ^4)*cot(d*x+c)-240*d*(A*a^3*b-A*a*b^3+1/4*B*a^4-3/2*B*a^2*b^2+1/4*B*b^4)*x )/d
Time = 0.27 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.08 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{6} + 5 \, {\left (11 \, A a^{4} - 36 \, B a^{3} b - 54 \, A a^{2} b^{2} + 24 \, B a b^{3} + 6 \, A b^{4} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{6} + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{6}} \]
-1/60*(30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(d* x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^6 + 5*(11*A*a^4 - 36*B*a^3*b - 54*A*a^2*b^2 + 24*B*a*b^3 + 6*A*b^4 + 12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*d*x)*tan(d*x + c)^6 + 60*(B*a^4 + 4*A*a^3*b - 6*B*a^ 2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c)^5 + 10*A*a^4 + 30*(A*a^4 - 4*B*a^3 *b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 - 20*(B*a^4 + 4*A*a^3 *b - 6*B*a^2*b^2 - 4*A*a*b^3)*tan(d*x + c)^3 - 15*(A*a^4 - 4*B*a^3*b - 6*A *a^2*b^2)*tan(d*x + c)^2 + 12*(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/(d*tan(d*x + c)^6)
Time = 12.58 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.99 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{7}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{4} x & \text {for}\: c = - d x \\\frac {A a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {A a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} - \frac {A a^{4}}{6 d \tan ^{6}{\left (c + d x \right )}} - 4 A a^{3} b x - \frac {4 A a^{3} b}{d \tan {\left (c + d x \right )}} + \frac {4 A a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {4 A a^{3} b}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 A a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 A a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 A a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - \frac {3 A a^{2} b^{2}}{2 d \tan ^{4}{\left (c + d x \right )}} + 4 A a b^{3} x + \frac {4 A a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {4 A a b^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {A b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - B a^{4} x - \frac {B a^{4}}{d \tan {\left (c + d x \right )}} + \frac {B a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {B a^{4}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {2 B a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 B a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {2 B a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3} b}{d \tan ^{4}{\left (c + d x \right )}} + 6 B a^{2} b^{2} x + \frac {6 B a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 B a^{2} b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {2 B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 B a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 B a b^{3}}{d \tan ^{2}{\left (c + d x \right )}} - B b^{4} x - \frac {B b^{4}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*A*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(A + B*tan(c))*(a + b*ta n(c))**4*cot(c)**7, Eq(d, 0)), (zoo*A*a**4*x, Eq(c, -d*x)), (A*a**4*log(ta n(c + d*x)**2 + 1)/(2*d) - A*a**4*log(tan(c + d*x))/d - A*a**4/(2*d*tan(c + d*x)**2) + A*a**4/(4*d*tan(c + d*x)**4) - A*a**4/(6*d*tan(c + d*x)**6) - 4*A*a**3*b*x - 4*A*a**3*b/(d*tan(c + d*x)) + 4*A*a**3*b/(3*d*tan(c + d*x) **3) - 4*A*a**3*b/(5*d*tan(c + d*x)**5) - 3*A*a**2*b**2*log(tan(c + d*x)** 2 + 1)/d + 6*A*a**2*b**2*log(tan(c + d*x))/d + 3*A*a**2*b**2/(d*tan(c + d* x)**2) - 3*A*a**2*b**2/(2*d*tan(c + d*x)**4) + 4*A*a*b**3*x + 4*A*a*b**3/( d*tan(c + d*x)) - 4*A*a*b**3/(3*d*tan(c + d*x)**3) + A*b**4*log(tan(c + d* x)**2 + 1)/(2*d) - A*b**4*log(tan(c + d*x))/d - A*b**4/(2*d*tan(c + d*x)** 2) - B*a**4*x - B*a**4/(d*tan(c + d*x)) + B*a**4/(3*d*tan(c + d*x)**3) - B *a**4/(5*d*tan(c + d*x)**5) - 2*B*a**3*b*log(tan(c + d*x)**2 + 1)/d + 4*B* a**3*b*log(tan(c + d*x))/d + 2*B*a**3*b/(d*tan(c + d*x)**2) - B*a**3*b/(d* tan(c + d*x)**4) + 6*B*a**2*b**2*x + 6*B*a**2*b**2/(d*tan(c + d*x)) - 2*B* a**2*b**2/(d*tan(c + d*x)**3) + 2*B*a*b**3*log(tan(c + d*x)**2 + 1)/d - 4* B*a*b**3*log(tan(c + d*x))/d - 2*B*a*b**3/(d*tan(c + d*x)**2) - B*b**4*x - B*b**4/(d*tan(c + d*x)), True))
Time = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.03 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} - 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]
-1/60*(60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) - 30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(d*x + c )^2 + 1) + 60*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(ta n(d*x + c)) + (60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*ta n(d*x + c)^5 + 10*A*a^4 + 30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 - 20*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3) *tan(d*x + c)^3 - 15*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2)*tan(d*x + c)^2 + 12 *(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/tan(d*x + c)^6)/d
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (313) = 626\).
Time = 1.85 (sec) , antiderivative size = 943, normalized size of antiderivative = 2.92 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
-1/1920*(5*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 12*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 60*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 12 0*B*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 180*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 140*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 560*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 48 0*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 320*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 435*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 1440*B*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 2 160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 960*B*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 240*A*b^4*tan(1/2*d*x + 1/2*c)^2 - 1320*B*a^4*tan(1/2*d*x + 1/2*c) - 528 0*A*a^3*b*tan(1/2*d*x + 1/2*c) + 7200*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 480 0*A*a*b^3*tan(1/2*d*x + 1/2*c) - 960*B*b^4*tan(1/2*d*x + 1/2*c) + 1920*(B* a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) - 1920*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 1920*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(abs (tan(1/2*d*x + 1/2*c))) - (4704*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 18816*B*a^3 *b*tan(1/2*d*x + 1/2*c)^6 - 28224*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 18816 *B*a*b^3*tan(1/2*d*x + 1/2*c)^6 + 4704*A*b^4*tan(1/2*d*x + 1/2*c)^6 - 1320 *B*a^4*tan(1/2*d*x + 1/2*c)^5 - 5280*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 7200 *B*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 4800*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 960*B*b^4*tan(1/2*d*x + 1/2*c)^5 - 435*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 1440 *B*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 2160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4...
Time = 8.11 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^6\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{5}+\frac {4\,A\,b\,a^3}{5}\right )+\frac {A\,a^4}{6}-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,a^3\,b}{3}-2\,B\,a^2\,b^2-\frac {4\,A\,a\,b^3}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^4}{4}+B\,a^3\,b+\frac {3\,A\,a^2\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {A\,a^4}{2}-2\,B\,a^3\,b-3\,A\,a^2\,b^2+2\,B\,a\,b^3+\frac {A\,b^4}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3+B\,b^4\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^4-4\,B\,a^3\,b-6\,A\,a^2\,b^2+4\,B\,a\,b^3+A\,b^4\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4}{2\,d} \]
(log(tan(c + d*x) + 1i)*(A - B*1i)*(a*1i + b)^4)/(2*d) - (log(tan(c + d*x) )*(A*a^4 + A*b^4 - 6*A*a^2*b^2 + 4*B*a*b^3 - 4*B*a^3*b))/d - (cot(c + d*x) ^6*(tan(c + d*x)*((B*a^4)/5 + (4*A*a^3*b)/5) + (A*a^4)/6 - tan(c + d*x)^3* ((B*a^4)/3 - 2*B*a^2*b^2 - (4*A*a*b^3)/3 + (4*A*a^3*b)/3) + tan(c + d*x)^2 *((3*A*a^2*b^2)/2 - (A*a^4)/4 + B*a^3*b) + tan(c + d*x)^4*((A*a^4)/2 + (A* b^4)/2 - 3*A*a^2*b^2 + 2*B*a*b^3 - 2*B*a^3*b) + tan(c + d*x)^5*(B*a^4 + B* b^4 - 6*B*a^2*b^2 - 4*A*a*b^3 + 4*A*a^3*b)))/d + (log(tan(c + d*x) - 1i)*( A + B*1i)*(a*1i - b)^4)/(2*d)