Integrand size = 33, antiderivative size = 421 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
2/21*a*(7*A*a^2-18*A*b^2-21*B*a*b)*cot(d*x+c)^(3/2)/d-2/35*a^2*(11*A*b+7*B *a)*cot(d*x+c)^(5/2)/d-2/7*a*A*cot(d*x+c)^(3/2)*(b+a*cot(d*x+c))^2/d-1/2*( 3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(-1+2^(1/2)*cot(d*x +c)^(1/2))/d*2^(1/2)-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B)) *arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/4*(a^3*(A-B)-3*a*b^2*(A-B) -3*a^2*b*(A+B)+b^3*(A+B))*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1 /2)+1/4*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*ln(1+cot(d*x+c)+ 2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+2*(3*A*a^2*b-A*b^3+B*a^3-3*B*a*b^2)*co t(d*x+c)^(1/2)/d
Time = 3.91 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.77 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt {\cot (c+d x)} \left (-\frac {\left (3 a^2 b (A-B)+b^3 (-A+B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{2 \sqrt {2}}-\frac {\left (a^3 (A-B)+3 a b^2 (-A+B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{4 \sqrt {2}}-\frac {a^3 A}{7 \tan ^{\frac {7}{2}}(c+d x)}-\frac {a^2 (3 A b+a B)}{5 \tan ^{\frac {5}{2}}(c+d x)}+\frac {a \left (a^2 A-3 A b^2-3 a b B\right )}{3 \tan ^{\frac {3}{2}}(c+d x)}+\frac {3 a^2 A b-A b^3+a^3 B-3 a b^2 B}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{d} \]
(2*Sqrt[Cot[c + d*x]]*(-1/2*((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + S qrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] - ((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A + B) + b^3*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2] ) - (a^3*A)/(7*Tan[c + d*x]^(7/2)) - (a^2*(3*A*b + a*B))/(5*Tan[c + d*x]^( 5/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/(3*Tan[c + d*x]^(3/2)) + (3*a^2*A* b - A*b^3 + a^3*B - 3*a*b^2*B)/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d
Time = 1.59 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.84, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.697, Rules used = {3042, 4064, 3042, 4090, 27, 3042, 4120, 27, 3042, 4113, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{9/2} (a+b \tan (c+d x))^3 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^3 (A \cot (c+d x)+B)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4090 |
| \(\displaystyle -\frac {2}{7} \int \frac {1}{2} \sqrt {\cot (c+d x)} (b+a \cot (c+d x)) \left (-a (11 A b+7 a B) \cot ^2(c+d x)+7 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (3 a A-7 b B)\right )dx-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{7} \int \sqrt {\cot (c+d x)} (b+a \cot (c+d x)) \left (-a (11 A b+7 a B) \cot ^2(c+d x)+7 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (3 a A-7 b B)\right )dx-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{7} \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-a (11 A b+7 a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2-7 \left (A a^2-2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b (3 a A-7 b B)\right )dx-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {1}{7} \left (-\frac {2}{5} \int \frac {5}{2} \sqrt {\cot (c+d x)} \left ((3 a A-7 b B) b^2+a \left (7 A a^2-21 b B a-18 A b^2\right ) \cot ^2(c+d x)+7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )dx-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {\cot (c+d x)} \left ((3 a A-7 b B) b^2+a \left (7 A a^2-21 b B a-18 A b^2\right ) \cot ^2(c+d x)+7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )dx-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left ((3 a A-7 b B) b^2+a \left (7 A a^2-21 b B a-18 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2-7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {\cot (c+d x)} \left (7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)-7 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )\right )dx+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-7 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )-7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{7} \left (-\int \frac {-7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right )-7 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \frac {7 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan \left (c+d x+\frac {\pi }{2}\right )-7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{7} \left (-\frac {2 \int \frac {7 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3+\left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \int \frac {B a^3+3 A b a^2-3 b^2 B a-A b^3+\left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{7} \left (\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {14 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {14 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}\right )-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d}\) |
(-2*a*A*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x])^2)/(7*d) + ((14*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cot[c + d*x]])/d + (2*a*(7*a^2*A - 18*A *b^2 - 21*a*b*B)*Cot[c + d*x]^(3/2))/(3*d) - (2*a^2*(11*A*b + 7*a*B)*Cot[c + d*x]^(5/2))/(5*d) - (14*(((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + Ar cTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 - ((a^3*(A - B) - 3*a*b^2 *(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Co t[c + d*x]]/(2*Sqrt[2])))/2))/d)/7
3.6.83.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b *(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2 , 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
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*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1191\) vs. \(2(379)=758\).
Time = 1.18 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.83
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1192\) |
| default | \(\text {Expression too large to display}\) | \(1192\) |
1/420/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(-840*A*tan(d*x+c)^3*b^3-168*B*tan (d*x+c)*a^3-630*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7 /2)*a*b^2-630*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/ 2)*a^2*b-630*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2 )*a*b^2-315*B*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d *x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2-315*A*ln(-(1+2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2 ^(1/2)*tan(d*x+c)^(7/2)*a*b^2+630*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^( 1/2)*tan(d*x+c)^(7/2)*a^2*b-630*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/ 2)*tan(d*x+c)^(7/2)*a*b^2+630*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2 )*tan(d*x+c)^(7/2)*a^2*b-630*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2) *tan(d*x+c)^(7/2)*a*b^2+315*A*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/ (1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-31 5*B*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)- tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-630*B*arctan(1+2^(1/2)*tan(d *x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b+2520*A*tan(d*x+c)^3*a^2*b-120* A*a^3-2520*B*tan(d*x+c)^3*a*b^2-840*A*tan(d*x+c)^2*a*b^2-840*B*tan(d*x+c)^ 2*a^2*b-504*A*tan(d*x+c)*a^2*b+840*B*tan(d*x+c)^3*a^3+280*A*tan(d*x+c)^2*a ^3+210*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^3- 210*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3-...
Leaf count of result is larger than twice the leaf count of optimal. 6214 vs. \(2 (379) = 758\).
Time = 6.61 (sec) , antiderivative size = 6214, normalized size of antiderivative = 14.76 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 0.45 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.87 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {120 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}} - \frac {840 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}}{\sqrt {\tan \left (d x + c\right )}} - \frac {280 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {168 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{420 \, d} \]
-1/420*(210*sqrt(2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 210*sqrt( 2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*arctan( -1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 105*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*log(sqrt(2)/sqrt(tan(d* x + c)) + 1/tan(d*x + c) + 1) + 105*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan( d*x + c) + 1) + 120*A*a^3/tan(d*x + c)^(7/2) - 840*(B*a^3 + 3*A*a^2*b - 3* B*a*b^2 - A*b^3)/sqrt(tan(d*x + c)) - 280*(A*a^3 - 3*B*a^2*b - 3*A*a*b^2)/ tan(d*x + c)^(3/2) + 168*(B*a^3 + 3*A*a^2*b)/tan(d*x + c)^(5/2))/d
\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]