Integrand size = 33, antiderivative size = 342 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {\left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{64 a^{3/2} d}+\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{64 a d}+\frac {\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{96 d}-\frac {a (11 A b+8 a B) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d} \]
-1/64*(128*A*a^4-240*A*a^2*b^2-5*A*b^4-320*B*a^3*b+40*B*a*b^3)*arctanh((a+ b*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+(a-I*b)^(5/2)*(A-I*B)*arctanh((a+b* tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(5/2)*(A+I*B)*arctanh((a+b*tan( d*x+c))^(1/2)/(a+I*b)^(1/2))/d+1/64*(144*A*a^2*b-5*A*b^3+64*B*a^3-88*B*a*b ^2)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a/d+1/96*(48*A*a^2-59*A*b^2-104*B*a* b)*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d-1/24*a*(11*A*b+8*B*a)*cot(d*x+c)^ 3*(a+b*tan(d*x+c))^(1/2)/d-1/4*a*A*cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)/d
Time = 6.58 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.82 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {2 b B \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac {2}{5} \left (\frac {b (5 A b+2 a B) \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)}}{7 d}-\frac {2}{7} \left (-\frac {\left (35 a^2 A-40 A b^2-72 a b B\right ) \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)}}{16 d}-\frac {\frac {7 a \left (85 a A b+40 a^2 B-48 b^2 B\right ) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{24 d}-\frac {\frac {35 a^2 \left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{32 d}-\frac {-\frac {-\frac {105 a^{5/2} \left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{32 d}+\frac {i \sqrt {a-i b} \left (210 a^4 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+210 i a^4 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(-a+i b) d}-\frac {i \sqrt {a+i b} \left (210 a^4 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-210 i a^4 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(-a-i b) d}}{a}-\frac {105 a^2 \left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{32 d}}{2 a}}{3 a}}{4 a}\right )\right ) \]
(-2*b*B*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2))/(5*d) - (2*((b*(5*A*b + 2*a*B)*Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]])/(7*d) - (2*(-1/16*((35*a^ 2*A - 40*A*b^2 - 72*a*b*B)*Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]])/d - (( 7*a*(85*a*A*b + 40*a^2*B - 48*b^2*B)*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x ]])/(24*d) - ((35*a^2*(48*a^2*A - 59*A*b^2 - 104*a*b*B)*Cot[c + d*x]^2*Sqr t[a + b*Tan[c + d*x]])/(32*d) - (-(((-105*a^(5/2)*(128*a^4*A - 240*a^2*A*b ^2 - 5*A*b^4 - 320*a^3*b*B + 40*a*b^3*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/ Sqrt[a]])/(32*d) + (I*Sqrt[a - I*b]*(210*a^4*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) + (210*I)*a^4*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))*ArcTanh[ Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b] *(210*a^4*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (210*I)*a^4*(a^3*A - 3 *a*A*b^2 - 3*a^2*b*B + b^3*B))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I *b]])/((-a - I*b)*d))/a) - (105*a^2*(144*a^2*A*b - 5*A*b^3 + 64*a^3*B - 88 *a*b^2*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(32*d))/(2*a))/(3*a))/(4* a)))/7))/5
Time = 3.04 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.04, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.758, Rules used = {3042, 4088, 27, 3042, 4128, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} \left (-b (5 a A-8 b B) \tan ^2(c+d x)-8 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (11 A b+8 a B)\right )dx-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} \left (-b (5 a A-8 b B) \tan ^2(c+d x)-8 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (11 A b+8 a B)\right )dx-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \tan (c+d x)} \left (-b (5 a A-8 b B) \tan (c+d x)^2-8 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (11 A b+8 a B)\right )}{\tan (c+d x)^4}dx-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int -\frac {\cot ^3(c+d x) \left (b \left (40 B a^2+85 A b a-48 b^2 B\right ) \tan ^2(c+d x)+48 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (48 A a^2-104 b B a-59 A b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (-\frac {1}{6} \int \frac {\cot ^3(c+d x) \left (b \left (40 B a^2+85 A b a-48 b^2 B\right ) \tan ^2(c+d x)+48 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (48 A a^2-104 b B a-59 A b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (-\frac {1}{6} \int \frac {b \left (40 B a^2+85 A b a-48 b^2 B\right ) \tan (c+d x)^2+48 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (48 A a^2-104 b B a-59 A b^2\right )}{\tan (c+d x)^3 \sqrt {a+b \tan (c+d x)}}dx-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int -\frac {3 \cot ^2(c+d x) \left (-a b \left (48 A a^2-104 b B a-59 A b^2\right ) \tan ^2(c+d x)-64 a \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)+a \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{2 a}+\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \int \frac {\cot ^2(c+d x) \left (-a b \left (48 A a^2-104 b B a-59 A b^2\right ) \tan ^2(c+d x)-64 a \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)+a \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \int \frac {-a b \left (48 A a^2-104 b B a-59 A b^2\right ) \tan (c+d x)^2-64 a \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)+a \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right )}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \frac {\cot (c+d x) \left (128 A a^5-320 b B a^4-240 A b^2 a^3+40 b^3 B a^2+128 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x) a^2-5 A b^4 a+b \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right ) \tan ^2(c+d x) a\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \frac {\cot (c+d x) \left (128 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x) a^2+b \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right ) \tan ^2(c+d x) a+\left (128 A a^4-320 b B a^3-240 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \frac {128 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x) a^2+b \left (64 B a^3+144 A b a^2-88 b^2 B a-5 A b^3\right ) \tan (c+d x)^2 a+\left (128 A a^4-320 b B a^3-240 A b^2 a^2+40 b^3 B a-5 A b^4\right ) a}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\int \frac {128 \left (a^2 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right )-a^2 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx+a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {128 \int \frac {a^2 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right )-a^2 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {128 \int \frac {a^2 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right )-a^2 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}\right )-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\right )-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (\frac {1}{2} a^2 (a-i b)^3 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^2 (a+i b)^3 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (\frac {1}{2} a^2 (a-i b)^3 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^2 (a+i b)^3 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (\frac {i a^2 (a-i b)^3 (B+i A) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {i a^2 (a+i b)^3 (-B+i A) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (-\frac {i a^2 (a-i b)^3 (B+i A) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^2 (a+i b)^3 (-B+i A) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (\frac {a^2 (a-i b)^3 (B+i A) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {a^2 (a+i b)^3 (-B+i A) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+128 \left (\frac {a^2 (a-i b)^{5/2} (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {a^2 (a+i b)^{5/2} (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+128 \left (\frac {a^2 (a-i b)^{5/2} (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {a^2 (a+i b)^{5/2} (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {2 a \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+128 \left (\frac {a^2 (a-i b)^{5/2} (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {a^2 (a+i b)^{5/2} (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac {1}{8} \left (-\frac {a (8 a B+11 A b) \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {\left (64 a^3 B+144 a^2 A b-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {-\frac {2 \sqrt {a} \left (128 a^4 A-320 a^3 b B-240 a^2 A b^2+40 a b^3 B-5 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+128 \left (\frac {a^2 (a-i b)^{5/2} (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {a^2 (a+i b)^{5/2} (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\right )\) |
-1/4*(a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2))/d + (-1/3*(a*(11*A*b + 8*a*B)*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/d + (((48*a^2*A - 59*A*b ^2 - 104*a*b*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d) - (3*(-1/2* (128*((a^2*(a - I*b)^(5/2)*(I*A + B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d - (a^2*(a + I*b)^(5/2)*(I*A - B)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d) - (2*Sqrt[a]*(128*a^4*A - 240*a^2*A*b^2 - 5*A*b^4 - 320*a^3*b*B + 40*a*b^3* B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d)/a - ((144*a^2*A*b - 5*A*b ^3 + 64*a^3*B - 88*a*b^2*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d))/(4* a))/6)/8
3.4.39.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
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]
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*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2913\) vs. \(2(300)=600\).
Time = 0.30 (sec) , antiderivative size = 2914, normalized size of antiderivative = 8.52
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2914\) |
| default | \(\text {Expression too large to display}\) | \(2914\) |
-55/24/d/b/tan(d*x+c)^4*B*(a+b*tan(d*x+c))^(3/2)*a^2-2*a^(5/2)*A*arctanh(( a+b*tan(d*x+c))^(1/2)/a^(1/2))/d-2/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta n((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1 /2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)*a+1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*( a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^( 1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x +c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^( 1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+2/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*ar ctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2) ^(1/2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)*a-3/d*b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2 +b^2)^(1/2)-2*a)^(1/2))*A*a-3/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2 *(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2 *a)^(1/2))*B*a^2-1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a) ^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3 +1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^ (1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+5/64/d*b^4/a^(3 /2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*A+55/192/d/tan(d*x+c)^4*A*(a+b *tan(d*x+c))^(3/2)*a-5/64/d/tan(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*A*a^2-5/64 /d/tan(d*x+c)^4/a*(a+b*tan(d*x+c))^(7/2)*A-1/4/d*b^2*ln(b*tan(d*x+c)+a+...
Leaf count of result is larger than twice the leaf count of optimal. 5045 vs. \(2 (294) = 588\).
Time = 55.78 (sec) , antiderivative size = 10107, normalized size of antiderivative = 29.55 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
Timed out. \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Time = 12.43 (sec) , antiderivative size = 36736, normalized size of antiderivative = 107.42 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
atan(((((((10240*A*a*b^14*d^4 + 436224*A*a^3*b^12*d^4 + 229376*A*a^5*b^10* d^4 - 196608*A*a^7*b^8*d^4 - 81920*B*a^2*b^13*d^4 + 442368*B*a^4*b^11*d^4 + 524288*B*a^6*b^9*d^4)/(512*a^2*d^5) - ((131072*a^2*b^10*d^4 + 196608*a^4 *b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 8 0*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*( A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^1 0 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^ 4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a ^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1 /2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/(256*a^2*d^4))*((((8*B^2*a^5*d^2 - 8*A^ 2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40* A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2) ^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a ^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2 *a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*...