Integrand size = 35, antiderivative size = 422 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {(i a-b)^{3/2} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d} \]
(I*a-b)^(3/2)*(I*A-B)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c ))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-(I*a+b)^(3/2)*(I*A+B)*arctan h((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)* tan(d*x+c)^(1/2)/d+2/315*(126*A*a^2*b+4*A*b^3+105*B*a^3-9*B*a*b^2)*cot(d*x +c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/a^2/d+2/105*(21*A*a^2-A*b^2-24*B*a*b)*cot (d*x+c)^(5/2)*(a+b*tan(d*x+c))^(1/2)/a/d-2/63*(10*A*b+9*B*a)*cot(d*x+c)^(7 /2)*(a+b*tan(d*x+c))^(1/2)/d-2/9*a*A*cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^(1/ 2)/d-2/315*(315*A*a^4-63*A*a^2*b^2+8*A*b^4-420*B*a^3*b-18*B*a*b^3)*cot(d*x +c)^(1/2)*(a+b*tan(d*x+c))^(1/2)/a^3/d
Time = 6.73 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.17 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {b B \sqrt {a+b \tan (c+d x)}}{4 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{4} \left (-\frac {(8 a A-9 b B) \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (-\frac {4 a (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \left (-\frac {6 a \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (\frac {a \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (-\frac {945 a^4 \left (\sqrt [4]{-1} (-a+i b)^{3/2} (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-\sqrt [4]{-1} (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{4 d}+\frac {3 a \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{2 d \sqrt {\tan (c+d x)}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )\right ) \]
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-1/4*(b*B*Sqrt[a + b*Tan[c + d*x]]) /(d*Tan[c + d*x]^(9/2)) + (-1/9*((8*a*A - 9*b*B)*Sqrt[a + b*Tan[c + d*x]]) /(d*Tan[c + d*x]^(9/2)) + (2*((-4*a*(10*A*b + 9*a*B)*Sqrt[a + b*Tan[c + d* x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*((-6*a*(21*a^2*A - A*b^2 - 24*a*b*B)*Sq rt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*((a*(126*a^2*A*b + 4 *A*b^3 + 105*a^3*B - 9*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Tan[c + d*x]^ (3/2)) - (2*((-945*a^4*((-1)^(1/4)*(-a + I*b)^(3/2)*(A - I*B)*ArcTan[((-1) ^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1) ^(1/4)*(a + I*b)^(3/2)*(A + I*B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan [c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/(4*d) + (3*a*(315*a^4*A - 63*a^2*A *b^2 + 8*A*b^4 - 420*a^3*b*B - 18*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(2*d* Sqrt[Tan[c + d*x]])))/(3*a)))/(5*a)))/(7*a)))/(9*a))/4)
Time = 3.20 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.07, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 4729, 3042, 4088, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 216, 219}
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.
| \(\displaystyle \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{11/2} (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan (c+d x)^{11/2}}dx\) |
| \(\Big \downarrow \) 4088 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2}{9} \int \frac {-b (8 a A-9 b B) \tan ^2(c+d x)-9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (10 A b+9 a B)}{2 \tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \int \frac {-b (8 a A-9 b B) \tan ^2(c+d x)-9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (10 A b+9 a B)}{\tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \int \frac {-b (8 a A-9 b B) \tan (c+d x)^2-9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (10 A b+9 a B)}{\tan (c+d x)^{9/2} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {2 \int \frac {3 \left (2 a b (10 A b+9 a B) \tan ^2(c+d x)+21 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (21 A a^2-24 b B a-A b^2\right )\right )}{2 \tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \int \frac {2 a b (10 A b+9 a B) \tan ^2(c+d x)+21 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (21 A a^2-24 b B a-A b^2\right )}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \int \frac {2 a b (10 A b+9 a B) \tan (c+d x)^2+21 a \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (21 A a^2-24 b B a-A b^2\right )}{\tan (c+d x)^{7/2} \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (-\frac {2 \int -\frac {-105 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-4 b \left (21 A a^2-24 b B a-A b^2\right ) \tan ^2(c+d x) a+\left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) a}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {\int \frac {-105 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-4 b \left (21 A a^2-24 b B a-A b^2\right ) \tan ^2(c+d x) a+\left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) a}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {\int \frac {-105 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x) a^2-4 b \left (21 A a^2-24 b B a-A b^2\right ) \tan (c+d x)^2 a+\left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) a}{\tan (c+d x)^{5/2} \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {2 \int \frac {315 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^3+2 b \left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) \tan ^2(c+d x) a+\left (315 A a^4-420 b B a^3-63 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\int \frac {315 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^3+2 b \left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) \tan ^2(c+d x) a+\left (315 A a^4-420 b B a^3-63 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\int \frac {315 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x) a^3+2 b \left (105 B a^3+126 A b a^2-9 b^2 B a+4 A b^3\right ) \tan (c+d x)^2 a+\left (315 A a^4-420 b B a^3-63 A b^2 a^2-18 b^3 B a+8 A b^4\right ) a}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {-\frac {2 \int -\frac {315 \left (a^4 \left (B a^2+2 A b a-b^2 B\right )-a^4 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\frac {315 \int \frac {a^4 \left (B a^2+2 A b a-b^2 B\right )-a^4 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\frac {315 \int \frac {a^4 \left (B a^2+2 A b a-b^2 B\right )-a^4 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {1}{2} a^4 (a-i b)^2 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (a+i b)^2 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {1}{2} a^4 (a-i b)^2 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (a+i b)^2 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (a-i b)^2 (B+i A) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}-\frac {a^4 (a+i b)^2 (-B+i A) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (a-i b)^2 (B+i A) \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {a^4 (a+i b)^2 (-B+i A) \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (a-i b)^2 (B+i A) \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {a^4 (a+i b)^2 (-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (a-i b)^2 (B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}-\frac {a^4 (a+i b)^2 (-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*a*A*Sqrt[a + b*Tan[c + d*x]])/( 9*d*Tan[c + d*x]^(9/2)) + ((-2*(10*A*b + 9*a*B)*Sqrt[a + b*Tan[c + d*x]])/ (7*d*Tan[c + d*x]^(7/2)) - (3*((-2*(21*a^2*A - A*b^2 - 24*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) + ((-2*(126*a^2*A*b + 4*A*b^3 + 105*a^3*B - 9*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - ((315*(-((a^4*(a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d *x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + (a^4*(a - I*b)^2*(I* A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]] ])/(Sqrt[I*a + b]*d)))/a - (2*(315*a^4*A - 63*a^2*A*b^2 + 8*A*b^4 - 420*a^ 3*b*B - 18*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a ))/(5*a)))/(7*a))/9)
3.7.19.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 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[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
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[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.38 (sec) , antiderivative size = 2403002, normalized size of antiderivative = 5694.32
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 12342 vs. \(2 (356) = 712\).
Time = 2.36 (sec) , antiderivative size = 12342, normalized size of antiderivative = 29.25 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
\[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]