Integrand size = 25, antiderivative size = 358 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {(i a-b)^{5/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 b \left (231 a^2-5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2-25 b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {38 a b \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d} \] Output:
(I*a-b)^(5/2)*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)^(5/2)*arctanh((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*(315*a^4-483*a^2*b^2-10*b^4)*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1 /2)/a^2/d+2/315*b*(231*a^2-5*b^2)*cot(d*x+c)^(3/2)*(a+b*tan(d*x+c))^(1/2)/ a/d+2/105*(21*a^2-25*b^2)*cot(d*x+c)^(5/2)*(a+b*tan(d*x+c))^(1/2)/d-38/63* a*b*cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^(1/2)/d-2/9*a^2*cot(d*x+c)^(9/2)*(a+ b*tan(d*x+c))^(1/2)/d
Time = 3.67 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.92 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {\cot ^{\frac {9}{2}}(c+d x) \left (-315 \sqrt [4]{-1} a^2 (-a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {9}{2}}(c+d x)-315 \sqrt [4]{-1} a^2 (a+i b)^{5/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \tan ^{\frac {9}{2}}(c+d x)+\frac {1}{4} \sec ^4(c+d x) \left (-987 a^4+1374 a^2 b^2+30 b^4+4 \left (280 a^4-483 a^2 b^2-10 b^4\right ) \cos (2 (c+d x))+\left (-413 a^4+558 a^2 b^2+10 b^4\right ) \cos (4 (c+d x))+272 a^3 b \sin (2 (c+d x))-10 a b^3 \sin (2 (c+d x))-326 a^3 b \sin (4 (c+d x))+5 a b^3 \sin (4 (c+d x))\right ) \sqrt {a+b \tan (c+d x)}\right )}{315 a^2 d} \] Input:
Integrate[Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2),x]
Output:
(Cot[c + d*x]^(9/2)*(-315*(-1)^(1/4)*a^2*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/ 4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d* x]^(9/2) - 315*(-1)^(1/4)*a^2*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(9/2) + (S ec[c + d*x]^4*(-987*a^4 + 1374*a^2*b^2 + 30*b^4 + 4*(280*a^4 - 483*a^2*b^2 - 10*b^4)*Cos[2*(c + d*x)] + (-413*a^4 + 558*a^2*b^2 + 10*b^4)*Cos[4*(c + d*x)] + 272*a^3*b*Sin[2*(c + d*x)] - 10*a*b^3*Sin[2*(c + d*x)] - 326*a^3* b*Sin[4*(c + d*x)] + 5*a*b^3*Sin[4*(c + d*x)])*Sqrt[a + b*Tan[c + d*x]])/4 ))/(315*a^2*d)
Time = 2.82 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.05, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 4729, 3042, 4048, 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.
| \(\displaystyle \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{11/2} (a+b \tan (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(a+b \tan (c+d x))^{5/2}}{\tan (c+d x)^{11/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2}{9} \int \frac {19 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (8 a^2-9 b^2\right ) \tan ^2(c+d x)}{2 \tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \int \frac {19 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (8 a^2-9 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \int \frac {19 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (8 a^2-9 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^{9/2} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {2 \int \frac {3 \left (38 b^2 \tan ^2(c+d x) a^2+\left (21 a^2-25 b^2\right ) a^2+21 b \left (3 a^2-b^2\right ) \tan (c+d x) a\right )}{2 \tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \int \frac {38 b^2 \tan ^2(c+d x) a^2+\left (21 a^2-25 b^2\right ) a^2+21 b \left (3 a^2-b^2\right ) \tan (c+d x) a}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \int \frac {38 b^2 \tan (c+d x)^2 a^2+\left (21 a^2-25 b^2\right ) a^2+21 b \left (3 a^2-b^2\right ) \tan (c+d x) a}{\tan (c+d x)^{7/2} \sqrt {a+b \tan (c+d x)}}dx}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (-\frac {2 \int -\frac {-105 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-4 b \left (21 a^2-25 b^2\right ) \tan ^2(c+d x) a^2+b \left (231 a^2-5 b^2\right ) a^2}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {\int \frac {-105 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-4 b \left (21 a^2-25 b^2\right ) \tan ^2(c+d x) a^2+b \left (231 a^2-5 b^2\right ) a^2}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {\int \frac {-105 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-4 b \left (21 a^2-25 b^2\right ) \tan (c+d x)^2 a^2+b \left (231 a^2-5 b^2\right ) a^2}{\tan (c+d x)^{5/2} \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {2 \int \frac {315 b \left (3 a^2-b^2\right ) \tan (c+d x) a^3+2 b^2 \left (231 a^2-5 b^2\right ) \tan ^2(c+d x) a^2+\left (315 a^4-483 b^2 a^2-10 b^4\right ) a^2}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\int \frac {315 b \left (3 a^2-b^2\right ) \tan (c+d x) a^3+2 b^2 \left (231 a^2-5 b^2\right ) \tan ^2(c+d x) a^2+\left (315 a^4-483 b^2 a^2-10 b^4\right ) a^2}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\int \frac {315 b \left (3 a^2-b^2\right ) \tan (c+d x) a^3+2 b^2 \left (231 a^2-5 b^2\right ) \tan (c+d x)^2 a^2+\left (315 a^4-483 b^2 a^2-10 b^4\right ) a^2}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {-\frac {2 \int -\frac {315 \left (a^4 b \left (3 a^2-b^2\right )-a^5 \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\frac {315 \int \frac {a^4 b \left (3 a^2-b^2\right )-a^5 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{9} \left (-\frac {3 \left (\frac {-\frac {\frac {315 \int \frac {a^4 b \left (3 a^2-b^2\right )-a^5 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )}{7 a}-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {1}{2} a^4 (-b+i a)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (b+i a)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {1}{2} a^4 (-b+i a)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (b+i a)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (-b+i a)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}-\frac {a^4 (b+i a)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (-b+i a)^3 \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}-\frac {a^4 (b+i a)^3 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (-b+i a)^{5/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a^4 (b+i a)^3 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 a^2 \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (-\frac {38 a b \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {3 \left (-\frac {2 a \left (21 a^2-25 b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {-\frac {2 a b \left (231 a^2-5 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {-\frac {2 a \left (315 a^4-483 a^2 b^2-10 b^4\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {315 \left (\frac {a^4 (-b+i a)^{5/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a^4 (b+i a)^{5/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\right )}{a}}{3 a}}{5 a}\right )}{7 a}\right )\right )\) |
Input:
Int[Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2),x]
Output:
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*a^2*Sqrt[a + b*Tan[c + d*x]])/( 9*d*Tan[c + d*x]^(9/2)) + ((-38*a*b*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (3*((-2*a*(21*a^2 - 25*b^2)*Sqrt[a + b*Tan[c + d*x]])/(5*d* Tan[c + d*x]^(5/2)) + ((-2*a*b*(231*a^2 - 5*b^2)*Sqrt[a + b*Tan[c + d*x]]) /(3*d*Tan[c + d*x]^(3/2)) - ((315*((a^4*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (a^4*(I*a + b)^(5/2 )*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) )/a - (2*a*(315*a^4 - 483*a^2*b^2 - 10*b^4)*Sqrt[a + b*Tan[c + d*x]])/(d*S qrt[Tan[c + d*x]]))/(3*a))/(5*a)))/(7*a))/9)
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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (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]
Leaf count of result is larger than twice the leaf count of optimal. \(2833\) vs. \(2(300)=600\).
Time = 5.08 (sec) , antiderivative size = 2834, normalized size of antiderivative = 7.92
Input:
int(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/1260/d*2^(1/2)/a^2/(-b+(a^2+b^2)^(1/2))^(1/2)*((630*cos(d*x+c)-630)*sin( d*x+c)^4*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1 /2)*a^2*b*ln(-(a*cot(d*x+c)*cos(d*x+c)+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+ c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c )-2*a*cot(d*x+c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a+ a*csc(d*x+c)-2*(a^2+b^2)^(1/2)-2*b)/(-1+cos(d*x+c)))+(315*cos(d*x+c)-315)* sin(d*x+c)^4*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a^4*ln(- (a*cot(d*x+c)*cos(d*x+c)+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c) /(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*a*cot(d*x+ c)+2*(a^2+b^2)^(1/2)*cos(d*x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a+a*csc(d*x+c)-2 *(a^2+b^2)^(1/2)-2*b)/(-1+cos(d*x+c)))+(-945*cos(d*x+c)+945)*sin(d*x+c)^4* (b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a^2*b^2*ln(-(a*cot(d* x+c)*cos(d*x+c)+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos(d*x +c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*a*cot(d*x+c)+2*(a^2 +b^2)^(1/2)*cos(d*x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a+a*csc(d*x+c)-2*(a^2+b^2 )^(1/2)-2*b)/(-1+cos(d*x+c)))+(-630*cos(d*x+c)+630)*sin(d*x+c)^4*(a^2+b^2) ^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*a^2*b*ln((a*co t(d*x+c)*cos(d*x+c)-2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+c)/(cos (d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*a*cot(d*x+c)+2* (a^2+b^2)^(1/2)*cos(d*x+c)+2*cos(d*x+c)*b-sin(d*x+c)*a+a*csc(d*x+c)-2*(...
Leaf count of result is larger than twice the leaf count of optimal. 5248 vs. \(2 (296) = 592\).
Time = 0.79 (sec) , antiderivative size = 5248, normalized size of antiderivative = 14.66 \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(11/2)*(a+b*tan(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((b*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^(11/2), x)
Exception generated. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,14,5]%%%}+%%%{6,[0,12,5]%%%}+%%%{15,[0,10,5]%%%}+ %%%{20,[0
Timed out. \[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cot(c + d*x)^(11/2)*(a + b*tan(c + d*x))^(5/2),x)
Output:
int(cot(c + d*x)^(11/2)*(a + b*tan(c + d*x))^(5/2), x)
\[ \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int \cot \left (d x +c \right )^{\frac {11}{2}} \left (a +\tan \left (d x +c \right ) b \right )^{\frac {5}{2}}d x \] Input:
int(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x)
Output:
int(cot(d*x+c)^(11/2)*(a+b*tan(d*x+c))^(5/2),x)