Integrand size = 25, antiderivative size = 450 \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\frac {a^{5/2} \left (15 a^4+46 a^2 b^2+63 b^4\right ) e^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (15 a^4+31 a^2 b^2+8 b^4\right ) e^4 \sqrt {e \cot (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {a^2 \left (5 a^2+13 b^2\right ) e^3 (e \cot (c+d x))^{3/2}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))} \] Output:
1/4*a^(5/2)*(15*a^4+46*a^2*b^2+63*b^4)*e^(9/2)*arctan(b^(1/2)*(e*cot(d*x+c ))^(1/2)/a^(1/2)/e^(1/2))/b^(7/2)/(a^2+b^2)^3/d+1/2*(a-b)*(a^2+4*a*b+b^2)* e^(9/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/(a^2+b^2)^3 /d-1/2*(a-b)*(a^2+4*a*b+b^2)*e^(9/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2) /e^(1/2))*2^(1/2)/(a^2+b^2)^3/d+1/2*(a+b)*(a^2-4*a*b+b^2)*e^(9/2)*arctanh( 2^(1/2)*(e*cot(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*cot(d*x+c)))*2^(1/2)/(a^2+b^ 2)^3/d-1/4*(15*a^4+31*a^2*b^2+8*b^4)*e^4*(e*cot(d*x+c))^(1/2)/b^3/(a^2+b^2 )^2/d+1/2*a^2*e^2*(e*cot(d*x+c))^(5/2)/b/(a^2+b^2)/d/(a+b*cot(d*x+c))^2+1/ 4*a^2*(5*a^2+13*b^2)*e^3*(e*cot(d*x+c))^(3/2)/b^2/(a^2+b^2)^2/d/(a+b*cot(d *x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.18 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.39 \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=-\frac {(e \cot (c+d x))^{9/2} \left (-\frac {2 a^{9/2} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right )^3}+\frac {2 a^4 \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{b^3 \left (a^2+b^2\right )^3}-\frac {2 a^3 \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^3}+\frac {2 a^2 \left (3 a^2-b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 b \left (a^2+b^2\right )^3}-\frac {2 a \left (3 a^2-b^2\right ) \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {9}{2}}(c+d x)}{9 \left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \left (7 \cot ^{\frac {3}{2}}(c+d x)-3 \cot ^{\frac {7}{2}}(c+d x)-7 \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{21 \left (a^2+b^2\right )^3}+\frac {4 b^2 \cot ^{\frac {11}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {11}{2},\frac {13}{2},-\frac {b \cot (c+d x)}{a}\right )}{11 a \left (a^2+b^2\right )^2}+\frac {2 b^2 \cot ^{\frac {11}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (3,\frac {11}{2},\frac {13}{2},-\frac {b \cot (c+d x)}{a}\right )}{11 a^3 \left (a^2+b^2\right )}-\frac {b \left (3 a^2-b^2\right ) \left (90 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-90 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+360 \sqrt {\cot (c+d x)}-72 \cot ^{\frac {5}{2}}(c+d x)+40 \cot ^{\frac {9}{2}}(c+d x)+45 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-45 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{180 \left (a^2+b^2\right )^3}\right )}{d \cot ^{\frac {9}{2}}(c+d x)} \] Input:
Integrate[(e*Cot[c + d*x])^(9/2)/(a + b*Cot[c + d*x])^3,x]
Output:
-(((e*Cot[c + d*x])^(9/2)*((-2*a^(9/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[ Cot[c + d*x]])/Sqrt[a]])/(b^(7/2)*(a^2 + b^2)^3) + (2*a^4*(3*a^2 - b^2)*Sq rt[Cot[c + d*x]])/(b^3*(a^2 + b^2)^3) - (2*a^3*(3*a^2 - b^2)*Cot[c + d*x]^ (3/2))/(3*b^2*(a^2 + b^2)^3) + (2*a^2*(3*a^2 - b^2)*Cot[c + d*x]^(5/2))/(5 *b*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^ 3) + (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(9/2))/(9*(a^2 + b^2)^3) - (2*a*(a^2 - 3*b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/ 2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((b*Cot[c + d *x])/a)])/(11*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(11/2)*Hypergeometric 2F1[3, 11/2, 13/2, -((b*Cot[c + d*x])/a)])/(11*a^3*(a^2 + b^2)) - (b*(3*a^ 2 - b^2)*(90*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 90*Sqrt[2]*A rcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 360*Sqrt[Cot[c + d*x]] - 72*Cot[c + d*x]^(5/2) + 40*Cot[c + d*x]^(9/2) + 45*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot [c + d*x]] + Cot[c + d*x]] - 45*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(180*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(9/2)))
Time = 2.76 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.10, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.080, Rules used = {3042, 4048, 27, 3042, 4128, 27, 3042, 4130, 27, 3042, 4136, 27, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
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.
| \(\displaystyle \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {\int -\frac {(e \cot (c+d x))^{3/2} \left (5 a^2 e^3+\left (5 a^2+4 b^2\right ) \cot ^2(c+d x) e^3-4 a b \cot (c+d x) e^3\right )}{2 (a+b \cot (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(e \cot (c+d x))^{3/2} \left (5 a^2 e^3+\left (5 a^2+4 b^2\right ) \cot ^2(c+d x) e^3-4 a b \cot (c+d x) e^3\right )}{(a+b \cot (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 a^2 e^3+\left (5 a^2+4 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^3+4 a b \tan \left (c+d x+\frac {\pi }{2}\right ) e^3\right )}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac {\int -\frac {\sqrt {e \cot (c+d x)} \left (\left (15 a^4+31 b^2 a^2+8 b^4\right ) \cot ^2(c+d x) e^4+3 a^2 \left (5 a^2+13 b^2\right ) e^4-16 a b^3 \cot (c+d x) e^4\right )}{2 (a+b \cot (c+d x))}dx}{b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {e \cot (c+d x)} \left (\left (15 a^4+31 b^2 a^2+8 b^4\right ) \cot ^2(c+d x) e^4+3 a^2 \left (5 a^2+13 b^2\right ) e^4-16 a b^3 \cot (c+d x) e^4\right )}{a+b \cot (c+d x)}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 a^4+31 b^2 a^2+8 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^4+3 a^2 \left (5 a^2+13 b^2\right ) e^4+16 a b^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^4\right )}{a-b \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {a \left (15 a^4+31 b^2 a^2+24 b^4\right ) \cot ^2(c+d x) e^5+a \left (15 a^4+31 b^2 a^2+8 b^4\right ) e^5-8 b^3 \left (a^2-b^2\right ) \cot (c+d x) e^5}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (15 a^4+31 b^2 a^2+24 b^4\right ) \cot ^2(c+d x) e^5+a \left (15 a^4+31 b^2 a^2+8 b^4\right ) e^5-8 b^3 \left (a^2-b^2\right ) \cot (c+d x) e^5}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (15 a^4+31 b^2 a^2+24 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^5+a \left (15 a^4+31 b^2 a^2+8 b^4\right ) e^5+8 b^3 \left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^5}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int -\frac {8 \left (b^4 \left (3 a^2-b^2\right ) e^5+a b^3 \left (a^2-3 b^2\right ) \cot (c+d x) e^5\right )}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {b^4 \left (3 a^2-b^2\right ) e^5+a b^3 \left (a^2-3 b^2\right ) \cot (c+d x) e^5}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {b^4 \left (3 a^2-b^2\right ) e^5-a b^3 \left (a^2-3 b^2\right ) e^5 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {b^3 e^5 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 \int \frac {b^3 e^5 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \int \frac {b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 e^5 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 a^3 e^4 \left (15 a^4+46 a^2 b^2+63 b^4\right ) \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a^2 e^2 (e \cot (c+d x))^{5/2}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {\frac {a^2 e^3 \left (5 a^2+13 b^2\right ) (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}+\frac {-\frac {2 e^4 \left (15 a^4+31 a^2 b^2+8 b^4\right ) \sqrt {e \cot (c+d x)}}{b d}-\frac {\frac {16 b^3 e^5 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 a^{5/2} e^{9/2} \left (15 a^4+46 a^2 b^2+63 b^4\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}}{4 b \left (a^2+b^2\right )}\) |
Input:
Int[(e*Cot[c + d*x])^(9/2)/(a + b*Cot[c + d*x])^3,x]
Output:
(a^2*e^2*(e*Cot[c + d*x])^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2 ) + ((a^2*(5*a^2 + 13*b^2)*e^3*(e*Cot[c + d*x])^(3/2))/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((-2*(15*a^4 + 31*a^2*b^2 + 8*b^4)*e^4*Sqrt[e*Cot[c + d*x]])/(b*d) - ((2*a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*e^(9/2)*ArcTan [(Sqrt[b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d) + (16* b^3*e^5*(((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 - ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2 *Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*S qrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/( 2*Sqrt[2]*Sqrt[e])))/2))/((a^2 + b^2)*d))/b)/(2*b*(a^2 + b^2)))/(4*b*(a^2 + b^2))
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, 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((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[((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_.)*((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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*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[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !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]
Time = 1.45 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(-\frac {2 e^{4} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b^{3}}+\frac {e \left (\frac {\left (3 a^{2} b e -b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{3} e \left (\frac {\left (-\frac {9}{8} a^{4} b -\frac {13}{4} a^{2} b^{3}-\frac {17}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {a e \left (7 a^{4}+22 a^{2} b^{2}+15 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (15 a^{4}+46 a^{2} b^{2}+63 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(471\) |
| default | \(-\frac {2 e^{4} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b^{3}}+\frac {e \left (\frac {\left (3 a^{2} b e -b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{3} e \left (\frac {\left (-\frac {9}{8} a^{4} b -\frac {13}{4} a^{2} b^{3}-\frac {17}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {a e \left (7 a^{4}+22 a^{2} b^{2}+15 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (15 a^{4}+46 a^{2} b^{2}+63 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(471\) |
Input:
int((e*cot(d*x+c))^(9/2)/(a+b*cot(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/d*e^4*(1/b^3*(e*cot(d*x+c))^(1/2)+e/(a^2+b^2)^3*(1/8*(3*a^2*b*e-b^3*e)* (e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2) *2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/ 2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*ar ctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/8*(a^3-3*a*b^2)/(e^2) ^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+ (e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^ (1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^( 1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-a^3*e/b^3/(a^2+b^2)^3*(((-9/8*a ^4*b-13/4*a^2*b^3-17/8*b^5)*(e*cot(d*x+c))^(3/2)-1/8*a*e*(7*a^4+22*a^2*b^2 +15*b^4)*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)*b+a*e)^2+1/8*(15*a^4+46*a^2*b ^2+63*b^4)/(a*e*b)^(1/2)*arctan(b*(e*cot(d*x+c))^(1/2)/(a*e*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 4519 vs. \(2 (392) = 784\).
Time = 1.77 (sec) , antiderivative size = 9101, normalized size of antiderivative = 20.22 \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate((e*cot(d*x+c))^(9/2)/(a+b*cot(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((e*cot(d*x+c))**(9/2)/(a+b*cot(d*x+c))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*cot(d*x+c))^(9/2)/(a+b*cot(d*x+c))^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((e*cot(d*x+c))^(9/2)/(a+b*cot(d*x+c))^3,x, algorithm="giac")
Output:
integrate((e*cot(d*x + c))^(9/2)/(b*cot(d*x + c) + a)^3, x)
Time = 19.19 (sec) , antiderivative size = 20651, normalized size of antiderivative = 45.89 \[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int((e*cot(c + d*x))^(9/2)/(a + b*cot(c + d*x))^3,x)
Output:
atan(((((((128*a*b^26*d^4*e^15 + 3648*a^3*b^24*d^4*e^15 + 25536*a^5*b^22*d ^4*e^15 + 88320*a^7*b^20*d^4*e^15 + 182784*a^9*b^18*d^4*e^15 + 244608*a^11 *b^16*d^4*e^15 + 217728*a^13*b^14*d^4*e^15 + 128256*a^15*b^12*d^4*e^15 + 4 8000*a^17*b^10*d^4*e^15 + 10304*a^19*b^8*d^4*e^15 + 960*a^21*b^6*d^4*e^15) /(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b ^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d ^5) + ((e*cot(c + d*x))^(1/2)*(-(e^9*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2 *6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^ (1/2)*(512*b^30*d^4*e^10 + 4608*a^2*b^28*d^4*e^10 + 17920*a^4*b^26*d^4*e^1 0 + 38400*a^6*b^24*d^4*e^10 + 46080*a^8*b^22*d^4*e^10 + 21504*a^10*b^20*d^ 4*e^10 - 21504*a^12*b^18*d^4*e^10 - 46080*a^14*b^16*d^4*e^10 - 38400*a^16* b^14*d^4*e^10 - 17920*a^18*b^12*d^4*e^10 - 4608*a^20*b^10*d^4*e^10 - 512*a ^22*b^8*d^4*e^10))/(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b ^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^ 7*d^4 + a^16*b^5*d^4))*(-(e^9*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a ^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(1800*a^23*b*d^2*e^19 - 1472*a*b^23*d^2*e^19 - 10 24*a^3*b^21*d^2*e^19 + 8448*a^5*b^19*d^2*e^19 + 46088*a^7*b^17*d^2*e^19 + 177344*a^9*b^15*d^2*e^19 + 402912*a^11*b^13*d^2*e^19 + 541632*a^13*b^11*d^ 2*e^19 + 455472*a^15*b^9*d^2*e^19 + 248064*a^17*b^7*d^2*e^19 + 87008*a^...
\[ \int \frac {(e \cot (c+d x))^{9/2}}{(a+b \cot (c+d x))^3} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{4}}{\cot \left (d x +c \right )^{3} b^{3}+3 \cot \left (d x +c \right )^{2} a \,b^{2}+3 \cot \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) e^{4} \] Input:
int((e*cot(d*x+c))^(9/2)/(a+b*cot(d*x+c))^3,x)
Output:
sqrt(e)*int((sqrt(cot(c + d*x))*cot(c + d*x)**4)/(cot(c + d*x)**3*b**3 + 3 *cot(c + d*x)**2*a*b**2 + 3*cot(c + d*x)*a**2*b + a**3),x)*e**4