Integrand size = 23, antiderivative size = 338 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{7/2} \left (9 a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a^3 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))} \] Output:
1/2*(a^2-2*a*b-b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b^2)^ 2/d+1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b^ 2)^2/d-b^(7/2)*(9*a^2+5*b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(7 /2)/(a^2+b^2)^2/d-1/2*(a^2+2*a*b-b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+ cot(d*x+c)))*2^(1/2)/(a^2+b^2)^2/d+b*(4*a^2+5*b^2)*cot(d*x+c)^(1/2)/a^3/(a ^2+b^2)/d-1/3*(2*a^2+5*b^2)*cot(d*x+c)^(3/2)/a^2/(a^2+b^2)/d+b^2*cot(d*x+c )^(5/2)/a/(a^2+b^2)/d/(b+a*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.63 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.37 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {1540 a^{7/2} b^2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-420 a^{11/2} \left (a^2+b^2\right ) \cot ^{\frac {11}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {11}{2},\frac {13}{2},-\frac {a \cot (c+d x)}{b}\right )+11 b^2 \left (210 \sqrt {2} a^{9/2} b \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-210 \sqrt {2} a^{9/2} b \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+840 b^{11/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )+840 a^{9/2} b \sqrt {\cot (c+d x)}-840 \sqrt {a} b^5 \sqrt {\cot (c+d x)}-140 a^{11/2} \cot ^{\frac {3}{2}}(c+d x)+140 a^{7/2} b^2 \cot ^{\frac {3}{2}}(c+d x)+280 a^{3/2} b^4 \cot ^{\frac {3}{2}}(c+d x)-168 a^{9/2} b \cot ^{\frac {5}{2}}(c+d x)-168 a^{5/2} b^3 \cot ^{\frac {5}{2}}(c+d x)+60 a^{11/2} \cot ^{\frac {7}{2}}(c+d x)+60 a^{7/2} b^2 \cot ^{\frac {7}{2}}(c+d x)+105 \sqrt {2} a^{9/2} b \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-105 \sqrt {2} a^{9/2} b \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{2310 a^{7/2} b^2 \left (a^2+b^2\right )^2 d} \] Input:
Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2,x]
Output:
(1540*a^(7/2)*b^2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 420*a^(11/2)*(a^2 + b^2)*Cot[c + d*x]^(11/2)*Hype rgeometric2F1[2, 11/2, 13/2, -((a*Cot[c + d*x])/b)] + 11*b^2*(210*Sqrt[2]* a^(9/2)*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 210*Sqrt[2]*a^(9/2)*b*A rcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 840*b^(11/2)*ArcTan[(Sqrt[a]*Sqrt[ Cot[c + d*x]])/Sqrt[b]] + 840*a^(9/2)*b*Sqrt[Cot[c + d*x]] - 840*Sqrt[a]*b ^5*Sqrt[Cot[c + d*x]] - 140*a^(11/2)*Cot[c + d*x]^(3/2) + 140*a^(7/2)*b^2* Cot[c + d*x]^(3/2) + 280*a^(3/2)*b^4*Cot[c + d*x]^(3/2) - 168*a^(9/2)*b*Co t[c + d*x]^(5/2) - 168*a^(5/2)*b^3*Cot[c + d*x]^(5/2) + 60*a^(11/2)*Cot[c + d*x]^(7/2) + 60*a^(7/2)*b^2*Cot[c + d*x]^(7/2) + 105*Sqrt[2]*a^(9/2)*b*L og[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 105*Sqrt[2]*a^(9/2)*b* Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2310*a^(7/2)*b^2*(a^ 2 + b^2)^2*d)
Time = 2.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.07, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.261, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4130, 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.
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{5/2}}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {9}{2}}(c+d x)}{(a \cot (c+d x)+b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {\cot ^{\frac {3}{2}}(c+d x) \left (5 b^2-2 a \cot (c+d x) b+\left (2 a^2+5 b^2\right ) \cot ^2(c+d x)\right )}{2 (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (5 b^2-2 a \cot (c+d x) b+\left (2 a^2+5 b^2\right ) \cot ^2(c+d x)\right )}{b+a \cot (c+d x)}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 b^2+2 a \tan \left (c+d x+\frac {\pi }{2}\right ) b+\left (2 a^2+5 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )}{b-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 \sqrt {\cot (c+d x)} \left (2 \cot (c+d x) a^3+b \left (4 a^2+5 b^2\right ) \cot ^2(c+d x)+b \left (2 a^2+5 b^2\right )\right )}{2 (b+a \cot (c+d x))}dx}{3 a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {\cot (c+d x)} \left (2 \cot (c+d x) a^3+b \left (4 a^2+5 b^2\right ) \cot ^2(c+d x)+b \left (2 a^2+5 b^2\right )\right )}{b+a \cot (c+d x)}dx}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+b \left (4 a^2+5 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 a^2+5 b^2\right )\right )}{b-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int \frac {2 b \cot (c+d x) a^3-\left (2 a^4-4 b^2 a^2-5 b^4\right ) \cot ^2(c+d x)+b^2 \left (4 a^2+5 b^2\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {2 b \cot (c+d x) a^3-\left (2 a^4-4 b^2 a^2-5 b^4\right ) \cot ^2(c+d x)+b^2 \left (4 a^2+5 b^2\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-2 b \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (-2 a^4+4 b^2 a^2+5 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b^2 \left (4 a^2+5 b^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {2 \left (2 a^4 b-a^3 \left (a^2-b^2\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {2 \int \frac {2 a^4 b-a^3 \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {2 \int \frac {2 b a^4+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 \int -\frac {a^3 \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int \frac {a^3 \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \int \frac {2 a b-\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b^4 \left (9 a^2+5 b^2\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {2 b^4 \left (9 a^2+5 b^2\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}}{a}-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b^2 \cot ^{\frac {5}{2}}(c+d x)}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {-\frac {2 \left (2 a^2+5 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a d}-\frac {-\frac {2 b \left (4 a^2+5 b^2\right ) \sqrt {\cot (c+d x)}}{a d}-\frac {\frac {2 b^{7/2} \left (9 a^2+5 b^2\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {4 a^3 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}}{a}}{2 a \left (a^2+b^2\right )}\) |
Input:
Int[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2,x]
Output:
(b^2*Cot[c + d*x]^(5/2))/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((-2*(2* a^2 + 5*b^2)*Cot[c + d*x]^(3/2))/(3*a*d) - ((-2*b*(4*a^2 + 5*b^2)*Sqrt[Cot [c + d*x]])/(a*d) - ((2*b^(7/2)*(9*a^2 + 5*b^2)*ArcTan[(Sqrt[a]*Cot[c + d* x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - (4*a^3*(-1/2*((a^2 - 2*a*b - b^2)* (-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sq rt[Cot[c + d*x]]]/Sqrt[2])) + ((a^2 + 2*a*b - b^2)*(-1/2*Log[1 - Sqrt[2]*S qrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d *x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/a)/a)/(2*a*(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[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]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(1196\) vs. \(2(302)=604\).
Time = 0.74 (sec) , antiderivative size = 1197, normalized size of antiderivative = 3.54
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1197\) |
| default | \(\text {Expression too large to display}\) | \(1197\) |
Input:
int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/12/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(108*tan(d*x+c)^2*(a*b)^(1/2)*a^2*b ^4+108*tan(d*x+c)^(3/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^3*b^4+60* tan(d*x+c)^(3/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a*b^6+40*tan(d*x+c )*(a*b)^(1/2)*a^5*b+80*tan(d*x+c)*(a*b)^(1/2)*a^3*b^3+40*tan(d*x+c)*(a*b)^ (1/2)*a*b^5+48*tan(d*x+c)^2*(a*b)^(1/2)*a^4*b^2+108*tan(d*x+c)^(5/2)*arcta n(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a^2*b^5-3*tan(d*x+c)^(5/2)*ln(-(tan(d*x+ c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^ (1/2)*(a*b)^(1/2)*a^5*b+3*tan(d*x+c)^(5/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x +c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*(a*b)^(1/2)* a^3*b^3-6*tan(d*x+c)^(5/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b )^(1/2)*a^5*b+12*tan(d*x+c)^(5/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/ 2)*(a*b)^(1/2)*a^4*b^2+6*tan(d*x+c)^(5/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2 ))*2^(1/2)*(a*b)^(1/2)*a^3*b^3-6*tan(d*x+c)^(5/2)*arctan(-1+2^(1/2)*tan(d* x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^5*b+12*tan(d*x+c)^(5/2)*arctan(-1+2^(1/2 )*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^4*b^2+6*tan(d*x+c)^(5/2)*arctan( -1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^3*b^3+6*tan(d*x+c)^(5/2 )*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+ c)^(1/2)+1))*2^(1/2)*(a*b)^(1/2)*a^4*b^2+6*tan(d*x+c)^(3/2)*arctan(-1+2^(1 /2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^4*b^2+6*tan(d*x+c)^(3/2)*ln(-( 2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(...
Leaf count of result is larger than twice the leaf count of optimal. 1630 vs. \(2 (302) = 604\).
Time = 0.79 (sec) , antiderivative size = 3290, normalized size of antiderivative = 9.73 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**(5/2)/(a+b*tan(d*x+c))**2,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {12 \, b^{4}}{{\left (a^{5} b + a^{3} b^{3} + \frac {a^{6} + a^{4} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} - \frac {12 \, {\left (9 \, a^{2} b^{4} + 5 \, b^{6}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a b}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {8 \, {\left (\frac {6 \, b}{\sqrt {\tan \left (d x + c\right )}} - \frac {a}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )}}{a^{3}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/12*(12*b^4/((a^5*b + a^3*b^3 + (a^6 + a^4*b^2)/tan(d*x + c))*sqrt(tan(d* x + c))) - 12*(9*a^2*b^4 + 5*b^6)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c)))) /((a^7 + 2*a^5*b^2 + a^3*b^4)*sqrt(a*b)) + 3*(2*sqrt(2)*(a^2 - 2*a*b - b^2 )*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt( 2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1 ) + sqrt(2)*(a^2 + 2*a*b - b^2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d* x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4) + 8*(6*b/sqrt(tan(d*x + c)) - a/tan(d *x + c)^(3/2))/a^3)/d
\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate(cot(d*x + c)^(5/2)/(b*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \] Input:
int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^2,x)
Output:
int(cot(c + d*x)^(5/2)/(a + b*tan(c + d*x))^2, x)
\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\cot \left (d x +c \right )^{\frac {5}{2}}}{\left (a +\tan \left (d x +c \right ) b \right )^{2}}d x \] Input:
int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x)
Output:
int(cot(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x)