Integrand size = 23, antiderivative size = 257 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{3/2} \left (5 a^2+b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))} \] 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^(3/2)*(5*a^2+b^2)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(3/ 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+c ot(d*x+c)))*2^(1/2)/(a^2+b^2)^2/d+b^2*cot(d*x+c)^(1/2)/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 = 1.26 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\frac {-28 a^{3/2} b^2 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-12 a^{7/2} \left (a^2+b^2\right ) \cot ^{\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},-\frac {a \cot (c+d x)}{b}\right )+7 b^2 \left (-6 \sqrt {2} a^{5/2} b \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+6 \sqrt {2} a^{5/2} b \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+24 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )-24 a^{5/2} b \sqrt {\cot (c+d x)}-24 \sqrt {a} b^3 \sqrt {\cot (c+d x)}+4 a^{7/2} \cot ^{\frac {3}{2}}(c+d x)+4 a^{3/2} b^2 \cot ^{\frac {3}{2}}(c+d x)-3 \sqrt {2} a^{5/2} b \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )+3 \sqrt {2} a^{5/2} b \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{42 a^{3/2} b^2 \left (a^2+b^2\right )^2 d} \] Input:
Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^2,x]
Output:
(-28*a^(3/2)*b^2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 12*a^(7/2)*(a^2 + b^2)*Cot[c + d*x]^(7/2)*Hypergeo metric2F1[2, 7/2, 9/2, -((a*Cot[c + d*x])/b)] + 7*b^2*(-6*Sqrt[2]*a^(5/2)* b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 6*Sqrt[2]*a^(5/2)*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x] ])/Sqrt[b]] - 24*a^(5/2)*b*Sqrt[Cot[c + d*x]] - 24*Sqrt[a]*b^3*Sqrt[Cot[c + d*x]] + 4*a^(7/2)*Cot[c + d*x]^(3/2) + 4*a^(3/2)*b^2*Cot[c + d*x]^(3/2) - 3*Sqrt[2]*a^(5/2)*b*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 3*Sqrt[2]*a^(5/2)*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/ (42*a^(3/2)*b^2*(a^2 + b^2)^2*d)
Time = 1.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.10, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4156, 3042, 4048, 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.
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a \cot (c+d x)+b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {b^2-2 a \cot (c+d x) b+\left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^2-2 a \cot (c+d x) b+\left (2 a^2+b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b^2+2 a \tan \left (c+d x+\frac {\pi }{2}\right ) b+\left (2 a^2+b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {\int -\frac {2 \left (2 a^2 b-a \left (a^2-b^2\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {2 \int \frac {2 a^2 b-a \left (a^2-b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {2 b a^2+\left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int -\frac {a \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 \int \frac {a \left (2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 a \int \frac {2 a b-\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {b^2 \left (5 a^2+b^2\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^2 \left (5 a^2+b^2\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {4 a \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{3/2} \left (5 a^2+b^2\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}\) |
Input:
Int[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^2,x]
Output:
(b^2*Sqrt[Cot[c + d*x]])/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((2*b^(3 /2)*(5*a^2 + b^2)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) + (4*a*(-1/2*((a^2 - 2*a*b - b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) + (( a^2 + 2*a*b - b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[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))/(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[(((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. \(970\) vs. \(2(228)=456\).
Time = 0.25 (sec) , antiderivative size = 971, normalized size of antiderivative = 3.78
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(971\) |
| default | \(\text {Expression too large to display}\) | \(971\) |
Input:
int(cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/4/d*(1/tan(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*(2^(1/2)*(a*b)^(1/2)*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) )*a^3*b*tan(d*x+c)-2^(1/2)*(a*b)^(1/2)*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))*a*b^3*tan(d*x+c)+2*2^(1/ 2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b*tan(d*x+c)-4*2^(1/ 2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^2*tan(d*x+c)-2*2^( 1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^3*tan(d*x+c)+2*2^( 1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b*tan(d*x+c)-4*2^ (1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^2*tan(d*x+c)-2 *2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^3*tan(d*x+c)- 2*2^(1/2)*(a*b)^(1/2)*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))*a^2*b^2*tan(d*x+c)+2^(1/2)*(a*b)^(1/2)*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))*a^4-2^(1/2)*(a*b)^(1/2)*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))*a^2*b^2+2*2^(1/2)*(a*b)^(1/2)* arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^4-4*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1 /2)*tan(d*x+c)^(1/2))*a^3*b-2*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x +c)^(1/2))*a^2*b^2+2*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2 ))*a^4-4*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3*b-2*2 ^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b^2-2*2^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 1494 vs. \(2 (228) = 456\).
Time = 0.32 (sec) , antiderivative size = 3018, normalized size of antiderivative = 11.74 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\cot {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cot(d*x+c)**(1/2)/(a+b*tan(d*x+c))**2,x)
Output:
Integral(sqrt(cot(c + d*x))/(a + b*tan(c + d*x))**2, x)
Time = 0.13 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (5 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a b}} - \frac {4 \, b^{2}}{{\left (a^{3} b + a b^{3} + \frac {a^{4} + a^{2} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}} + \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \] Input:
integrate(cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-1/4*(4*(5*a^2*b^2 + b^4)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(a*b)) - 4*b^2/((a^3*b + a*b^3 + (a^4 + a^2*b^2)/t an(d*x + c))*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)))) + 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))/d
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\int { \frac {\sqrt {\cot \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
integrate(sqrt(cot(d*x + c))/(b*tan(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \] Input:
int(cot(c + d*x)^(1/2)/(a + b*tan(c + d*x))^2,x)
Output:
int(cot(c + d*x)^(1/2)/(a + b*tan(c + d*x))^2, x)
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )}}{\tan \left (d x +c \right )^{2} b^{2}+2 \tan \left (d x +c \right ) a b +a^{2}}d x \] Input:
int(cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x)
Output:
int(sqrt(cot(c + d*x))/(tan(c + d*x)**2*b**2 + 2*tan(c + d*x)*a*b + a**2), x)