Integrand size = 36, antiderivative size = 319 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+7 i) A-(23+2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {5 (i A-5 B)}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {3 A+7 i B}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}+\frac {i A-B}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d} \]
(1/32-1/32*I)*((2+7*I)*A-(23+2*I)*B)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a ^2/d*2^(1/2)+1/32*((9+5*I)*A+(-25+21*I)*B)*arctan(1+2^(1/2)*cot(d*x+c)^(1/ 2))/a^2/d*2^(1/2)+(-1/64+1/64*I)*((7+2*I)*A+(2+23*I)*B)*ln(1+cot(d*x+c)-2^ (1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(1/64-1/64*I)*((7+2*I)*A+(2+23*I)*B) *ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+5/8*(I*A-5*B)/a^2 /d/cot(d*x+c)^(1/2)+1/8*(3*A+7*I*B)/a^2/d/(I+cot(d*x+c))/cot(d*x+c)^(1/2)+ 1/4*(I*A-B)/d/(I*a+a*cot(d*x+c))^2/cot(d*x+c)^(1/2)
Time = 4.94 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-2 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+(-1)^{3/4} (7 A+23 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\tan (c+d x)} \left (5 i (A+5 i B)-(7 A+43 i B) \tan (c+d x)+16 B \tan ^2(c+d x)\right )\right )}{8 a^2 d (-i+\tan (c+d x))^2} \]
-1/8*(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2*(-1)^(1/4)*(I*A + B)*ArcTa n[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[ 2*(c + d*x)]) + (-1)^(3/4)*(7*A + (23*I)*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Tan [c + d*x]]*((5*I)*(A + (5*I)*B) - (7*A + (43*I)*B)*Tan[c + d*x] + 16*B*Tan [c + d*x]^2)))/(a^2*d*(-I + Tan[c + d*x])^2)
Time = 1.27 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.90, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.639, Rules used = {3042, 4064, 3042, 4079, 27, 3042, 4079, 25, 3042, 4012, 25, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot (c+d x)^{5/2} (a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {A \cot (c+d x)+B}{\cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B-A \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\int -\frac {a (A+9 i B)+5 a (i A-B) \cot (c+d x)}{2 \cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{4 a^2}+\frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {\int \frac {a (A+9 i B)+5 a (i A-B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {\int \frac {a (A+9 i B)-5 a (i A-B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{8 a^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {\frac {\int -\frac {5 a^2 (i A-5 B)-3 a^2 (3 A+7 i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\int \frac {5 a^2 (i A-5 B)-3 a^2 (3 A+7 i B) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\int \frac {5 (i A-5 B) a^2+3 (3 A+7 i B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\int -\frac {3 (3 A+7 i B) a^2+5 (i A-5 B) \cot (c+d x) a^2}{\sqrt {\cot (c+d x)}}dx+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}-\int \frac {3 (3 A+7 i B) a^2+5 (i A-5 B) \cot (c+d x) a^2}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}-\int \frac {3 a^2 (3 A+7 i B)-5 a^2 (i A-5 B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}-\frac {2 \int -\frac {a^2 (3 (3 A+7 i B)+5 (i A-5 B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 \int \frac {a^2 (3 (3 A+7 i B)+5 (i A-5 B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \int \frac {3 (3 A+7 i B)+5 (i A-5 B) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-B+i A}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {2 a^2 \left (\frac {1}{2} ((9+5 i) A-(25-21 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) ((7+2 i) A+(2+23 i) B) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {10 a^2 (-5 B+i A)}{d \sqrt {\cot (c+d x)}}}{2 a^2}-\frac {3 A+7 i B}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}\) |
(I*A - B)/(4*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - (-((3*A + (7 *I)*B)/(d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x]))) - ((10*a^2*(I*A - 5*B))/ (d*Sqrt[Cot[c + d*x]]) + (2*a^2*((((9 + 5*I)*A - (25 - 21*I)*B)*(-(ArcTan[ 1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + (1/2 - I/2)*((7 + 2*I)*A + (2 + 23*I)*B)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sq rt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))))/d)/(2*a^2))/(8*a^2)
3.6.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^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_) + (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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp [g^(m + n) Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c *Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !Integer Q[p] && IntegerQ[m] && IntegerQ[n]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 0.45 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.49
| method | result | size |
| derivativedivides | \(\frac {-\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {2 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (\frac {5 i A}{2}-\frac {9 B}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7 A}{2}-\frac {11 i B}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{2} d}\) | \(157\) |
| default | \(\frac {-\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {2 B}{\sqrt {\cot \left (d x +c \right )}}+\frac {\left (\frac {5 i A}{2}-\frac {9 B}{2}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-\frac {7 A}{2}-\frac {11 i B}{2}\right ) \sqrt {\cot \left (d x +c \right )}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{2} d}\) | \(157\) |
1/a^2/d*(-1/2*I*(A-I*B)/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^( 1/2)-I*2^(1/2)))-2*B/cot(d*x+c)^(1/2)+1/4*((5/2*I*A-9/2*B)*cot(d*x+c)^(3/2 )+(-7/2*A-11/2*I*B)*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^2+1/4*(7*I*A-23*B)/(2 ^(1/2)+I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (234) = 468\).
Time = 0.27 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.39 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 2 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (-i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} + 7 i \, A - 23 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} - 7 i \, A + 23 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (6 \, {\left (A + 7 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (A + 33 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (3 \, A + 5 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{32 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
-1/32*(2*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A ^2 + 2*A*B - I*B^2)/(a^4*d^2))*log(-2*((I*a^2*d*e^(2*I*d*x + 2*I*c) - I*a^ 2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((I*A ^2 + 2*A*B - I*B^2)/(a^4*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d* x - 2*I*c)/(I*A + B)) - 2*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*log(-2*((-I*a^2*d*e^(2*I*d *x + 2*I*c) + I*a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I* c) - 1))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - (a^2*d*e^(6*I*d*x + 6*I*c) + a^2 *d*e^(4*I*d*x + 4*I*c))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2))* log(1/8*((a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4 *d^2)) + 7*I*A - 23*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) + (a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2 )/(a^4*d^2))*log(-1/8*((a^2*d*e^(2*I*d*x + 2*I*c) - a^2*d)*sqrt((I*e^(2*I* d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((-49*I*A^2 + 322*A*B + 5 29*I*B^2)/(a^4*d^2)) - 7*I*A + 23*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 2*(6* (A + 7*I*B)*e^(6*I*d*x + 6*I*c) - (A + 33*I*B)*e^(4*I*d*x + 4*I*c) - 2*(3* A + 5*I*B)*e^(2*I*d*x + 2*I*c) + A + I*B)*sqrt((I*e^(2*I*d*x + 2*I*c) + I) /(e^(2*I*d*x + 2*I*c) - 1)))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*...
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]