Integrand size = 36, antiderivative size = 318 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {((-7+5 i) A+2 i B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((-7+5 i) A+2 i B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {5 A \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((7+5 i) A-2 i B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((7+5 i) A-2 i B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d} \]
1/6*(A+I*B)*cot(d*x+c)^(5/2)/d/(I*a+a*cot(d*x+c))^3+1/12*(4*A+I*B)*cot(d*x +c)^(3/2)/a/d/(I*a+a*cot(d*x+c))^2+1/32*((-7+5*I)*A+2*I*B)*arctan(-1+2^(1/ 2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/32*((-7+5*I)*A+2*I*B)*arctan(1+2^(1/2 )*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-1/64*((7+5*I)*A-2*I*B)*ln(1+cot(d*x+c)-2 ^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/64*((7+5*I)*A-2*I*B)*ln(1+cot(d*x +c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+5/8*A*cot(d*x+c)^(1/2)/d/(I*a^ 3+a^3*cot(d*x+c))
Time = 4.59 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=-\frac {i \sqrt {\cot (c+d x)} \sec ^3(c+d x) \left (12 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))+12 \sqrt [4]{-1} (6 A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-4 \cos (c+d x) (6 A+3 i B+3 (7 A+i B) \cos (2 (c+d x))+(19 i A-B) \sin (2 (c+d x))) \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}}{96 a^3 d (-i+\tan (c+d x))^3} \]
((-1/96*I)*Sqrt[Cot[c + d*x]]*Sec[c + d*x]^3*(12*(-1)^(1/4)*(A - I*B)*ArcT an[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) + 12*(-1)^(1/4)*(6*A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*(Cos[3* (c + d*x)] + I*Sin[3*(c + d*x)]) - 4*Cos[c + d*x]*(6*A + (3*I)*B + 3*(7*A + I*B)*Cos[2*(c + d*x)] + ((19*I)*A - B)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d* x]])*Sqrt[Tan[c + d*x]])/(a^3*d*(-I + Tan[c + d*x])^3)
Time = 1.33 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.92, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3042, 4064, 3042, 4078, 27, 3042, 4078, 27, 3042, 4078, 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.
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+i a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\int -\frac {\cot ^{\frac {3}{2}}(c+d x) (5 a (i A-B)-a (11 A-i B) \cot (c+d x))}{2 (\cot (c+d x) a+i a)^2}dx}{6 a^2}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) (5 a (i A-B)-a (11 A-i B) \cot (c+d x))}{(\cot (c+d x) a+i a)^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 a (i A-B)+a (11 A-i B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {6 \sqrt {\cot (c+d x)} \left (a^2 (4 i A-B)-a^2 (6 A-i B) \cot (c+d x)\right )}{\cot (c+d x) a+i a}dx}{4 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {\sqrt {\cot (c+d x)} \left (a^2 (4 i A-B)-a^2 (6 A-i B) \cot (c+d x)\right )}{\cot (c+d x) a+i a}dx}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left ((4 i A-B) a^2+(6 A-i B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{i a-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {5 i a^3 A-a^3 (7 A-2 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {5 i A a^3+(7 A-2 i B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int -\frac {a^3 (5 i A-(7 A-2 i B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {\int \frac {a^3 (5 i A-(7 A-2 i B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \int \frac {5 i A-(7 A-2 i B) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 i B-(7-5 i) A) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} ((7+5 i) A-2 i B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {5 a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}-\frac {a \left (\frac {1}{2} (2 i B-(7-5 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} ((7+5 i) A-2 i B) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )}{2 a^2}-\frac {a (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
((A + I*B)*Cot[c + d*x]^(5/2))/(6*d*(I*a + a*Cot[c + d*x])^3) - (-((a*(4*A + I*B)*Cot[c + d*x]^(3/2))/(d*(I*a + a*Cot[c + d*x])^2)) + (3*((-5*a^2*A* Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])) - (a*((((-7 + 5*I)*A + (2*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 + (((7 + 5*I)*A - (2*I)*B)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqr t[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d))/(2*a^2))/(12*a^2)
3.6.31.3.1 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[((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*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f* x])^(n - 1)*Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a *A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 = 162, normalized size of antiderivative = 0.51
| method | result | size |
| derivativedivides | \(\frac {\frac {i \left (\frac {-i \left (2 i B +9 A \right ) \cot \left (d x +c \right )^{\frac {5}{2}}+\left (\frac {2 i B}{3}+\frac {38 A}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+5 i A \sqrt {\cot \left (d x +c \right )}}{\left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {2 \left (6 i A +B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{8}+\frac {4 \left (-\frac {A}{16}+\frac {i B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}}{a^{3} d}\) | \(162\) |
| default | \(\frac {\frac {i \left (\frac {-i \left (2 i B +9 A \right ) \cot \left (d x +c \right )^{\frac {5}{2}}+\left (\frac {2 i B}{3}+\frac {38 A}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+5 i A \sqrt {\cot \left (d x +c \right )}}{\left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {2 \left (6 i A +B \right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{\sqrt {2}+i \sqrt {2}}\right )}{8}+\frac {4 \left (-\frac {A}{16}+\frac {i B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}}{a^{3} d}\) | \(162\) |
1/a^3/d*(1/8*I*((-I*(2*I*B+9*A)*cot(d*x+c)^(5/2)+(2/3*I*B+38/3*A)*cot(d*x+ c)^(3/2)+5*I*A*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^3+2*(6*I*A+B)/(2^(1/2)+I*2 ^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2))))+4*(-1/16*A+1/16*I* 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. 683 vs. \(2 (247) = 494\).
Time = 0.27 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {{\left (3 \, a^{3} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} 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 ) - 3 \, a^{3} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {2 \, {\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} 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 ) - 3 \, a^{3} d \sqrt {\frac {36 i \, A^{2} + 12 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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 {36 i \, A^{2} + 12 \, A B - i \, B^{2}}{a^{6} d^{2}}} + 6 \, A - i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 3 \, a^{3} d \sqrt {\frac {36 i \, A^{2} + 12 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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 {36 i \, A^{2} + 12 \, A B - i \, B^{2}}{a^{6} d^{2}}} - 6 \, A + i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 2 \, {\left (2 \, {\left (10 i \, A - B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (14 i \, A + B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (5 i \, A - 2 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A + 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}}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
1/96*(3*a^3*d*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c) *log(-2*((a^3*d*e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) + (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 3*a^3*d*s qrt((-I*A^2 - 2*A*B + I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(2*((a^3*d* e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) - (A - I*B)*e^(2*I *d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 3*a^3*d*sqrt((36*I*A^2 + 12*A*B - I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/8*((a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((36*I*A^2 + 12*A*B - I*B^2)/(a^6*d^2)) + 6*A - I*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 3*a^3*d*sqrt((36*I*A^2 + 12*A*B - I*B^2)/(a^6*d^2))*e^ (6*I*d*x + 6*I*c)*log(1/8*((a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^( 2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((36*I*A^2 + 12*A*B - I*B^2)/(a^6*d^2)) - 6*A + I*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 2*(2*(10*I *A - B)*e^(6*I*d*x + 6*I*c) - (14*I*A + B)*e^(4*I*d*x + 4*I*c) - (5*I*A - 2*B)*e^(2*I*d*x + 2*I*c) - I*A + B)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2 *I*d*x + 2*I*c) - 1)))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
\[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \left (\int \frac {A \sqrt {\cot {\left (c + d x \right )}}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx + \int \frac {B \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx\right )}{a^{3}} \]
I*(Integral(A*sqrt(cot(c + d*x))/(tan(c + d*x)**3 - 3*I*tan(c + d*x)**2 - 3*tan(c + d*x) + I), x) + Integral(B*tan(c + d*x)*sqrt(cot(c + d*x))/(tan( c + d*x)**3 - 3*I*tan(c + d*x)**2 - 3*tan(c + d*x) + I), x))/a**3
Exception generated. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]