Integrand size = 36, antiderivative size = 308 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {((1+i) A+2 B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((1+i) A+2 B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac {A \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((-1+i) A+2 B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((-1+i) A+2 B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d} \]
1/32*((1+I)*A+2*B)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/32* ((1+I)*A+2*B)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-1/64*((-1+I )*A+2*B)*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/64*((-1 +I)*A+2*B)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/6*(A+ I*B)*cot(d*x+c)^(1/2)/d/(I*a+a*cot(d*x+c))^3+1/12*(2*I*A+B)*cot(d*x+c)^(1/ 2)/a/d/(I*a+a*cot(d*x+c))^2+1/8*A*cot(d*x+c)^(1/2)/d/(I*a^3+a^3*cot(d*x+c) )
Time = 4.33 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (3 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^3(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-3 \sqrt [4]{-1} B \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^3(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-i \sqrt {\tan (c+d x)} (-i+3 \tan (c+d x)) (-3 i A+(A-2 i B) \tan (c+d x))\right )}{24 a^3 d (-i+\tan (c+d x))^3} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*(-1)^(1/4)*(I*A + B)*ArcTan[(-1) ^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^3*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - 3*(-1)^(1/4)*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d* x]^3*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - I*Sqrt[Tan[c + d*x]]*(-I + 3*Tan[c + d*x])*((-3*I)*A + (A - (2*I)*B)*Tan[c + d*x])))/(24*a^3*d*(-I + Tan[c + d*x])^3)
Time = 1.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93, 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, 4079, 27, 3042, 4079, 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.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\cot (c+d x)^{3/2} (a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A \cot (c+d x)+B)}{(a \cot (c+d x)+i a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\int -\frac {a (i A-B)-a (7 A-5 i B) \cot (c+d x)}{2 \sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)^2}dx}{6 a^2}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {a (i A-B)-a (7 A-5 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {a (i A-B)+a (7 A-5 i B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{12 a^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {6 \left (i B a^2+(2 i A+B) \cot (c+d x) a^2\right )}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {i B a^2+(2 i A+B) \cot (c+d x) a^2}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \int \frac {i a^2 B-a^2 (2 i A+B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {(i A+2 B) a^3+A \cot (c+d x) a^3}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int \frac {a^3 (i A+2 B)-a^3 A \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (\frac {\int -\frac {a^3 (\cot (c+d x) A+i A+2 B)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {\int \frac {a^3 (\cot (c+d x) A+i A+2 B)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \int \frac {\cot (c+d x) A+i A+2 B}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 B+(1+i) A) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 B+(1+i) A) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 B+(1+i) A) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (2 B+(1+i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (2 B+(1+i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (2 B+(1+i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a \left (\frac {1}{2} (2 B-(1-i) A) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (2 B+(1+i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(A+i B) \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {3 \left (-\frac {a^2 A \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}-\frac {a \left (\frac {1}{2} (2 B+(1+i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (2 B-(1-i) A) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )}{2 a^2}-\frac {a (B+2 i A) \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}\) |
((A + I*B)*Sqrt[Cot[c + d*x]])/(6*d*(I*a + a*Cot[c + d*x])^3) - (-((a*((2* I)*A + B)*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^2)) + (3*(-((a^2*A *Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x]))) - (a*((((1 + I)*A + 2*B)* (-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sq rt[Cot[c + d*x]]]/Sqrt[2]))/2 + (((-1 + I)*A + 2*B)*(-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))/d))/(2*a^2))/(12*a^2)
3.6.33.3.1 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]
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 = 0.52 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.49
| method | result | size |
| derivativedivides | \(\frac {-\frac {-A \cot \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {10 i A}{3}-\frac {2 B}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-2 i B +A \right ) \sqrt {\cot \left (d x +c \right )}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i B \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}+\frac {4 \left (\frac {A}{16}-\frac {i B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}}{a^{3} d}\) | \(150\) |
| default | \(\frac {-\frac {-A \cot \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {10 i A}{3}-\frac {2 B}{3}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}+\left (-2 i B +A \right ) \sqrt {\cot \left (d x +c \right )}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i B \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}+\frac {4 \left (\frac {A}{16}-\frac {i B}{16}\right ) \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}}{a^{3} d}\) | \(150\) |
1/a^3/d*(-1/8*(-A*cot(d*x+c)^(5/2)+(-10/3*I*A-2/3*B)*cot(d*x+c)^(3/2)+(A-2 *I*B)*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^3+1/4*I*B/(2^(1/2)+I*2^(1/2))*arcta n(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. 637 vs. \(2 (247) = 494\).
Time = 0.26 (sec) , antiderivative size = 637, normalized size of antiderivative = 2.07 \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(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 ) - 24 \, a^{3} d \sqrt {-\frac {i \, B^{2}}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (8 \, {\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 \, B^{2}}{64 \, a^{6} d^{2}}} + i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 24 \, a^{3} d \sqrt {-\frac {i \, B^{2}}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left (8 \, {\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 \, B^{2}}{64 \, a^{6} d^{2}}} - i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 2 \, {\left (2 \, {\left (2 i \, A + B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (4 i \, A + 5 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (i \, A - 4 \, 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* 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)) - 24*a^3*d*sqrt(-1/64*I*B^ 2/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/8*(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(-1/ 64*I*B^2/(a^6*d^2)) + I*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 24*a^3*d*sqrt(- 1/64*I*B^2/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/8*(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(-1/64*I*B^2/(a^6*d^2)) - I*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 2*(2* (2*I*A + B)*e^(6*I*d*x + 6*I*c) - (4*I*A + 5*B)*e^(4*I*d*x + 4*I*c) - (I*A - 4*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)
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]