Integrand size = 36, antiderivative size = 284 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {((9-5 i) A+(1-3 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((-2+7 i) A+(1+2 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((-7+2 i) A+(2+i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {((9+5 i) A-(1+3 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d} \]
1/4*(A+I*B)*cot(d*x+c)^(3/2)/d/(I*a+a*cot(d*x+c))^2-1/32*((9-5*I)*A+(1-3*I )*B)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(1/32+1/32*I)*((-2+ 7*I)*A+(1+2*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+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*((9+5*I)*A-(1+3*I)*B)*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1 /2))/a^2/d*2^(1/2)+1/8*(5*A+I*B)*cot(d*x+c)^(1/2)/a^2/d/(I+cot(d*x+c))
Time = 4.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (2 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt [4]{-1} (7 A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\tan (c+d x)} (-7 A-3 i B+(-5 i A+B) \tan (c+d x))\right )}{8 a^2 d (-i+\tan (c+d x))^2} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(2*(-1)^(1/4)*(A - I*B)*ArcTan[(-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)^(1/4)*(7*A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Se c[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + Sqrt[Tan[c + d*x]]* (-7*A - (3*I)*B + ((-5*I)*A + B)*Tan[c + d*x])))/(8*a^2*d*(-I + Tan[c + d* x])^2)
Time = 1.00 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.87, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {3042, 4064, 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.
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A \cot (c+d x)+B)}{(a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\int -\frac {\sqrt {\cot (c+d x)} (3 a (i A-B)-a (7 A-i B) \cot (c+d x))}{2 (\cot (c+d x) a+i a)}dx}{4 a^2}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\int \frac {\sqrt {\cot (c+d x)} (3 a (i A-B)-a (7 A-i B) \cot (c+d x))}{\cot (c+d x) a+i a}dx}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (3 a (i A-B)+a (7 A-i B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{i a-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{8 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\int \frac {a^2 (5 i A-B)-3 a^2 (3 A-i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\int \frac {(5 i A-B) a^2+3 (3 A-i B) \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {\frac {\int -\frac {a^2 (5 i A-B-3 (3 A-i B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\int \frac {a^2 (5 i A-B-3 (3 A-i B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\int \frac {5 i A-B-3 (3 A-i B) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}-\frac {-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) ((1+2 i) B-(2-7 i) A) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )}{d}-\frac {(5 A+i B) \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}\) |
((A + I*B)*Cot[c + d*x]^(3/2))/(4*d*(I*a + a*Cot[c + d*x])^2) - (-(((5*A + I*B)*Sqrt[Cot[c + d*x]])/(d*(I + Cot[c + d*x]))) - ((1/2 + I/2)*((-2 + 7* I)*A + (1 + 2*I)*B)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + A rcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (((9 + 5*I)*A - (1 + 3*I) *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)/(8*a^2 )
3.6.26.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (235 ) = 470\).
Time = 0.42 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.40
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(681\) |
| default | \(\text {Expression too large to display}\) | \(681\) |
-1/16/a^2/d*(1/tan(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*(-B*arctan((1/2+1/2*I)*t an(d*x+c)^(1/2)*2^(1/2))+2*A*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+ 2*B*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+7*A*arctan((1/2+1/2*I)*ta n(d*x+c)^(1/2)*2^(1/2))-7*I*A*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2)) +2*I*A*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-2*I*B*arctan((1/2-1/2* I)*tan(d*x+c)^(1/2)*2^(1/2))+7*A*tan(d*x+c)^(1/2)*2^(1/2)-I*B*arctan((1/2+ 1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+2*B*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2 ^(1/2))*tan(d*x+c)+B*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+ c)^2-B*tan(d*x+c)^(3/2)*2^(1/2)-7*A*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^ (1/2))*tan(d*x+c)^2-2*A*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d *x+c)^2-2*B*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)^2+14*A *arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)-4*A*arctan((1/2-1 /2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)+4*B*arctan((1/2-1/2*I)*tan(d*x+ c)^(1/2)*2^(1/2))*tan(d*x+c)+5*I*A*tan(d*x+c)^(3/2)*2^(1/2)+7*I*A*arctan(( 1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)^2-2*I*A*arctan((1/2-1/2*I) *tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)^2+2*I*B*arctan((1/2-1/2*I)*tan(d*x+c )^(1/2)*2^(1/2))*tan(d*x+c)^2+14*I*A*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2 ^(1/2))*tan(d*x+c)+4*I*A*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan( d*x+c)+3*I*B*tan(d*x+c)^(1/2)*2^(1/2)-2*I*B*arctan((1/2+1/2*I)*tan(d*x+c)^ (1/2)*2^(1/2))*tan(d*x+c)+4*I*B*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (211) = 422\).
Time = 0.26 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (2 \, a^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {2 \, {\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 {-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 \, a^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {2 \, {\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 {-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 ) - a^{2} d \sqrt {\frac {49 i \, A^{2} + 14 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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} + 14 \, A B - i \, B^{2}}{a^{4} d^{2}}} + 7 \, A - i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + a^{2} d \sqrt {\frac {49 i \, A^{2} + 14 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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} + 14 \, A B - i \, B^{2}}{a^{4} d^{2}}} - 7 \, A + i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (2 \, {\left (3 i \, A - B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (5 i \, A - 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 (-4 i \, d x - 4 i \, c\right )}}{32 \, a^{2} d} \]
1/32*(2*a^2*d*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c) *log(-2*((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((-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*s qrt((-I*A^2 - 2*A*B + I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(2*((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((-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*sqrt((49*I*A^2 + 14 *A*B - I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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 + 14*A*B - I*B^2)/(a^4*d^2)) + 7*A - I*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) + a^2*d*sqrt((49*I*A^2 + 14*A*B - I*B^2)/(a^4*d^2))*e^(4*I *d*x + 4*I*c)*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 + 14*A*B - I*B ^2)/(a^4*d^2)) - 7*A + I*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 2*(2*(3*I*A - B)*e^(4*I*d*x + 4*I*c) - (5*I*A - 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^(-4*I*d*x - 4*I*c )/(a^2*d)
\[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {A \sqrt {\cot {\left (c + d x \right )}}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx + \int \frac {B \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
-(Integral(A*sqrt(cot(c + d*x))/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x) + Integral(B*tan(c + d*x)*sqrt(cot(c + d*x))/(tan(c + d*x)**2 - 2*I*tan (c + d*x) - 1), x))/a**2
Exception generated. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\cot \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]