Integrand size = 36, antiderivative size = 285 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((-2+7 i) A+(1+2 i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((9-5 i) A+(1-3 i) B) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((-7+2 i) A+(2+i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {((9+5 i) A-(1+3 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{8 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2} \]
(-1/32-1/32*I)*((-2+7*I)*A+(1+2*I)*B)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/ a^2/d*2^(1/2)+1/32*((9-5*I)*A+(1-3*I)*B)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2) )/a^2/d*2^(1/2)+(1/64+1/64*I)*((-7+2*I)*A+(2+I)*B)*ln(1-2^(1/2)*tan(d*x+c) ^(1/2)+tan(d*x+c))/a^2/d*2^(1/2)+1/64*((9+5*I)*A-(1+3*I)*B)*ln(1+2^(1/2)*t an(d*x+c)^(1/2)+tan(d*x+c))/a^2/d*2^(1/2)+1/8*(5*A+I*B)*tan(d*x+c)^(1/2)/a ^2/d/(1+I*tan(d*x+c))+1/4*(A+I*B)*tan(d*x+c)^(1/2)/d/(a+I*a*tan(d*x+c))^2
Time = 2.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt [4]{-1} (7 A-i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\tan (c+d x)} (-7 A-3 i B+(-5 i A+B) \tan (c+d x))}{8 a^2 d (-i+\tan (c+d x))^2} \]
(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)*ArcTan h[(-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]]*(-7*A - (3*I)*B + ((-5*I)*A + B)*Tan[c + d*x]))/(8*a^2*d*(-I + Tan[c + d*x])^2)
Time = 0.87 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.85, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4079, 27, 3042, 4079, 3042, 4017, 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.
\(\displaystyle \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {\int \frac {a (7 A-i B)-3 a (i A-B) \tan (c+d x)}{2 \sqrt {\tan (c+d x)} (i \tan (c+d x) a+a)}dx}{4 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a (7 A-i B)-3 a (i A-B) \tan (c+d x)}{\sqrt {\tan (c+d x)} (i \tan (c+d x) a+a)}dx}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (7 A-i B)-3 a (i A-B) \tan (c+d x)}{\sqrt {\tan (c+d x)} (i \tan (c+d x) a+a)}dx}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {\frac {\int \frac {3 a^2 (3 A-i B)-a^2 (5 i A-B) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{2 a^2}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {3 a^2 (3 A-i B)-a^2 (5 i A-B) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{2 a^2}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {\frac {\int \frac {a^2 (3 (3 A-i B)-(5 i A-B) \tan (c+d x))}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{a^2 d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {3 (3 A-i B)-(5 i A-B) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {\frac {1}{2} ((9-5 i) A+(1-3 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} ((9+5 i) A-(1+3 i) B) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )}{d}+\frac {(5 A+i B) \sqrt {\tan (c+d x)}}{d (1+i \tan (c+d x))}}{8 a^2}+\frac {(A+i B) \sqrt {\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\) |
(((((9 - 5*I)*A + (1 - 3*I)*B)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/S qrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 + (((9 + 5*I) *A - (1 + 3*I)*B)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] /Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])) )/2)/d + ((5*A + I*B)*Sqrt[Tan[c + d*x]])/(d*(1 + I*Tan[c + d*x])))/(8*a^2 ) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(4*d*(a + I*a*Tan[c + d*x])^2)
3.2.43.3.1 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[((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 = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.51
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\frac {5 i A}{2}-\frac {B}{2}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (\frac {7 A}{2}+\frac {3 i B}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{4 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {\left (7 i A +B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \sqrt {2}+2 i \sqrt {2}}}{d \,a^{2}}\) | \(144\) |
default | \(\frac {-\frac {\left (\frac {5 i A}{2}-\frac {B}{2}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (\frac {7 A}{2}+\frac {3 i B}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{4 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {\left (7 i A +B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {i \left (-i B +A \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \sqrt {2}+2 i \sqrt {2}}}{d \,a^{2}}\) | \(144\) |
1/d/a^2*(-1/4*((5/2*I*A-1/2*B)*tan(d*x+c)^(3/2)+(7/2*A+3/2*I*B)*tan(d*x+c) ^(1/2))/(tan(d*x+c)-I)^2-1/4*(7*I*A+B)/(2^(1/2)-I*2^(1/2))*arctan(2*tan(d* x+c)^(1/2)/(2^(1/2)-I*2^(1/2)))+1/2*I*(A-I*B)/(2^(1/2)+I*2^(1/2))*arctan(2 *tan(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. 664 vs. \(2 (212) = 424\).
Time = 0.27 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.33 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\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 (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 \, 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 (-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 ) - 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 i \, A + 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 i \, A - B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (2 \, {\left (3 \, A + i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (7 \, A + 3 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}}\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*((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*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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*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*I*A + 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)*s qrt((-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*I*A - B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 2*(2*(3*A + I*B)*e^(4*I*d*x + 4*I*c) + (7*A + 3*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)))*e ^(-4*I*d*x - 4*I*c)/(a^2*d)
Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: TypeError} \]
Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.87 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.44 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} {\left (7 i \, A + B\right )} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{2} d} + \frac {\left (i - 1\right ) \, \sqrt {2} {\left (-i \, A - B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{8 \, a^{2} d} - \frac {5 i \, A \tan \left (d x + c\right )^{\frac {3}{2}} - B \tan \left (d x + c\right )^{\frac {3}{2}} + 7 \, A \sqrt {\tan \left (d x + c\right )} + 3 i \, B \sqrt {\tan \left (d x + c\right )}}{8 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} \]
-(1/16*I + 1/16)*sqrt(2)*(7*I*A + B)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan (d*x + c)))/(a^2*d) + (1/8*I - 1/8)*sqrt(2)*(-I*A - B)*arctan(-(1/2*I - 1/ 2)*sqrt(2)*sqrt(tan(d*x + c)))/(a^2*d) - 1/8*(5*I*A*tan(d*x + c)^(3/2) - B *tan(d*x + c)^(3/2) + 7*A*sqrt(tan(d*x + c)) + 3*I*B*sqrt(tan(d*x + c)))/( a^2*d*(tan(d*x + c) - I)^2)
Time = 11.11 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx=-\frac {-\frac {5\,A\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{8\,a^2\,d}+\frac {A\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,7{}\mathrm {i}}{8\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+\frac {\frac {3\,B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{8\,a^2\,d}+\frac {B\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,1{}\mathrm {i}}{8\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+\mathrm {atan}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{A}\right )\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {A^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}}{7\,A}\right )\,\sqrt {\frac {A^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}\,2{}\mathrm {i}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{B}\right )\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{256\,a^4\,d^2}}}{B}\right )\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{256\,a^4\,d^2}} \]
((3*B*tan(c + d*x)^(1/2))/(8*a^2*d) + (B*tan(c + d*x)^(3/2)*1i)/(8*a^2*d)) /(2*tan(c + d*x) + tan(c + d*x)^2*1i - 1i) - ((A*tan(c + d*x)^(1/2)*7i)/(8 *a^2*d) - (5*A*tan(c + d*x)^(3/2))/(8*a^2*d))/(2*tan(c + d*x) + tan(c + d* x)^2*1i - 1i) + atan((8*a^2*d*tan(c + d*x)^(1/2)*(-(A^2*1i)/(64*a^4*d^2))^ (1/2))/A)*(-(A^2*1i)/(64*a^4*d^2))^(1/2)*2i - atan((16*a^2*d*tan(c + d*x)^ (1/2)*((A^2*49i)/(256*a^4*d^2))^(1/2))/(7*A))*((A^2*49i)/(256*a^4*d^2))^(1 /2)*2i - 2*atanh((8*a^2*d*tan(c + d*x)^(1/2)*((B^2*1i)/(64*a^4*d^2))^(1/2) )/B)*((B^2*1i)/(64*a^4*d^2))^(1/2) + 2*atanh((16*a^2*d*tan(c + d*x)^(1/2)* (-(B^2*1i)/(256*a^4*d^2))^(1/2))/B)*(-(B^2*1i)/(256*a^4*d^2))^(1/2)