Integrand size = 36, antiderivative size = 297 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) ((6+i) A+(1+4 i) B) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}+\frac {((7-5 i) A+(5+3 i) B) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{4 \sqrt {2} a d}+\frac {5 (i A-B) \sqrt {\cot (c+d x)}}{2 a d}-\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{6 a d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (i a+a \cot (c+d x))}+\frac {((7+5 i) A-(5-3 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d}+\frac {((-7-5 i) A+(5-3 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{8 \sqrt {2} a d} \]
-1/6*(7*A+3*I*B)*cot(d*x+c)^(3/2)/a/d+1/2*(A+I*B)*cot(d*x+c)^(5/2)/d/(I*a+ a*cot(d*x+c))+(1/8-1/8*I)*((6+I)*A+(1+4*I)*B)*arctan(-1+2^(1/2)*cot(d*x+c) ^(1/2))/a/d*2^(1/2)+1/8*((7-5*I)*A+(5+3*I)*B)*arctan(1+2^(1/2)*cot(d*x+c)^ (1/2))/a/d*2^(1/2)+1/16*((7+5*I)*A+(-5+3*I)*B)*ln(1+cot(d*x+c)-2^(1/2)*cot (d*x+c)^(1/2))/a/d*2^(1/2)+1/16*((-7-5*I)*A+(5-3*I)*B)*ln(1+cot(d*x+c)+2^( 1/2)*cot(d*x+c)^(1/2))/a/d*2^(1/2)+5/2*(I*A-B)*cot(d*x+c)^(1/2)/a/d
Time = 3.47 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\frac {\cot ^{\frac {3}{2}}(c+d x) \left (4 i A+8 A \tan (c+d x)+12 i B \tan (c+d x)+15 i A \tan ^2(c+d x)-15 B \tan ^2(c+d x)+3 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (-i+\tan (c+d x))+6 \sqrt [4]{-1} (3 A+2 i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x) (-i+\tan (c+d x))\right )}{6 a d (-i+\tan (c+d x))} \]
(Cot[c + d*x]^(3/2)*((4*I)*A + 8*A*Tan[c + d*x] + (12*I)*B*Tan[c + d*x] + (15*I)*A*Tan[c + d*x]^2 - 15*B*Tan[c + d*x]^2 + 3*(-1)^(1/4)*(A - I*B)*Arc Tan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2)*(-I + Tan[c + d*x]) + 6*(-1)^(1/4)*(3*A + (2*I)*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Tan[ c + d*x]^(3/2)*(-I + Tan[c + d*x])))/(6*a*d*(-I + Tan[c + d*x]))
Time = 1.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.89, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4064, 3042, 4078, 27, 3042, 4011, 3042, 4011, 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.
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{5/2} (A+B \tan (c+d x))}{a+i a \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4064 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A \cot (c+d x)+B)}{a \cot (c+d x)+i a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a}dx\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\int -\frac {1}{2} \cot ^{\frac {3}{2}}(c+d x) (5 a (i A-B)-a (7 A+3 i B) \cot (c+d x))dx}{2 a^2}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \cot ^{\frac {3}{2}}(c+d x) (5 a (i A-B)-a (7 A+3 i B) \cot (c+d x))dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 a (i A-B)+a (7 A+3 i B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx}{4 a^2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \sqrt {\cot (c+d x)} (a (7 A+3 i B)+5 a (i A-B) \cot (c+d x))dx+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (a (7 A+3 i B)-5 a (i A-B) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}}{4 a^2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \frac {a (7 A+3 i B) \cot (c+d x)-5 a (i A-B)}{\sqrt {\cot (c+d x)}}dx+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\int \frac {-5 a (i A-B)-a (7 A+3 i B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 \int \frac {a (5 (i A-B)-(7 A+3 i B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \int \frac {5 (i A-B)-(7 A+3 i B) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac {\frac {2 a \left (\frac {1}{2} ((7+5 i) A-(5-3 i) B) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) ((6+i) A+(1+4 i) B) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a (7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 a (-B+i A) \sqrt {\cot (c+d x)}}{d}}{4 a^2}\) |
((A + I*B)*Cot[c + d*x]^(5/2))/(2*d*(I*a + a*Cot[c + d*x])) - ((-10*a*(I*A - B)*Sqrt[Cot[c + d*x]])/d + (2*a*(7*A + (3*I)*B)*Cot[c + d*x]^(3/2))/(3* d) + (2*a*((-1/2 + I/2)*((6 + I)*A + (1 + 4*I)*B)*(-(ArcTan[1 - Sqrt[2]*Sq rt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2 ]) + (((7 + 5*I)*A - (5 - 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)/(4*a^2)
3.6.19.3.1 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[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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. 542 vs. \(2 (246 ) = 492\).
Time = 0.41 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.83
| method | result | size |
| derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (-12 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+18 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-12 i B \tan \left (d x +c \right ) \sqrt {2}+3 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+15 B \tan \left (d x +c \right )^{2} \sqrt {2}-15 i A \tan \left (d x +c \right )^{2} \sqrt {2}-8 A \tan \left (d x +c \right ) \sqrt {2}\right ) \sqrt {2}}{12 a d \left (-\tan \left (d x +c \right )+i\right )}\) | \(543\) |
| default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {5}{2}} \tan \left (d x +c \right ) \left (-12 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+18 A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 i A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+3 i A \tan \left (d x +c \right )^{\frac {5}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-12 i B \tan \left (d x +c \right ) \sqrt {2}+3 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-18 A \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-3 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )+12 B \tan \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\tan \left (d x +c \right )}\, \sqrt {2}\right )-4 i A \sqrt {2}+15 B \tan \left (d x +c \right )^{2} \sqrt {2}-15 i A \tan \left (d x +c \right )^{2} \sqrt {2}-8 A \tan \left (d x +c \right ) \sqrt {2}\right ) \sqrt {2}}{12 a d \left (-\tan \left (d x +c \right )+i\right )}\) | \(543\) |
1/12/a/d*(1/tan(d*x+c))^(5/2)*tan(d*x+c)*(-12*I*B*tan(d*x+c)^(3/2)*arctan( (1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+12*I*B*tan(d*x+c)^(5/2)*arctan((1/2+ 1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-3*I*B*tan(d*x+c)^(5/2)*arctan((1/2-1/2*I) *tan(d*x+c)^(1/2)*2^(1/2))-18*I*A*tan(d*x+c)^(5/2)*arctan((1/2+1/2*I)*tan( d*x+c)^(1/2)*2^(1/2))+3*A*tan(d*x+c)^(5/2)*arctan((1/2-1/2*I)*tan(d*x+c)^( 1/2)*2^(1/2))+18*A*tan(d*x+c)^(5/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^ (1/2))+3*B*tan(d*x+c)^(5/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+1 2*B*tan(d*x+c)^(5/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-3*I*B*ta n(d*x+c)^(3/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-18*I*A*tan(d*x +c)^(3/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-3*I*A*tan(d*x+c)^(3 /2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+3*I*A*tan(d*x+c)^(5/2)*ar ctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-12*I*B*tan(d*x+c)*2^(1/2)+3*A*t an(d*x+c)^(3/2)*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-18*A*tan(d*x+ c)^(3/2)*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-3*B*tan(d*x+c)^(3/2) *arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+12*B*tan(d*x+c)^(3/2)*arctan ((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-4*I*A*2^(1/2)+15*B*tan(d*x+c)^2*2^( 1/2)-15*I*A*tan(d*x+c)^2*2^(1/2)-8*A*tan(d*x+c)*2^(1/2))*2^(1/2)/(-tan(d*x +c)+I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (222) = 444\).
Time = 0.27 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.41 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=-\frac {3 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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^{2} 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 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{2} d^{2}}} \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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^{2} 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 ) - 6 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} + 3 \, A + 2 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) + 6 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a 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 {9 i \, A^{2} - 12 \, A B - 4 i \, B^{2}}{a^{2} d^{2}}} - 3 \, A - 2 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - 2 \, {\left ({\left (19 i \, A - 27 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (19 i \, A - 15 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, A - 3 \, 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}}}{24 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
-1/24*(3*(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^2*d^2))*log(-2*((a*d*e^(2*I*d*x + 2*I*c) - a*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^2*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*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))*sqrt(( -I*A^2 - 2*A*B + I*B^2)/(a^2*d^2))*log(2*((a*d*e^(2*I*d*x + 2*I*c) - a*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^2*d^2)) - (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 6*(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))* sqrt((9*I*A^2 - 12*A*B - 4*I*B^2)/(a^2*d^2))*log(((a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(( 9*I*A^2 - 12*A*B - 4*I*B^2)/(a^2*d^2)) + 3*A + 2*I*B)*e^(-2*I*d*x - 2*I*c) /(a*d)) + 6*(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))*sqrt((9*I* A^2 - 12*A*B - 4*I*B^2)/(a^2*d^2))*log(-((a*d*e^(2*I*d*x + 2*I*c) - a*d)*s qrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt((9*I*A^2 - 12*A*B - 4*I*B^2)/(a^2*d^2)) - 3*A - 2*I*B)*e^(-2*I*d*x - 2*I*c)/(a*d)) - 2*((19*I*A - 27*B)*e^(4*I*d*x + 4*I*c) - 2*(19*I*A - 15*B)*e^(2*I*d*x + 2 *I*c) + 3*I*A - 3*B)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {5}{2}}}{i \, a \tan \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]