Integrand size = 26, antiderivative size = 254 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}-\frac {25}{8 a^2 d \sqrt {\cot (c+d x)}}+\frac {7 i}{8 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))}-\frac {1}{4 d \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac {\left (\frac {25}{32}+\frac {21 i}{32}\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {25}{32}+\frac {21 i}{32}\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d} \]
(-25/32+21/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(-25/32 +21/32*I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)-(25/64+21/64*I) *ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(25/64+21/64*I)*l n(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)-25/8/a^2/d/cot(d*x+ c)^(1/2)+7/8*I/a^2/d/(I+cot(d*x+c))/cot(d*x+c)^(1/2)-1/4/d/(I*a+a*cot(d*x+ c))^2/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {-16+(i+\cot (c+d x)) \left (56 i-\frac {(i+\cot (c+d x)) \left (25 \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+8 \sqrt {\tan (c+d x)}\right )+56 i \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{\sqrt {\tan (c+d x)}}\right )}{64 a^2 d \sqrt {\cot (c+d x)} (i+\cot (c+d x))^2} \]
(-16 + (I + Cot[c + d*x])*(56*I - ((I + Cot[c + d*x])*(25*(2*Sqrt[2]*ArcTa n[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[ c + d*x]]] + Sqrt[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]] + 8*Sqrt[Tan[c + d*x]]) + (56*I)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/Sqrt[Tan[c + d*x]]))/(64*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])^2)
Time = 0.95 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {3042, 4156, 3042, 4042, 27, 3042, 4079, 3042, 4012, 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.
| \(\displaystyle \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (c+d x)^{7/2} (a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int -\frac {9 i a-5 a \cot (c+d x)}{2 \cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {9 i a-5 a \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (\cot (c+d x) a+i a)}dx}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {5 \tan \left (c+d x+\frac {\pi }{2}\right ) a+9 i a}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {\int \frac {21 i \cot (c+d x) a^2+25 a^2}{\cot ^{\frac {3}{2}}(c+d x)}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {25 a^2-21 i a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\int \frac {21 i a^2-25 a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\int \frac {25 \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+21 i a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}+\frac {2 \int -\frac {a^2 (21 i-25 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 \int \frac {a^2 (21 i-25 \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \int \frac {21 i-25 \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\frac {50 a^2}{d \sqrt {\cot (c+d x)}}-\frac {2 a^2 \left (\left (\frac {25}{2}+\frac {21 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\left (\frac {25}{2}-\frac {21 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {7 i}{d \sqrt {\cot (c+d x)} (\cot (c+d x)+i)}}{8 a^2}-\frac {1}{4 d \sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}\) |
-1/4*1/(d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - ((-7*I)/(d*Sqrt[C ot[c + d*x]]*(I + Cot[c + d*x])) + ((50*a^2)/(d*Sqrt[Cot[c + d*x]]) - (2*a ^2*((-25/2 + (21*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + (25/2 + (21*I)/2)*(-1/ 2*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqr t[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))))/d)/(2*a^2))/(8*a^2)
3.8.45.3.1 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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(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)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*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*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]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Time = 2.02 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\left (-16 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) \sqrt {2}+43 i \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+2 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-23 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-4 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-46 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )+2 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+23 \left (\tan ^{2}\left (d x +c \right )\right ) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-2 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )+23 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )+25 \sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+4 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-46 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-2 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-23 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )\right ) \sqrt {2}}{16 a^{2} d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right )^{\frac {7}{2}} \left (-\tan \left (d x +c \right )+i\right )^{2}}\) | \(362\) |
| default | \(\frac {\left (-16 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right ) \sqrt {2}+43 i \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+2 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-23 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-4 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-46 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )+2 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+23 \left (\tan ^{2}\left (d x +c \right )\right ) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-2 i \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )+23 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )+25 \sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+4 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-46 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right ) \tan \left (d x +c \right )-2 \arctan \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-23 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )\right ) \sqrt {2}}{16 a^{2} d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right )^{\frac {7}{2}} \left (-\tan \left (d x +c \right )+i\right )^{2}}\) | \(362\) |
1/16/a^2/d*(-16*tan(d*x+c)^(5/2)*2^(1/2)+43*I*tan(d*x+c)^(3/2)*2^(1/2)+2*I *arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)^2-23*I*arctan((1/ 2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)^2-4*I*arctan((1/2-1/2*I)*tan (d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)-46*I*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2) *2^(1/2))*tan(d*x+c)+2*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d* x+c)^2+23*tan(d*x+c)^2*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-2*I*ar ctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+23*I*arctan((1/2+1/2*I)*tan(d*x +c)^(1/2)*2^(1/2))+25*2^(1/2)*tan(d*x+c)^(1/2)+4*arctan((1/2-1/2*I)*tan(d* x+c)^(1/2)*2^(1/2))*tan(d*x+c)-46*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1 /2))*tan(d*x+c)-2*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))-23*arctan(( 1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2)))*2^(1/2)/(1/tan(d*x+c))^(7/2)/tan(d*x +c)^(7/2)/(-tan(d*x+c)+I)^2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (187) = 374\).
Time = 0.27 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.37 \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {4 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {-\frac {i}{16 \, a^{4} d^{2}}} \log \left (-2 \, {\left (4 \, {\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}{16 \, a^{4} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 4 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {-\frac {i}{16 \, a^{4} d^{2}}} \log \left (-2 \, {\left (4 \, {\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}{16 \, a^{4} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 4 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {529 i}{64 \, a^{4} d^{2}}} \log \left (-\frac {{\left (8 \, {\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 {529 i}{64 \, a^{4} d^{2}}} + 23\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 4 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {529 i}{64 \, a^{4} d^{2}}} \log \left (\frac {{\left (8 \, {\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 {529 i}{64 \, a^{4} d^{2}}} - 23\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, 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}} {\left (42 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 33 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 10 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}}{16 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
-1/16*(4*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/1 6*I/(a^4*d^2))*log(-2*(4*(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(-1/16*I/(a^4*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 4*(a^2*d*e^(6*I*d*x + 6* I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(-1/16*I/(a^4*d^2))*log(-2*(4*(-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(-1/16*I/(a^4*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(- 2*I*d*x - 2*I*c)) + 4*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I* c))*sqrt(529/64*I/(a^4*d^2))*log(-1/8*(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(529/64 *I/(a^4*d^2)) + 23)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 4*(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(529/64*I/(a^4*d^2))*log(1/8*(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(529/64*I/(a^4*d^2)) - 23)*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))*(42*I *e^(6*I*d*x + 6*I*c) - 33*I*e^(4*I*d*x + 4*I*c) - 10*I*e^(2*I*d*x + 2*I*c) + I))/(a^2*d*e^(6*I*d*x + 6*I*c) + a^2*d*e^(4*I*d*x + 4*I*c))
Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]