Integrand size = 26, antiderivative size = 234 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{16}+\frac {3 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {i \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {1}{32}-\frac {3 i}{32}\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a^2 d} \]
(1/32+3/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(1/32+3/32 *I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(1/64-3/64*I)*ln(1+co t(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+(-1/64+3/64*I)*ln(1+cot(d *x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^2/d*2^(1/2)+1/8*cot(d*x+c)^(1/2)/a^2/d/( I+cot(d*x+c))+1/4*I*cot(d*x+c)^(1/2)/d/(I*a+a*cot(d*x+c))^2
Time = 1.33 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\frac {3 i+\cot (c+d x)}{(i+\cot (c+d x))^2}+2 \sqrt [4]{-1} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}-\sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}\right )}{8 a^2 d} \]
(Sqrt[Cot[c + d*x]]*((3*I + Cot[c + d*x])/(I + Cot[c + d*x])^2 + 2*(-1)^(1 /4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - (-1)^(1/4)* ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]]))/(8*a^2*d)
Time = 0.79 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.91, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {3042, 4156, 3042, 4040, 3042, 4079, 25, 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.
| \(\displaystyle \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (c+d x)^{3/2} (a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4040 |
| \(\displaystyle \frac {\int \frac {a-3 i a \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 i \tan \left (c+d x+\frac {\pi }{2}\right ) a+a}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int -\frac {\cot (c+d x) a^2+3 i a^2}{\sqrt {\cot (c+d x)}}dx}{2 a^2}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}-\frac {\int \frac {\cot (c+d x) a^2+3 i a^2}{\sqrt {\cot (c+d x)}}dx}{2 a^2}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}-\frac {\int \frac {3 i a^2-a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}-\frac {\int -\frac {a^2 (\cot (c+d x)+3 i)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (\cot (c+d x)+3 i)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cot (c+d x)+3 i}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\left (\frac {1}{2}+\frac {3 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\left (\frac {1}{2}-\frac {3 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )}{d}+\frac {\sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
((I/4)*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^2) + (Sqrt[Cot[c + d* x]]/(d*(I + Cot[c + d*x])) + ((1/2 + (3*I)/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[C ot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) - (1/2 - (3*I)/2)*(-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]))) /d)/(8*a^2)
3.8.43.3.1 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_)*Sqrt[(c_.) + (d_.)*tan[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[(-b)*(a + b*Tan[e + f*x])^m*(Sqrt[c + d *Tan[e + f*x]]/(2*a*f*m)), x] + Simp[1/(4*a^2*m) Int[(a + b*Tan[e + f*x]) ^(m + 1)*(Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/Sqrt[c + d*Ta n[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] & & EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2*m]
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 = 1.90 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(-\frac {\left (3 i \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+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 )+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 )+2 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 )-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 )-\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 \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 )+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-i \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 )+2 \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 )+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 )+\arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-2 \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 (-\tan \left (d x +c \right )+i\right )^{2} \tan \left (d x +c \right )^{\frac {3}{2}} \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}}}\) | \(346\) |
| default | \(-\frac {\left (3 i \sqrt {2}\, \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+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 )+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 )+2 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 )-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 )-\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 \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 )+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-i \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 )+2 \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 )+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 )+\arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right ) \sqrt {2}\right )-2 \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 (-\tan \left (d x +c \right )+i\right )^{2} \tan \left (d x +c \right )^{\frac {3}{2}} \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}}}\) | \(346\) |
-1/16/a^2/d*(3*I*2^(1/2)*tan(d*x+c)^(3/2)+I*arctan((1/2+1/2*I)*tan(d*x+c)^ (1/2)*2^(1/2))*tan(d*x+c)^2+2*I*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2 ))*tan(d*x+c)^2+2*I*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c )-4*I*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)-tan(d*x+c)^2 *arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))+2*arctan((1/2-1/2*I)*tan(d*x +c)^(1/2)*2^(1/2))*tan(d*x+c)^2+2^(1/2)*tan(d*x+c)^(1/2)-I*arctan((1/2+1/2 *I)*tan(d*x+c)^(1/2)*2^(1/2))-2*I*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1 /2))+2*arctan((1/2+1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)+4*arctan((1 /2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2))*tan(d*x+c)+arctan((1/2+1/2*I)*tan(d*x+ c)^(1/2)*2^(1/2))-2*arctan((1/2-1/2*I)*tan(d*x+c)^(1/2)*2^(1/2)))*2^(1/2)/ (-tan(d*x+c)+I)^2/tan(d*x+c)^(3/2)/(1/tan(d*x+c))^(3/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (171) = 342\).
Time = 0.27 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {{\left (4 \, a^{2} d \sqrt {-\frac {i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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 \, a^{2} d \sqrt {-\frac {i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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 \, a^{2} d \sqrt {\frac {i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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 {i}{64 \, a^{4} d^{2}}} + 1\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + 4 \, a^{2} d \sqrt {\frac {i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \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 {i}{64 \, a^{4} d^{2}}} - 1\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 (-2 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \]
1/16*(4*a^2*d*sqrt(-1/16*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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*sqrt(-1/16*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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*sqrt(1/64*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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(1/64*I/(a^4*d^2)) + 1)*e^ (-2*I*d*x - 2*I*c)/(a^2*d)) + 4*a^2*d*sqrt(1/64*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*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(1/64*I/(a^4*d^2)) - 1)*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))*(-2*I*e^(4*I*d*x + 4*I*c) + 3*I*e^(2*I*d*x + 2*I*c) - I))*e^(- 4*I*d*x - 4*I*c)/(a^2*d)
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {1}{\tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a^{2}} \]
-Integral(1/(tan(c + d*x)**2*cot(c + d*x)**(3/2) - 2*I*tan(c + d*x)*cot(c + d*x)**(3/2) - cot(c + d*x)**(3/2)), x)/a**2
Exception generated. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]