Integrand size = 26, antiderivative size = 222 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a^3 d}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a^3 d}+\frac {i \sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {\sqrt {\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}-\frac {\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt {2} a^3 d}+\frac {\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt {2} a^3 d} \]
1/16*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/16*arctan(1+2^(1/ 2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-1/32*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c) ^(1/2))/a^3/d*2^(1/2)+1/32*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d *2^(1/2)+1/6*I*cot(d*x+c)^(1/2)/d/(I*a+a*cot(d*x+c))^3+1/12*cot(d*x+c)^(1/ 2)/a/d/(I*a+a*cot(d*x+c))^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {\frac {\cot ^2(c+d x) (3 i+\cot (c+d x))}{(i+\cot (c+d x))^3}-\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )}{12 a^3 d \cot ^{\frac {3}{2}}(c+d x)} \]
((Cot[c + d*x]^2*(3*I + Cot[c + d*x]))/(I + Cot[c + d*x])^3 - Hypergeometr ic2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])/(12*a^3*d*Cot[c + d*x]^(3/2))
Time = 0.81 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.97, 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, 27, 2011, 3042, 3957, 266, 755, 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.
\(\displaystyle \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cot (c+d x)^{5/2} (a+i a \tan (c+d x))^3}dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a \cot (c+d x)+i a)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
\(\Big \downarrow \) 4040 |
\(\displaystyle \frac {\int \frac {a-5 i a \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)^2}dx}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {5 i \tan \left (c+d x+\frac {\pi }{2}\right ) a+a}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {\frac {\int -\frac {6 \left (\cot (c+d x) a^2+i a^2\right )}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{4 a^2}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}-\frac {3 \int \frac {\cot (c+d x) a^2+i a^2}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{2 a^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 2011 |
\(\displaystyle \frac {-\frac {3 \int \frac {1}{\sqrt {\cot (c+d x)}}dx}{2 a}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {3 \int \frac {1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{\sqrt {\cot (c+d x)} \left (\cot ^2(c+d x)+1\right )}d\cot (c+d x)}{2 a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {3 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{a d}+\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{12 a^2}+\frac {i \sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}\) |
((I/6)*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^3) + ((a*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^2) + (3*((-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[ c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])/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]))/2))/(a*d))/(12*a^ 2)
3.8.49.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
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[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
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 = 2.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\frac {-\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {\frac {2 \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 i \left (\sqrt {\cot }\left (d x +c \right )\right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{3} d}\) | \(120\) |
default | \(\frac {-\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {\frac {2 \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 i \left (\sqrt {\cot }\left (d x +c \right )\right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{3} d}\) | \(120\) |
1/a^3/d*(-1/4*I/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)-I*2 ^(1/2)))+1/8*(2/3*cot(d*x+c)^(3/2)+2*I*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^3+ 1/4*I/(2^(1/2)+I*2^(1/2))*arctan(2*cot(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. 520 vs. \(2 (175) = 350\).
Time = 0.27 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {{\left (12 \, a^{3} d \sqrt {\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (2 \, {\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} 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 ) - 12 \, a^{3} d \sqrt {\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} 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 ) + 12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} d^{2}}} + i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{3} 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^{6} d^{2}}} - i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} 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 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 5 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{48 \, a^{3} d} \]
1/48*(12*a^3*d*sqrt(1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(2*(8*(a^3*d* e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) + I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-2*(8 *(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^( -2*I*d*x - 2*I*c)) + 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)* log(1/8*(8*(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) + I)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*l og(-1/8*(8*(a^3*d*e^(2*I*d*x + 2*I*c) - a^3*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^6*d^2)) - I)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(2*e^(6*I*d*x + 6*I*c) - 5*e^(4*I*d*x + 4*I*c) + 4*e^(2*I*d*x + 2*I* c) - 1))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]