Integrand size = 26, antiderivative size = 267 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=-\frac {i \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a^3 d}+\frac {i \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{8 \sqrt {2} a^3 d}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {\sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {i \sqrt {\cot (c+d x)}}{4 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {i \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt {2} a^3 d}-\frac {i \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt {2} a^3 d} \]
1/6*cot(d*x+c)^(3/2)/d/(I*a+a*cot(d*x+c))^3+1/16*I*arctan(-1+2^(1/2)*cot(d *x+c)^(1/2))/a^3/d*2^(1/2)+1/16*I*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d *2^(1/2)+1/32*I*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)-1/ 32*I*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/a^3/d*2^(1/2)+1/4*cot(d*x+c )^(1/2)/a/d/(I*a+a*cot(d*x+c))^2+1/4*I*cot(d*x+c)^(1/2)/d/(I*a^3+a^3*cot(d *x+c))
Time = 1.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=\frac {\sqrt {\cot (c+d x)} \left (-\frac {16 \cot ^3(c+d x)}{(i+\cot (c+d x))^3}+i \sqrt {\tan (c+d x)} \left (6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+\frac {8 i}{\sqrt {\tan (c+d x)} (-i+\tan (c+d x))^2}-\frac {8}{\sqrt {\tan (c+d x)} (-i+\tan (c+d x))}\right )\right )}{96 a^3 d} \]
(Sqrt[Cot[c + d*x]]*((-16*Cot[c + d*x]^3)/(I + Cot[c + d*x])^3 + I*Sqrt[Ta n[c + d*x]]*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]* ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Ta n[c + d*x]] + Tan[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + (8*I)/(Sqrt[Tan[c + d*x]]*(-I + Tan[c + d*x])^2) - 8/(S qrt[Tan[c + d*x]]*(-I + Tan[c + d*x])))))/(96*a^3*d)
Time = 1.06 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4156, 3042, 4041, 27, 3042, 4078, 27, 3042, 4079, 27, 3042, 3957, 266, 826, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a \cot (c+d x)+i a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}dx\) |
| \(\Big \downarrow \) 4041 |
| \(\displaystyle \frac {\int -\frac {3 \sqrt {\cot (c+d x)} (i a-3 a \cot (c+d x))}{2 (\cot (c+d x) a+i a)^2}dx}{6 a^2}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\sqrt {\cot (c+d x)} (i a-3 a \cot (c+d x))}{(\cot (c+d x) a+i a)^2}dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (3 \tan \left (c+d x+\frac {\pi }{2}\right ) a+i a\right )}{\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a^2}\) |
| \(\Big \downarrow \) 4078 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {2 \left (i a^2-3 a^2 \cot (c+d x)\right )}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {i a^2-3 a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\int \frac {3 \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+i a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {\frac {\int 2 i a^3 \sqrt {\cot (c+d x)}dx}{2 a^2}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {i a \int \sqrt {\cot (c+d x)}dx-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {i a \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {i a \int \frac {\sqrt {\cot (c+d x)}}{\cot ^2(c+d x)+1}d\cot (c+d x)}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \int \frac {\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}-\frac {\frac {-\frac {2 i a^2 \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)}-\frac {2 i a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}}{2 a^2}-\frac {a \sqrt {\cot (c+d x)}}{d (a \cot (c+d x)+i a)^2}}{4 a^2}\) |
Cot[c + d*x]^(3/2)/(6*d*(I*a + a*Cot[c + d*x])^3) - (-((a*Sqrt[Cot[c + d*x ]])/(d*(I*a + a*Cot[c + d*x])^2)) + (((-2*I)*a^2*Sqrt[Cot[c + d*x]])/(d*(I *a + a*Cot[c + d*x])) - ((2*I)*a*((-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] ]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*S qrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2]))/2))/d)/(2*a^2))/(4*a^2)
3.8.47.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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[((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_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* ((c + d*Tan[e + f*x])^(n - 1)/(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 - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Ta n[e + f*x], 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] && GtQ[n, 1] && (In tegerQ[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*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]
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 = 2.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.45
| method | result | size |
| derivativedivides | \(\frac {\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \sqrt {2}-4 i \sqrt {2}}-\frac {-2 i \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )+\frac {2 \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{3} d}\) | \(120\) |
| default | \(\frac {\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \sqrt {2}-4 i \sqrt {2}}-\frac {-2 i \left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )+\frac {2 \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}+\frac {i \arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{4 \sqrt {2}+4 i \sqrt {2}}}{a^{3} d}\) | \(120\) |
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*I*cot(d*x+c)^(5/2)+2/3*cot(d*x+c)^(3/2))/(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. 518 vs. \(2 (206) = 412\).
Time = 0.27 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\sqrt {\cot (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 )} + e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, 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)* 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)) - 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) + e^(4*I*d*x + 4*I*c) - 2*e^(2*I*d*x + 2*I*c ) - 1))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=\frac {i \int \frac {1}{\tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 i \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} + i \sqrt {\cot {\left (c + d x \right )}}}\, dx}{a^{3}} \]
I*Integral(1/(tan(c + d*x)**3*sqrt(cot(c + d*x)) - 3*I*tan(c + d*x)**2*sqr t(cot(c + d*x)) - 3*tan(c + d*x)*sqrt(cot(c + d*x)) + I*sqrt(cot(c + d*x)) ), x)/a**3
Exception generated. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx=\int \frac {1}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]