Integrand size = 26, antiderivative size = 191 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\left (\frac {9}{16}+\frac {5 i}{16}\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {9}{16}+\frac {5 i}{16}\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {9}{16}-\frac {5 i}{16}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {5 i \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}-\frac {\sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2} \] Output:
(9/32+5/32*I)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(9/32+5/32 *I)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/a^2/d+(9/32-5/32*I)*arctanh (2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/a^2/d+5/8*I*cot(d*x+c)^( 1/2)/a^2/d/(I+cot(d*x+c))-1/4*cot(d*x+c)^(1/2)/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 = 1.42 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {i \sqrt {\cot (c+d x)} \left (16 i+(i+\cot (c+d x)) \left (40+(i+\cot (c+d x)) \sqrt {\tan (c+d x)} \left (5 \sqrt {2} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+24 i \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )\right )\right )}{64 a^2 d (i+\cot (c+d x))^2} \] Input:
Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]
Output:
((I/64)*Sqrt[Cot[c + d*x]]*(16*I + (I + Cot[c + d*x])*(40 + (I + Cot[c + d *x])*Sqrt[Tan[c + d*x]]*(5*Sqrt[2]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x] ]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d* x]]) + (24*I)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x] ^(3/2)))))/(a^2*d*(I + Cot[c + d*x])^2)
Time = 0.81 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.13, 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, 4042, 27, 3042, 4079, 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.
| \(\displaystyle \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot (c+d x)^{5/2} (a+i a \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a \cot (c+d x)+i a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int -\frac {7 i a-3 a \cot (c+d x)}{2 \sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{4 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {7 i a-3 a \cot (c+d x)}{\sqrt {\cot (c+d x)} (\cot (c+d x) a+i a)}dx}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {3 \tan \left (c+d x+\frac {\pi }{2}\right ) a+7 i a}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle -\frac {\frac {\int \frac {5 i \cot (c+d x) a^2+9 a^2}{\sqrt {\cot (c+d x)}}dx}{2 a^2}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {9 a^2-5 i a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a^2}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle -\frac {\frac {\int -\frac {a^2 (5 i \cot (c+d x)+9)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {a^2 (5 i \cot (c+d x)+9)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{a^2 d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {5 i \cot (c+d x)+9}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {-\frac {\left (\frac {9}{2}+\frac {5 i}{2}\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\left (\frac {9}{2}-\frac {5 i}{2}\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )}{d}-\frac {5 i \sqrt {\cot (c+d x)}}{d (\cot (c+d x)+i)}}{8 a^2}-\frac {\sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}\) |
Input:
Int[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]
Output:
-1/4*Sqrt[Cot[c + d*x]]/(d*(I*a + a*Cot[c + d*x])^2) - (((-5*I)*Sqrt[Cot[c + d*x]])/(d*(I + Cot[c + d*x])) - ((9/2 + (5*I)/2)*(-(ArcTan[1 - Sqrt[2]* Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt [2]) + (9/2 - (5*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)
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_)*((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 = 0.71 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {\frac {5 i \cot \left (d x +c \right )^{\frac {3}{2}}}{2}-\frac {7 \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {7 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\) | \(120\) |
| default | \(\frac {-\frac {i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{\sqrt {2}-i \sqrt {2}}\right )}{2 \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {\frac {5 i \cot \left (d x +c \right )^{\frac {3}{2}}}{2}-\frac {7 \sqrt {\cot \left (d x +c \right )}}{2}}{4 \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {7 i \arctan \left (\frac {2 \sqrt {\cot \left (d x +c \right )}}{i \sqrt {2}+\sqrt {2}}\right )}{4 \left (i \sqrt {2}+\sqrt {2}\right )}}{a^{2} d}\) | \(120\) |
Input:
int(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2/d*(-1/2*I/(2^(1/2)-I*2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(2^(1/2)-I*2 ^(1/2)))+1/4*(5/2*I*cot(d*x+c)^(3/2)-7/2*cot(d*x+c)^(1/2))/(I+cot(d*x+c))^ 2+7/4*I/(I*2^(1/2)+2^(1/2))*arctan(2*cot(d*x+c)^(1/2)/(I*2^(1/2)+2^(1/2))) )
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (140) = 280\).
Time = 0.10 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.66 \[ \int \frac {1}{\cot ^{\frac {5}{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 (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}{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 (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}{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 {49 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 {49 i}{64 \, a^{4} d^{2}}} + 7 i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 4 \, a^{2} d \sqrt {-\frac {49 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 {49 i}{64 \, a^{4} d^{2}}} - 7 i\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 (6 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 7 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \] Input:
integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
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*(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(1/16*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*( 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(1/16*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(-49/64*I/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*lo g(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(-49/64*I/(a^4*d^2)) + 7*I)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - 4*a^2*d*sqrt(-49/64*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(-49/64*I/(a^4*d^2)) - 7*I)*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))*(6*e^(4*I*d*x + 4*I*c) - 7*e^(2*I*d*x + 2*I*c) + 1))*e^(-4*I*d*x - 4*I*c)/(a^2*d)
Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(1/cot(d*x+c)**(5/2)/(a+I*a*tan(d*x+c))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.37 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {-\left (7 i + 7\right ) \, \sqrt {2} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {2 \, {\left (7 \, \tan \left (d x + c\right )^{\frac {3}{2}} - 5 i \, \sqrt {\tan \left (d x + c\right )}\right )}}{{\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{16 \, a^{2} d} \] Input:
integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
1/16*(-(7*I + 7)*sqrt(2)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c))) + (2*I - 2)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) + 2* (7*tan(d*x + c)^(3/2) - 5*I*sqrt(tan(d*x + c)))/(tan(d*x + c) - I)^2)/(a^2 *d)
Timed out. \[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:
int(1/(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2),x)
Output:
int(1/(cot(c + d*x)^(5/2)*(a + a*tan(c + d*x)*1i)^2), x)
\[ \int \frac {1}{\cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {1}{\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} \tan \left (d x +c \right ) i -\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}}d x}{a^{2}} \] Input:
int(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c))^2,x)
Output:
( - int(1/(sqrt(cot(c + d*x))*cot(c + d*x)**2*tan(c + d*x)**2 - 2*sqrt(cot (c + d*x))*cot(c + d*x)**2*tan(c + d*x)*i - sqrt(cot(c + d*x))*cot(c + d*x )**2),x))/a**2