Integrand size = 28, antiderivative size = 222 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {(4-4 i) a^{5/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {104 i a^2 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {32 a^2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {6 i a^2 \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d} \]
(4-4*I)*a^(5/2)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^ (1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+32/21*a^2*cot(d*x+c)^(3/2)*(a+I *a*tan(d*x+c))^(1/2)/d-6/7*I*a^2*cot(d*x+c)^(5/2)*(a+I*a*tan(d*x+c))^(1/2) /d-2/7*a^2*cot(d*x+c)^(7/2)*(a+I*a*tan(d*x+c))^(1/2)/d+104/21*I*a^2*cot(d* x+c)^(1/2)*(a+I*a*tan(d*x+c))^(1/2)/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(222)=444\).
Time = 7.04 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.25 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {4 i \sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {i a \tan (c+d x)}}{d}+\frac {4 i a^{5/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\cot (c+d x)} \sqrt {1+i \tan (c+d x)} \sqrt {i a \tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}+\frac {104 i a^2 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {32 a^2 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {6 i a^2 \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {5 (-1)^{3/4} a^2 \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d \sqrt {1+i \tan (c+d x)}}+\frac {i a^{3/2} \text {arcsinh}\left (\frac {\sqrt {i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\cot (c+d x)} \sqrt {i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d \sqrt {1+i \tan (c+d x)}} \]
((-4*I)*Sqrt[2]*a^2*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a* Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[I*a*Tan[c + d*x]])/d + ((4*I)*a^(5/ 2)*ArcSinh[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sqrt[Cot[c + d*x]]*Sqrt[1 + I*T an[c + d*x]]*Sqrt[I*a*Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((1 04*I)/21)*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d + (32*a^2*C ot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (((6*I)/7)*a^2*Cot[ c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a^2*Cot[c + d*x]^(7/2)*S qrt[a + I*a*Tan[c + d*x]])/(7*d) - (5*(-1)^(3/4)*a^2*ArcSinh[(-1)^(1/4)*Sq rt[Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]]) + (I*a^(3/2)*ArcSinh[Sqrt[I*a*Tan[c + d*x]]/Sqrt[a]]*Sqrt[Cot[c + d*x]]*Sqrt[I*a*Tan[c + d*x]]*Sqrt[a + I*a*T an[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]])
Time = 1.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {3042, 4729, 3042, 4036, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4027, 218}
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.
| \(\displaystyle \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{9/2} (a+i a \tan (c+d x))^{5/2}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{5/2}}{\tan ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {(i \tan (c+d x) a+a)^{5/2}}{\tan (c+d x)^{9/2}}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{7} \int -\frac {\sqrt {i \tan (c+d x) a+a} \left (15 i a^2-13 a^2 \tan (c+d x)\right )}{2 \tan ^{\frac {7}{2}}(c+d x)}dx-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \int \frac {\sqrt {i \tan (c+d x) a+a} \left (15 i a^2-13 a^2 \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)}dx-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \int \frac {\sqrt {i \tan (c+d x) a+a} \left (15 i a^2-13 a^2 \tan (c+d x)\right )}{\tan (c+d x)^{7/2}}dx-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (\frac {2 \int -\frac {10 \sqrt {i \tan (c+d x) a+a} \left (3 i \tan (c+d x) a^3+4 a^3\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 i \tan (c+d x) a^3+4 a^3\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 i \tan (c+d x) a^3+4 a^3\right )}{\tan (c+d x)^{5/2}}dx}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {2 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (13 i a^4-8 a^4 \tan (c+d x)\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (13 i a^4-8 a^4 \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (13 i a^4-8 a^4 \tan (c+d x)\right )}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {\frac {2 \int -\frac {21 a^5 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {26 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {-21 a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {-21 a^4 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {26 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {4 \left (\frac {\frac {42 i a^6 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {26 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{7} \left (-\frac {6 i a^2 \sqrt {a+i a \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \left (\frac {-\frac {(21-21 i) a^{9/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {26 i a^4 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {8 a^3 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )}{a}\right )-\frac {2 a^2 \sqrt {a+i a \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-2*a^2*Sqrt[a + I*a*Tan[c + d*x]]) /(7*d*Tan[c + d*x]^(7/2)) + (((-6*I)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Ta n[c + d*x]^(5/2)) - (4*((-8*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d *x]^(3/2)) + (((-21 + 21*I)*a^(9/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((26*I)*a^4*Sqrt[a + I*a*Tan[c + d *x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a)))/a)/7)
3.8.62.3.1 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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[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] && GtQ[m, 1] && LtQ[n, -1] && (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*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (179 ) = 358\).
Time = 1.88 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.10
| method | result | size |
| derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2} \left (21 i \sqrt {i a}\, \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )+84 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )-21 \sqrt {i a}\, \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )+32 \left (\tan ^{2}\left (d x +c \right )\right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}+104 i \sqrt {i a}\, \sqrt {-i a}\, \left (\tan ^{3}\left (d x +c \right )\right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-18 i \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}-6 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{21 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(467\) |
| default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2} \left (21 i \sqrt {i a}\, \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )+84 i \sqrt {-i a}\, \ln \left (\frac {2 i a \tan \left (d x +c \right )+2 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}+a}{2 \sqrt {i a}}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )-21 \sqrt {i a}\, \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 \tan \left (d x +c \right ) a}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )+32 \left (\tan ^{2}\left (d x +c \right )\right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}+104 i \sqrt {i a}\, \sqrt {-i a}\, \left (\tan ^{3}\left (d x +c \right )\right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-18 i \tan \left (d x +c \right ) \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}-6 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}\right )}{21 d \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {i a}\, \sqrt {-i a}}\) | \(467\) |
1/21/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(a*(1+I*tan(d*x+c)))^(1/2)*a^2*(21* I*(I*a)^(1/2)*2^(1/2)*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan( d*x+c)))^(1/2)+I*a-3*tan(d*x+c)*a)/(tan(d*x+c)+I))*a*tan(d*x+c)^4+84*I*(-I *a)^(1/2)*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2) *(I*a)^(1/2)+a)/(I*a)^(1/2))*a*tan(d*x+c)^4-21*(I*a)^(1/2)*2^(1/2)*ln(-(-2 *2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*tan(d*x+ c)*a)/(tan(d*x+c)+I))*a*tan(d*x+c)^4+32*tan(d*x+c)^2*(a*tan(d*x+c)*(1+I*ta n(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/2)+104*I*(I*a)^(1/2)*(-I*a)^(1/2)*t an(d*x+c)^3*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-18*I*tan(d*x+c)*(a*tan(d *x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/2)-6*(a*tan(d*x+c)*(1+ I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/2))/(a*tan(d*x+c)*(1+I*tan(d*x+ c)))^(1/2)/(I*a)^(1/2)/(-I*a)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (168) = 336\).
Time = 0.28 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.10 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {16 \, \sqrt {2} {\left (-40 i \, a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 77 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 21 i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - 21 \, \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (16 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{2}}\right ) + 21 \, \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (16 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {-\frac {128 i \, a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{2}}\right )}{84 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
-1/84*(16*sqrt(2)*(-40*I*a^2*e^(7*I*d*x + 7*I*c) + 77*I*a^2*e^(5*I*d*x + 5 *I*c) - 70*I*a^2*e^(3*I*d*x + 3*I*c) + 21*I*a^2*e^(I*d*x + I*c))*sqrt(a/(e ^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2* I*c) - 1)) - 21*sqrt(-128*I*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d *x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(1/4*(16*I*a^3*e^(I*d*x + I* c) + sqrt(2)*sqrt(-128*I*a^5/d^2)*(I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/( e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2 *I*c) - 1)))*e^(-I*d*x - I*c)/a^2) + 21*sqrt(-128*I*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(1/4 *(16*I*a^3*e^(I*d*x + I*c) + sqrt(2)*sqrt(-128*I*a^5/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2* I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-I*d*x - I*c)/a^2))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3165 vs. \(2 (168) = 336\).
Time = 0.64 (sec) , antiderivative size = 3165, normalized size of antiderivative = 14.26 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]
-2/105*(2*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c ) + 1)*(3*(-(35*I - 35)*a^2*cos(7*d*x + 7*c) + (35*I - 35)*a^2*cos(5*d*x + 5*c) - (21*I - 21)*a^2*cos(3*d*x + 3*c) + (I - 1)*a^2*cos(d*x + c) + (35* I + 35)*a^2*sin(7*d*x + 7*c) - (35*I + 35)*a^2*sin(5*d*x + 5*c) + (21*I + 21)*a^2*sin(3*d*x + 3*c) - (I + 1)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2 *d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 5*(13*((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (13*I - 13)*a^2*cos(d*x + c) + 13*((I - 1)*a^2*cos(d*x + c) - (I + 1)*a^2*sin(d*x + c))*sin(2*d*x + 2*c )^2 - (13*I + 13)*a^2*sin(d*x + c) + 21*(-(I - 1)*a^2*cos(2*d*x + 2*c)^2 - (I - 1)*a^2*sin(2*d*x + 2*c)^2 + (2*I - 2)*a^2*cos(2*d*x + 2*c) - (I - 1) *a^2)*cos(3*d*x + 3*c) + 26*(-(I - 1)*a^2*cos(d*x + c) + (I + 1)*a^2*sin(d *x + c))*cos(2*d*x + 2*c) + 21*((I + 1)*a^2*cos(2*d*x + 2*c)^2 + (I + 1)*a ^2*sin(2*d*x + 2*c)^2 - (2*I + 2)*a^2*cos(2*d*x + 2*c) + (I + 1)*a^2)*sin( 3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + 3 *(-(35*I + 35)*a^2*cos(7*d*x + 7*c) + (35*I + 35)*a^2*cos(5*d*x + 5*c) - ( 21*I + 21)*a^2*cos(3*d*x + 3*c) + (I + 1)*a^2*cos(d*x + c) - (35*I - 35)*a ^2*sin(7*d*x + 7*c) + (35*I - 35)*a^2*sin(5*d*x + 5*c) - (21*I - 21)*a^2*s in(3*d*x + 3*c) + (I - 1)*a^2*sin(d*x + c))*sin(7/2*arctan2(sin(2*d*x + 2* c), cos(2*d*x + 2*c) - 1)) + 5*(13*((I + 1)*a^2*cos(d*x + c) + (I - 1)*a^2 *sin(d*x + c))*cos(2*d*x + 2*c)^2 + (13*I + 13)*a^2*cos(d*x + c) + 13*(...
\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]