Integrand size = 28, antiderivative size = 198 \[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {a+i a \tan (c+d x)}}{5 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {23 i \sqrt {a+i a \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {61 \sqrt {a+i a \tan (c+d x)}}{15 a d \sqrt {\tan (c+d x)}} \]
(-1/2-1/2*I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/ 2))/d/a^(1/2)+61/15*(a+I*a*tan(d*x+c))^(1/2)/a/d/tan(d*x+c)^(1/2)+1/d/(a+I *a*tan(d*x+c))^(1/2)/tan(d*x+c)^(5/2)-7/5*(a+I*a*tan(d*x+c))^(1/2)/a/d/tan (d*x+c)^(5/2)+23/15*I*(a+I*a*tan(d*x+c))^(1/2)/a/d/tan(d*x+c)^(3/2)
Time = 2.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\frac {15 \sqrt {2} a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \tan ^4(c+d x)}{(i a \tan (c+d x))^{3/2}}+\frac {2 \left (-6+2 i \tan (c+d x)+38 \tan ^2(c+d x)+61 i \tan ^3(c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}}}{30 d \tan ^{\frac {5}{2}}(c+d x)} \]
((15*Sqrt[2]*a*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Tan[c + d*x]^4)/(I*a*Tan[c + d*x])^(3/2) + (2*(-6 + (2*I)*Tan[c + d*x] + 38*Tan[c + d*x]^2 + (61*I)*Tan[c + d*x]^3))/Sqrt[a + I*a*Tan[c + d*x]])/(30*d*Tan[c + d*x]^(5/2))
Time = 1.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {3042, 4042, 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.
\(\displaystyle \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{7/2} \sqrt {a+i a \tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4042 |
\(\displaystyle \frac {\int \frac {\sqrt {i \tan (c+d x) a+a} (7 a-6 i a \tan (c+d x))}{2 \tan ^{\frac {7}{2}}(c+d x)}dx}{a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {i \tan (c+d x) a+a} (7 a-6 i a \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)}dx}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {i \tan (c+d x) a+a} (7 a-6 i a \tan (c+d x))}{\tan (c+d x)^{7/2}}dx}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 4081 |
\(\displaystyle \frac {\frac {2 \int -\frac {\sqrt {i \tan (c+d x) a+a} \left (28 \tan (c+d x) a^2+23 i a^2\right )}{2 \tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (28 \tan (c+d x) a^2+23 i a^2\right )}{\tan ^{\frac {5}{2}}(c+d x)}dx}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (28 \tan (c+d x) a^2+23 i a^2\right )}{\tan (c+d x)^{5/2}}dx}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 4081 |
\(\displaystyle \frac {-\frac {\frac {2 \int \frac {\sqrt {i \tan (c+d x) a+a} \left (61 a^3-46 i a^3 \tan (c+d x)\right )}{2 \tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (61 a^3-46 i a^3 \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)}dx}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (61 a^3-46 i a^3 \tan (c+d x)\right )}{\tan (c+d x)^{3/2}}dx}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 4081 |
\(\displaystyle \frac {-\frac {\frac {\frac {2 \int \frac {15 i a^4 \sqrt {i \tan (c+d x) a+a}}{2 \sqrt {\tan (c+d x)}}dx}{a}-\frac {122 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {122 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {15 i a^3 \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {122 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 4027 |
\(\displaystyle \frac {-\frac {\frac {\frac {30 a^5 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {122 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {-\frac {\frac {\frac {(15+15 i) a^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {122 a^3 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {46 i a^2 \sqrt {a+i a \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {14 a \sqrt {a+i a \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}}{2 a^2}+\frac {1}{d \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}\) |
1/(d*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) + ((-14*a*Sqrt[a + I*a *Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - ((((-46*I)/3)*a^2*Sqrt[a + I*a* Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) + (((15 + 15*I)*a^(7/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (122*a^3 *Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a))/(5*a))/(2*a^2)
3.3.17.3.1 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*(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*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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (158 ) = 316\).
Time = 1.21 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.39
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (30 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )-15 \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \left (\tan ^{5}\left (d x +c \right )\right )-396 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{3}\left (d x +c \right )\right )+15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )+244 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{4}\left (d x +c \right )\right )+16 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )-144 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+24 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{60 d a \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{2} \sqrt {-i a}}\) | \(473\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (30 i \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{4}\left (d x +c \right )\right )-15 \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) \sqrt {2}\, a \left (\tan ^{5}\left (d x +c \right )\right )-396 i \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{3}\left (d x +c \right )\right )+15 \sqrt {2}\, \ln \left (-\frac {-2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}+i a -3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )+244 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\, \left (\tan ^{4}\left (d x +c \right )\right )+16 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )-144 \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (\tan ^{2}\left (d x +c \right )\right )+24 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{60 d a \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{2} \sqrt {-i a}}\) | \(473\) |
1/60/d*(a*(1+I*tan(d*x+c)))^(1/2)*(30*I*ln(-(-2*2^(1/2)*(-I*a)^(1/2)*(a*ta n(d*x+c)*(1+I*tan(d*x+c)))^(1/2)+I*a-3*a*tan(d*x+c))/(tan(d*x+c)+I))*2^(1/ 2)*a*tan(d*x+c)^4-15*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*a*tan(d*x+c))/(tan(d*x+c)+I))*2^(1/2)*a*tan(d*x+c)^5-3 96*I*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^3+15*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*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)^3+244*(a*tan(d*x+c)*(1+I*t an(d*x+c)))^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^4+16*I*tan(d*x+c)*(a*tan(d*x+c)* (1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)-144*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*t an(d*x+c)))^(1/2)*tan(d*x+c)^2+24*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(- I*a)^(1/2))/a/tan(d*x+c)^(5/2)/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(-tan (d*x+c)+I)^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. 456 vs. \(2 (148) = 296\).
Time = 0.27 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-103 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 102 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 40 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 150 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 15 i\right )} - 15 \, {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} - 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {2 i}{a d^{2}}} \log \left (\frac {1}{4} i \, a d \sqrt {\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) + 15 \, {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} - 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {2 i}{a d^{2}}} \log \left (-\frac {1}{4} i \, a d \sqrt {\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )}{60 \, {\left (a d e^{\left (7 i \, d x + 7 i \, c\right )} - 3 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}\right )}} \]
-1/60*(2*sqrt(2)*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))*(-103*I*e^(8*I*d*x + 8*I*c) + 102*I* e^(6*I*d*x + 6*I*c) + 40*I*e^(4*I*d*x + 4*I*c) - 150*I*e^(2*I*d*x + 2*I*c) + 15*I) - 15*(a*d*e^(7*I*d*x + 7*I*c) - 3*a*d*e^(5*I*d*x + 5*I*c) + 3*a*d *e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(2*I/(a*d^2))*log(1/4*I*a* d*sqrt(2*I/(a*d^2))*e^(I*d*x + I*c) + 1/4*sqrt(2)*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^ (2*I*d*x + 2*I*c) + 1)) + 15*(a*d*e^(7*I*d*x + 7*I*c) - 3*a*d*e^(5*I*d*x + 5*I*c) + 3*a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))*sqrt(2*I/(a*d^2 ))*log(-1/4*I*a*d*sqrt(2*I/(a*d^2))*e^(I*d*x + I*c) + 1/4*sqrt(2)*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^(2*I*d*x + 2*I*c) + 1)))/(a*d*e^(7*I*d*x + 7*I*c) - 3*a*d* e^(5*I*d*x + 5*I*c) + 3*a*d*e^(3*I*d*x + 3*I*c) - a*d*e^(I*d*x + I*c))
\[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \tan ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (148) = 296\).
Time = 2.30 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {{\left (i \, a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {1}{16} \, \sqrt {2} {\left ({\left (\frac {\sqrt {2} \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (-\frac {i \, {\left | a \right |}}{a} + 1\right )} {\left | a \right |}^{\frac {3}{2}}}{a^{2}} - \frac {\sqrt {-2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}} {\left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}{a^{2}}}} + 1\right )} {\left | a \right |}}{a^{2}}\right )}^{2} + 12 i\right )}\right )}{2 \, a^{2} d} - \frac {\sqrt {-2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}} {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (\frac {23 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}}{a^{3} d} - \frac {50}{a^{2} d}\right )} + \frac {30}{a d}\right )} \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}{a^{2}}}} + 1\right )} {\left | a \right |}}{15 \, a^{2} \tan \left (d x + c\right )^{3}} - \frac {4 \, \sqrt {2} {\left (i \, a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (-\frac {i \, {\left | a \right |}}{a} + 1\right )} {\left | a \right |}^{\frac {3}{2}}}{a^{2}} - \frac {\sqrt {-2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}} {\left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}{a^{2}}}} + 1\right )} {\left | a \right |}}{a^{2}}\right )}^{2} - 4 i\right )} a^{2} d} \]
1/2*(I*a*sqrt(abs(a)) - abs(a)^(3/2))*arctan(1/16*sqrt(2)*((sqrt(2)*sqrt(I *a*tan(d*x + c) + a)*(-I*abs(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan (d*x + c) + a)*a + 2*a^2)*(tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2 *(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/a^2)^2 + 12*I))/(a^2*d) - 1/15*sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*((I*a*tan(d*x + c) + a)*( 23*(I*a*tan(d*x + c) + a)/(a^3*d) - 50/(a^2*d)) + 30/(a*d))*sqrt(I*a*tan(d *x + c) + a)*(tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/(a^2*tan(d*x + c)^3) - 4*sqrt(2)*(I*a *sqrt(abs(a)) - abs(a)^(3/2))/(((sqrt(2)*sqrt(I*a*tan(d*x + c) + a)*(-I*ab s(a)/a + 1)*abs(a)^(3/2)/a^2 - sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*( tan(d*x + c)/sqrt(((I*a*tan(d*x + c) + a)^2 - 2*(I*a*tan(d*x + c) + a)*a + a^2)/a^2) + 1)*abs(a)/a^2)^2 - 4*I)*a^2*d)
Timed out. \[ \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{7/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]