Integrand size = 30, antiderivative size = 351 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 (c-i d)^{5/2} f}+\frac {\left (2 i c^2-14 c d-47 i d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{9/2} f}+\frac {d \left (6 c^2+27 i c d+49 d^2\right )}{24 a^2 (c-i d) (c+i d)^3 f (c+d \tan (e+f x))^{3/2}}+\frac {2 i c-9 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{8 a^2 (c-i d)^2 (c+i d)^4 f \sqrt {c+d \tan (e+f x)}} \]
-1/4*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^2/(c-I*d)^(5/2)/f+1 /8*(2*I*c^2-14*c*d-47*I*d^2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2)) /a^2/(c+I*d)^(9/2)/f+1/8*d*(2*c^3+9*I*c^2*d+88*c*d^2-45*I*d^3)/a^2/(c-I*d) ^2/(c+I*d)^4/f/(c+d*tan(f*x+e))^(1/2)+1/24*d*(6*c^2+27*I*c*d+49*d^2)/a^2/( c-I*d)/(c+I*d)^3/f/(c+d*tan(f*x+e))^(3/2)+1/8*(2*I*c-9*d)/a^2/(c+I*d)^2/f/ (1+I*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2)-1/4/(I*c-d)/f/(a+I*a*tan(f*x+e))^2 /(c+d*tan(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.40 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {-\frac {2 i (c+i d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}+\frac {i \left (2 c^2+14 i c d-47 d^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}+\frac {6 i (c+i d)}{(-i+\tan (e+f x))^2}-\frac {3 (2 c+9 i d)}{-i+\tan (e+f x)}}{24 a^2 (c+i d)^2 f (c+d \tan (e+f x))^{3/2}} \]
-1/24*(((-2*I)*(c + I*d)^2*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d) + (I*(2*c^2 + (14*I)*c*d - 47*d^2)*Hypergeome tric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d) + ((6*I) *(c + I*d))/(-I + Tan[e + f*x])^2 - (3*(2*c + (9*I)*d))/(-I + Tan[e + f*x] ))/(a^2*(c + I*d)^2*f*(c + d*Tan[e + f*x])^(3/2))
Time = 1.87 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.08, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 4042, 27, 3042, 4079, 3042, 4012, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle -\frac {\int -\frac {a (4 i c-11 d)+7 i a d \tan (e+f x)}{2 (i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (4 i c-11 d)+7 i a d \tan (e+f x)}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (4 i c-11 d)+7 i a d \tan (e+f x)}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+18 i d c-49 d^2\right ) a^2+5 (2 c+9 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+18 i d c-49 d^2\right ) a^2+5 (2 c+9 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 c^3+18 i d c^2-39 d^2 c+45 i d^3\right ) a^2+d \left (6 c^2+27 i d c+49 d^2\right ) \tan (e+f x) a^2}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 c^3+18 i d c^2-39 d^2 c+45 i d^3\right ) a^2+d \left (6 c^2+27 i d c+49 d^2\right ) \tan (e+f x) a^2}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 c^4+18 i d c^3-33 d^2 c^2+72 i d^3 c+49 d^4\right ) a^2+d \left (2 c^3+9 i d c^2+88 d^2 c-45 i d^3\right ) \tan (e+f x) a^2}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 c^4+18 i d c^3-33 d^2 c^2+72 i d^3 c+49 d^4\right ) a^2+d \left (2 c^3+9 i d c^2+88 d^2 c-45 i d^3\right ) \tan (e+f x) a^2}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {-\frac {\frac {\frac {a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^2 (c+i d)^4 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^2 (c+i d)^4 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {2 i a^2 (c+i d)^4 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}-\frac {i a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {i a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}-\frac {2 i a^2 (c+i d)^4 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {2 a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {4 a^2 (c+i d)^4 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {2 a^2 (c-i d)^2 \left (2 c^2+14 i c d-47 d^2\right ) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {4 a^2 (c+i d)^4 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}}{c^2+d^2}+\frac {2 a^2 d \left (2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^2 d \left (6 c^2+27 i c d+49 d^2\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {2 c+9 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{8 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}\) |
-1/4*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (-((2*c + (9*I)*d)/((c + I*d)*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x]) ^(3/2))) - ((2*a^2*d*(6*c^2 + (27*I)*c*d + 49*d^2))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (((4*a^2*(c + I*d)^4*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (2*a^2*(c - I*d)^2*(2*c^2 + (14*I)*c*d - 47*d^ 2)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f))/(c^2 + d^2) + (2 *a^2*d*(2*c^3 + (9*I)*c^2*d + 88*c*d^2 - (45*I)*d^3))/((c^2 + d^2)*f*Sqrt[ c + d*Tan[e + f*x]]))/(c^2 + d^2))/(2*a^2*(I*c - d)))/(8*a^2*(I*c - d))
3.12.35.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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]
Time = 0.53 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(\frac {2 d^{3} \left (\frac {i \left (\frac {-\frac {d \left (2 i c^{6}-9 i c^{4} d^{2}-24 i c^{2} d^{4}-13 i d^{6}-15 c^{5} d -30 c^{3} d^{3}-15 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{7}-28 i c^{5} d^{2}-62 i c^{3} d^{4}-32 i c \,d^{6}-19 c^{6} d -23 c^{4} d^{3}+11 c^{2} d^{5}+15 d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (16 i c^{6} d -15 i c^{4} d^{3}-78 i c^{2} d^{5}-47 i d^{7}+2 c^{7}-57 c^{5} d^{2}-120 c^{3} d^{4}-61 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 d^{3} \left (i d +c \right )^{5} \left (i d -c \right )^{2}}-\frac {-2 i c d -4 c^{2}-2 d^{2}}{\left (i d -c \right )^{2} \left (i d +c \right )^{5} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-c^{2}-d^{2}}{3 \left (i d +c \right )^{4} \left (i d -c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (5 i c^{4} d -10 i c^{2} d^{3}+i d^{5}+c^{5}-10 c^{3} d^{2}+5 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{5} d^{3}}\right )}{f \,a^{2}}\) | \(518\) |
| default | \(\frac {2 d^{3} \left (\frac {i \left (\frac {-\frac {d \left (2 i c^{6}-9 i c^{4} d^{2}-24 i c^{2} d^{4}-13 i d^{6}-15 c^{5} d -30 c^{3} d^{3}-15 c \,d^{5}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{7}-28 i c^{5} d^{2}-62 i c^{3} d^{4}-32 i c \,d^{6}-19 c^{6} d -23 c^{4} d^{3}+11 c^{2} d^{5}+15 d^{7}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (16 i c^{6} d -15 i c^{4} d^{3}-78 i c^{2} d^{5}-47 i d^{7}+2 c^{7}-57 c^{5} d^{2}-120 c^{3} d^{4}-61 c \,d^{6}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 d^{3} \left (i d +c \right )^{5} \left (i d -c \right )^{2}}-\frac {-2 i c d -4 c^{2}-2 d^{2}}{\left (i d -c \right )^{2} \left (i d +c \right )^{5} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-c^{2}-d^{2}}{3 \left (i d +c \right )^{4} \left (i d -c \right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {i \left (5 i c^{4} d -10 i c^{2} d^{3}+i d^{5}+c^{5}-10 c^{3} d^{2}+5 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{5} d^{3}}\right )}{f \,a^{2}}\) | \(518\) |
2/f/a^2*d^3*(1/8*I/d^3/(c+I*d)^5/(I*d-c)^2*((-1/2*d*(2*I*c^6-9*I*c^4*d^2-2 4*I*c^2*d^4-13*I*d^6-15*c^5*d-30*c^3*d^3-15*c*d^5)/(2*I*c*d+c^2-d^2)*(c+d* tan(f*x+e))^(3/2)+1/2*d*(2*I*c^7-28*I*c^5*d^2-62*I*c^3*d^4-32*I*c*d^6-19*c ^6*d-23*c^4*d^3+11*c^2*d^5+15*d^7)/(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(1/2 ))/(-d*tan(f*x+e)+I*d)^2-1/2*(-57*c^5*d^2-120*c^3*d^4-61*c*d^6+16*I*c^6*d- 15*I*c^4*d^3-78*I*c^2*d^5-47*I*d^7+2*c^7)/(2*I*c*d+c^2-d^2)/(-I*d-c)^(1/2) *arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2)))-1/(I*d-c)^2/(c+I*d)^5*(-2* I*c*d-4*c^2-2*d^2)/(c+d*tan(f*x+e))^(1/2)-1/3*(-c^2-d^2)/(c+I*d)^4/(I*d-c) ^2/(c+d*tan(f*x+e))^(3/2)+1/8*I/(I*d-c)^(5/2)/(c+I*d)^5*(5*c*d^4-10*I*c^2* d^3+I*d^5-10*c^3*d^2+5*I*c^4*d+c^5)/d^3*arctan((c+d*tan(f*x+e))^(1/2)/(I*d -c)^(1/2)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3246 vs. \(2 (281) = 562\).
Time = 5.31 (sec) , antiderivative size = 3246, normalized size of antiderivative = 9.25 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
-1/96*(24*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2* d^8)*f*e^(8*I*f*x + 8*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6*d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d^ 8)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I* a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4*I*a^2*c *d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(1/16*I/((-I*a^4*c^5 - 5*a^4*c^ 4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)*f^2))*l og(-2*(4*((I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^3 + 3*a^2*c^2*d - 3*I*a^2*c*d^2 - a^2*d^3)*f)*sqrt((( c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/ 16*I/((-I*a^4*c^5 - 5*a^4*c^4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I* a^4*c*d^4 - a^4*d^5)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f* x - 2*I*e)) - 24*((a^2*c^8 + 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 + 4*a^2*c^2*d^6 + a^2*d^8)*f*e^(8*I*f*x + 8*I*e) + 2*(a^2*c^8 + 2*I*a^2*c^7*d + 2*a^2*c^6 *d^2 + 6*I*a^2*c^5*d^3 + 6*I*a^2*c^3*d^5 - 2*a^2*c^2*d^6 + 2*I*a^2*c*d^7 - a^2*d^8)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^8 + 4*I*a^2*c^7*d - 4*a^2*c^6*d^2 + 4*I*a^2*c^5*d^3 - 10*a^2*c^4*d^4 - 4*I*a^2*c^3*d^5 - 4*a^2*c^2*d^6 - 4* I*a^2*c*d^7 + a^2*d^8)*f*e^(4*I*f*x + 4*I*e))*sqrt(1/16*I/((-I*a^4*c^5 - 5 *a^4*c^4*d + 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 - 5*I*a^4*c*d^4 - a^4*d^5)* f^2))*log(-2*(4*((-I*a^2*c^3 - 3*a^2*c^2*d + 3*I*a^2*c*d^2 + a^2*d^3)*f...
\[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=- \frac {\int \frac {1}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 4 i c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{4}{\left (e + f x \right )} - 2 i d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx}{a^{2}} \]
-Integral(1/(c**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 - 2*I*c**2*sqrt (c + d*tan(e + f*x))*tan(e + f*x) - c**2*sqrt(c + d*tan(e + f*x)) + 2*c*d* sqrt(c + d*tan(e + f*x))*tan(e + f*x)**3 - 4*I*c*d*sqrt(c + d*tan(e + f*x) )*tan(e + f*x)**2 - 2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x) + d**2*sqr t(c + d*tan(e + f*x))*tan(e + f*x)**4 - 2*I*d**2*sqrt(c + d*tan(e + f*x))* tan(e + f*x)**3 - d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x)/a**2
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (281) = 562\).
Time = 1.12 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {1}{24} \, d^{3} {\left (\frac {6 \, {\left (2 i \, c^{2} - 14 \, c d - 47 i \, d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (a^{2} c^{4} d^{3} f + 4 i \, a^{2} c^{3} d^{4} f - 6 \, a^{2} c^{2} d^{5} f - 4 i \, a^{2} c d^{6} f + a^{2} d^{7} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {16 \, {\left (12 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} - 6 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} d + d^{2}\right )}}{{\left (a^{2} c^{6} f + 2 i \, a^{2} c^{5} d f + a^{2} c^{4} d^{2} f + 4 i \, a^{2} c^{3} d^{3} f - a^{2} c^{2} d^{4} f + 2 i \, a^{2} c d^{5} f - a^{2} d^{6} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} + \frac {48 \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{-4 \, {\left (-i \, a^{2} c^{2} d^{3} f - 2 \, a^{2} c d^{4} f + i \, a^{2} d^{5} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {3 \, {\left (2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} + 13 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d - 17 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d + 15 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{2}\right )}}{{\left (a^{2} c^{4} d^{2} f + 4 i \, a^{2} c^{3} d^{3} f - 6 \, a^{2} c^{2} d^{4} f - 4 i \, a^{2} c d^{5} f + a^{2} d^{6} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}}\right )} \]
-1/24*d^3*(6*(2*I*c^2 - 14*c*d - 47*I*d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^ 2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/((a^2*c^4*d^3*f + 4*I*a^2*c^3*d^4*f - 6*a^2*c^2*d^ 5*f - 4*I*a^2*c*d^6*f + a^2*d^7*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - 16*(12*(d*tan(f*x + e) + c)*c + c^2 - 6*(I*d*ta n(f*x + e) + I*c)*d + d^2)/((a^2*c^6*f + 2*I*a^2*c^5*d*f + a^2*c^4*d^2*f + 4*I*a^2*c^3*d^3*f - a^2*c^2*d^4*f + 2*I*a^2*c*d^5*f - a^2*d^6*f)*(d*tan(f *x + e) + c)^(3/2)) + 48*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt (-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^ 2))))/((4*I*a^2*c^2*d^3*f + 8*a^2*c*d^4*f - 4*I*a^2*d^5*f)*sqrt(-2*c + 2*s qrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - 3*(2*(d*tan(f*x + e) + c)^(3/2)*c - 2*sqrt(d*tan(f*x + e) + c)*c^2 + 13*I*(d*tan(f*x + e) + c)^( 3/2)*d - 17*I*sqrt(d*tan(f*x + e) + c)*c*d + 15*sqrt(d*tan(f*x + e) + c)*d ^2)/((a^2*c^4*d^2*f + 4*I*a^2*c^3*d^3*f - 6*a^2*c^2*d^4*f - 4*I*a^2*c*d^5* f + a^2*d^6*f)*(d*tan(f*x + e) - I*d)^2))
Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]