Integrand size = 32, antiderivative size = 254 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac {(5 i c-3 d) \sqrt {c+d \tan (e+f x)}}{30 a (c+i d) f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (20 c d-i \left (15 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{60 a^2 (c+i d)^2 f \sqrt {a+i a \tan (e+f x)}} \] Output:
-1/8*I*(c-I*d)^(1/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d )^(1/2)/(a+I*a*tan(f*x+e))^(1/2))*2^(1/2)/a^(5/2)/f+1/5*I*(c+d*tan(f*x+e)) ^(1/2)/f/(a+I*a*tan(f*x+e))^(5/2)+1/30*(5*I*c-3*d)*(c+d*tan(f*x+e))^(1/2)/ a/(c+I*d)/f/(a+I*a*tan(f*x+e))^(3/2)-1/60*(20*c*d-I*(15*c^2+3*d^2))*(c+d*t an(f*x+e))^(1/2)/a^2/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^(1/2)
Time = 3.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\frac {i \left (\frac {15 \sqrt {2} \sqrt {-a (c-i d)} (c+i d) \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{a}+\frac {24 i a d \sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}}+\frac {2 \left (15 c^2+20 i c d+3 d^2\right ) \sqrt {c+d \tan (e+f x)}}{(c+i d) \sqrt {a+i a \tan (e+f x)}}+\frac {4 (-5 i c+3 d) \sqrt {c+d \tan (e+f x)}}{(-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}+\frac {24 a^2 (c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}}\right )}{120 a^2 (c+i d) f} \] Input:
Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2),x]
Output:
((I/120)*((15*Sqrt[2]*Sqrt[-(a*(c - I*d))]*(c + I*d)*ArcTan[(Sqrt[-(a*(c - I*d))]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])]) /a + ((24*I)*a*d*Sqrt[c + d*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^(3/2) + (2*(15*c^2 + (20*I)*c*d + 3*d^2)*Sqrt[c + d*Tan[e + f*x]])/((c + I*d)*Sqrt [a + I*a*Tan[e + f*x]]) + (4*((-5*I)*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(( -I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]]) + (24*a^2*(c + d*Tan[e + f* x])^(3/2))/(a + I*a*Tan[e + f*x])^(5/2)))/(a^2*(c + I*d)*f)
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.
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4040 |
| \(\displaystyle \frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}-\frac {\int -\frac {a (5 c-i d)+4 a d \tan (e+f x)}{(i \tan (e+f x) a+a)^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{10 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a (5 c-i d)+4 a d \tan (e+f x)}{(i \tan (e+f x) a+a)^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (5 c-i d)+4 a d \tan (e+f x)}{(i \tan (e+f x) a+a)^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{2 \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{3 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (15 i c^2-10 d c+9 i d^2\right ) a^2+2 (5 i c-3 d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {a^2 \left (10 c d-i \left (15 c^2+9 d^2\right )\right )-2 a^2 (5 i c-3 d) d \tan (e+f x)}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}dx}{6 a^2 (-d+i c)}+\frac {a (-3 d+5 i c) \sqrt {c+d \tan (e+f x)}}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2}}}{10 a^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}\) |
Input:
Int[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2),x]
Output:
$Aborted
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_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*Sqrt[(c_.) + (d_.)*tan[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[(-b)*(a + b*Tan[e + f*x])^m*(Sqrt[c + d *Tan[e + f*x]]/(2*a*f*m)), x] + Simp[1/(4*a^2*m) Int[(a + b*Tan[e + f*x]) ^(m + 1)*(Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/Sqrt[c + d*Ta n[e + f*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] && IntegersQ[2*m]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2225 vs. \(2 (206 ) = 412\).
Time = 0.53 (sec) , antiderivative size = 2226, normalized size of antiderivative = 8.76
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2226\) |
| default | \(\text {Expression too large to display}\) | \(2226\) |
Input:
int((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(5/2),x,method=_RETURNVERBOS E)
Output:
1/240/f*(c+d*tan(f*x+e))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^3*(220*I*c^2*( a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*tan(f*x+e)^2+12*I*(a*(c+d*tan(f *x+e))*(1+I*tan(f*x+e)))^(1/2)*d^2*tan(f*x+e)^2-304*(a*(c+d*tan(f*x+e))*(1 +I*tan(f*x+e)))^(1/2)*c*d*tan(f*x+e)^2+60*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d +3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*ta n(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*c^2*tan(f*x+e )^3-60*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I* d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2 ^(1/2)*(-a*(I*d-c))^(1/2)*d^2*tan(f*x+e)^3-60*ln((3*a*c+I*a*tan(f*x+e)*c-I *a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+ I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*c^2*tan(f *x+e)+60*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*( I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e))) *2^(1/2)*(-a*(I*d-c))^(1/2)*d^2*tan(f*x+e)-30*ln((3*a*c+I*a*tan(f*x+e)*c-I *a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+ I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*c*d-80*I* (a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*c*d*tan(f*x+e)^3+15*I*ln((3*a* c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a* (c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d -c))^(1/2)*c^2-15*I*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (191) = 382\).
Time = 0.12 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\frac {{\left (15 \, \sqrt {\frac {1}{2}} {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {c - i \, d}{a^{5} f^{2}}} e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (2 i \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {c - i \, d}{a^{5} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - 15 \, \sqrt {\frac {1}{2}} {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {c - i \, d}{a^{5} f^{2}}} e^{\left (5 i \, f x + 5 i \, e\right )} \log \left (-2 i \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {c - i \, d}{a^{5} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - \sqrt {2} {\left (-3 i \, c^{2} + 6 \, c d + 3 i \, d^{2} + {\left (-23 i \, c^{2} + 34 \, c d + 3 i \, d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 2 \, {\left (17 i \, c^{2} - 27 \, c d - 6 i \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (7 i \, c^{2} - 13 \, c d - 6 i \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{120 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f} \] Input:
integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="fr icas")
Output:
1/120*(15*sqrt(1/2)*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*sqrt(-(c - I*d)/(a ^5*f^2))*e^(5*I*f*x + 5*I*e)*log(2*I*sqrt(1/2)*a^3*f*sqrt(-(c - I*d)/(a^5* f^2))*e^(I*f*x + I*e) + sqrt(2)*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(a/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I* f*x + 2*I*e) + 1)) - 15*sqrt(1/2)*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*sqrt (-(c - I*d)/(a^5*f^2))*e^(5*I*f*x + 5*I*e)*log(-2*I*sqrt(1/2)*a^3*f*sqrt(- (c - I*d)/(a^5*f^2))*e^(I*f*x + I*e) + sqrt(2)*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(a/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)) - sqrt(2)*(-3*I*c^2 + 6*c*d + 3*I*d^2 + (-23*I*c^2 + 34*c*d + 3*I*d^2)*e^(6*I*f*x + 6*I*e) - 2*(17*I*c^2 - 27*c*d - 6*I*d^2)*e^(4*I*f*x + 4*I*e) - 2*(7*I*c^2 - 13*c*d - 6*I*d^2)*e^(2*I*f*x + 2*I*e))*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(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-5*I*f*x - 5*I*e)/((a^3*c ^2 + 2*I*a^3*c*d - a^3*d^2)*f)
\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((c+d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**(5/2),x)
Output:
Integral(sqrt(c + d*tan(e + f*x))/(I*a*(tan(e + f*x) - I))**(5/2), x)
Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="ma xima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeRecursive ass umption s
Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:
int((c + d*tan(e + f*x))^(1/2)/(a + a*tan(e + f*x)*1i)^(5/2),x)
Output:
int((c + d*tan(e + f*x))^(1/2)/(a + a*tan(e + f*x)*1i)^(5/2), x)
\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx=-\frac {\int \frac {\sqrt {d \tan \left (f x +e \right )+c}}{\sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}-2 \sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right ) i -\sqrt {\tan \left (f x +e \right ) i +1}}d x}{\sqrt {a}\, a^{2}} \] Input:
int((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(5/2),x)
Output:
( - int(sqrt(tan(e + f*x)*d + c)/(sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)**2 - 2*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x)*i - sqrt(tan(e + f*x)*i + 1)),x ))/(sqrt(a)*a**2)