Integrand size = 32, antiderivative size = 444 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} (c-i d)^{5/2} f}-\frac {1}{5 (i c-d) f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 i c-21 d}{30 a (c+i d)^2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {5 c^2+30 i c d-89 d^2}{20 a^2 (i c-d)^3 f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}+\frac {d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{60 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \] Output:
-1/8*I*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)/(c-I*d)^(5/2)/f-1/5/(I*c-d)/f/(a+I*a*t an(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2)+1/30*(5*I*c-21*d)/a/(c+I*d)^2/f/(a +I*a*tan(f*x+e))^(3/2)/(c+d*tan(f*x+e))^(3/2)+1/20*(5*c^2+30*I*c*d-89*d^2) /a^2/(I*c-d)^3/f/(a+I*a*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(3/2)+1/60*d*(1 5*c^3+85*I*c^2*d-221*c*d^2+361*I*d^3)*(a+I*a*tan(f*x+e))^(1/2)/a^3/(c-I*d) /(c+I*d)^4/f/(c+d*tan(f*x+e))^(3/2)+1/60*d*(15*c^4+80*I*c^3*d-182*c^2*d^2+ 1224*I*c*d^3+707*d^4)*(a+I*a*tan(f*x+e))^(1/2)/a^3/(c-I*d)^2/(c+I*d)^5/f/( c+d*tan(f*x+e))^(1/2)
Time = 5.81 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {\frac {3 a^4 (-i c+d)}{f (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}+\frac {a^3 (-5 i c+21 d)}{2 f (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac {3 a^2 \left (-5 i c^2+30 c d+89 i d^2\right )}{4 (c+i d) f \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {i a \left (\frac {15 \sqrt {2} a (c+i d)^5 \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a (c-i d)}}-\frac {2 d \left (c^2+d^2\right ) \left (-15 i c^3+85 c^2 d+221 i c d^2+361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}}+\frac {2 i d \left (15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{8 (c+i d) \left (c^2+d^2\right )^2 f}}{15 a^4 (c+i d)^2} \] Input:
Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]
Output:
-1/15*((3*a^4*((-I)*c + d))/(f*(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)) + (a^3*((-5*I)*c + 21*d))/(2*f*(a + I*a*Tan[e + f*x])^(3/2)* (c + d*Tan[e + f*x])^(3/2)) + (3*a^2*((-5*I)*c^2 + 30*c*d + (89*I)*d^2))/( 4*(c + I*d)*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + ((I /8)*a*((15*Sqrt[2]*a*(c + I*d)^5*ArcTan[(Sqrt[-(a*(c - I*d))]*Sqrt[a + I*a *Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-(a*(c - I*d)) ] - (2*d*(c^2 + d^2)*((-15*I)*c^3 + 85*c^2*d + (221*I)*c*d^2 + 361*d^3)*Sq rt[a + I*a*Tan[e + f*x]])/(c + d*Tan[e + f*x])^(3/2) + ((2*I)*d*(15*c^4 + (80*I)*c^3*d - 182*c^2*d^2 + (1224*I)*c*d^3 + 707*d^4)*Sqrt[a + I*a*Tan[e + f*x]])/Sqrt[c + d*Tan[e + f*x]]))/((c + I*d)*(c^2 + d^2)^2*f))/(a^4*(c + I*d)^2)
Time = 2.68 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.16, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 4042, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4081, 27, 3042, 4081, 27, 3042, 4027, 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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle -\frac {\int -\frac {a (5 i c-13 d)+8 i a d \tan (e+f x)}{2 (i \tan (e+f x) a+a)^{3/2} (c+d \tan (e+f x))^{5/2}}dx}{5 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (5 i c-13 d)+8 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^{3/2} (c+d \tan (e+f x))^{5/2}}dx}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (5 i c-13 d)+8 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^{3/2} (c+d \tan (e+f x))^{5/2}}dx}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-\frac {\int \frac {3 \left (\left (5 c^2+20 i d c-47 d^2\right ) a^2+2 (5 c+21 i d) d \tan (e+f x) a^2\right )}{2 \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{5/2}}dx}{3 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (5 c^2+20 i d c-47 d^2\right ) a^2+2 (5 c+21 i d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (5 c^2+20 i d c-47 d^2\right ) a^2+2 (5 c+21 i d) d \tan (e+f x) a^2}{\sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {-\frac {\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^3-35 d c^2-135 i d^2 c+361 d^3\right ) a^3+4 d \left (5 i c^2-30 d c-89 i d^2\right ) \tan (e+f x) a^3\right )}{2 (c+d \tan (e+f x))^{5/2}}dx}{a^2 (-d+i c)}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^3-35 d c^2-135 i d^2 c+361 d^3\right ) a^3+4 d \left (5 i c^2-30 d c-89 i d^2\right ) \tan (e+f x) a^3\right )}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (5 i c^3-35 d c^2-135 i d^2 c+361 d^3\right ) a^3+4 d \left (5 i c^2-30 d c-89 i d^2\right ) \tan (e+f x) a^3\right )}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (15 i c^4-90 d c^3-260 i d^2 c^2+502 d^3 c-707 i d^4\right ) a^4+2 d \left (15 i c^3-85 d c^2-221 i d^2 c-361 d^3\right ) \tan (e+f x) a^4\right )}{2 (c+d \tan (e+f x))^{3/2}}dx}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (15 i c^4-90 d c^3-260 i d^2 c^2+502 d^3 c-707 i d^4\right ) a^4+2 d \left (15 i c^3-85 d c^2-221 i d^2 c-361 d^3\right ) \tan (e+f x) a^4\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (\left (15 i c^4-90 d c^3-260 i d^2 c^2+502 d^3 c-707 i d^4\right ) a^4+2 d \left (15 i c^3-85 d c^2-221 i d^2 c-361 d^3\right ) \tan (e+f x) a^4\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4081 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {2 \int \frac {15 a^5 (i c-d)^5 \sqrt {i \tan (e+f x) a+a}}{2 \sqrt {c+d \tan (e+f x)}}dx}{a \left (c^2+d^2\right )}+\frac {2 a^4 d \left (15 i c^4-80 c^3 d-182 i c^2 d^2-1224 c d^3+707 i d^4\right ) \sqrt {a+i a \tan (e+f x)}}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {15 a^4 (-d+i c)^5 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^4 d \left (15 i c^4-80 c^3 d-182 i c^2 d^2-1224 c d^3+707 i d^4\right ) \sqrt {a+i a \tan (e+f x)}}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {15 a^4 (-d+i c)^5 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^4 d \left (15 i c^4-80 c^3 d-182 i c^2 d^2-1224 c d^3+707 i d^4\right ) \sqrt {a+i a \tan (e+f x)}}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\frac {2 a^4 d \left (15 i c^4-80 c^3 d-182 i c^2 d^2-1224 c d^3+707 i d^4\right ) \sqrt {a+i a \tan (e+f x)}}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {30 i a^6 (-d+i c)^5 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f \left (c^2+d^2\right )}}{3 a \left (c^2+d^2\right )}+\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {a^2 \left (5 i c^2-30 c d-89 i d^2\right )}{f (c+i d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 a^3 d \left (15 i c^3-85 c^2 d-221 i c d^2-361 d^3\right ) \sqrt {a+i a \tan (e+f x)}}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {2 a^4 d \left (15 i c^4-80 c^3 d-182 i c^2 d^2-1224 c d^3+707 i d^4\right ) \sqrt {a+i a \tan (e+f x)}}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {15 i \sqrt {2} a^{9/2} (-d+i c)^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d} \left (c^2+d^2\right )}}{3 a \left (c^2+d^2\right )}}{2 a^2 (-d+i c)}}{2 a^2 (-d+i c)}-\frac {a (5 c+21 i d)}{3 f (c+i d) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}}{10 a^2 (-d+i c)}-\frac {1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}\) |
Input:
Int[1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]
Output:
-1/5*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2 )) + (-1/3*(a*(5*c + (21*I)*d))/((c + I*d)*f*(a + I*a*Tan[e + f*x])^(3/2)* (c + d*Tan[e + f*x])^(3/2)) - ((a^2*((5*I)*c^2 - 30*c*d - (89*I)*d^2))/((c + I*d)*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + ((2*a^3 *d*((15*I)*c^3 - 85*c^2*d - (221*I)*c*d^2 - 361*d^3)*Sqrt[a + I*a*Tan[e + f*x]])/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (((-15*I)*Sqrt[2]*a^ (9/2)*(I*c - d)^5*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt [c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[c - I*d]*(c^2 + d^2)*f) + (2 *a^4*d*((15*I)*c^4 - 80*c^3*d - (182*I)*c^2*d^2 - 1224*c*d^3 + (707*I)*d^4 )*Sqrt[a + I*a*Tan[e + f*x]])/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/(3 *a*(c^2 + d^2)))/(2*a^2*(I*c - d)))/(2*a^2*(I*c - d)))/(10*a^2*(I*c - d))
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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Int[((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. 10144 vs. \(2 (374 ) = 748\).
Time = 0.67 (sec) , antiderivative size = 10145, normalized size of antiderivative = 22.85
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(10145\) |
| default | \(\text {Expression too large to display}\) | \(10145\) |
Input:
int(1/(a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERB OSE)
Output:
result too large to display
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1774 vs. \(2 (352) = 704\).
Time = 0.24 (sec) , antiderivative size = 1774, normalized size of antiderivative = 4.00 \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm=" fricas")
Output:
-1/120*(sqrt(2)*(3*c^6 + 6*I*c^5*d + 3*c^4*d^2 + 12*I*c^3*d^3 - 3*c^2*d^4 + 6*I*c*d^5 - 3*d^6 + (23*c^6 + 62*I*c^5*d + 55*c^4*d^2 + 860*I*c^3*d^3 + 3145*c^2*d^4 - 3298*I*c*d^5 - 983*d^6)*e^(10*I*f*x + 10*I*e) + 4*(20*c^6 + 71*I*c^5*d - 20*c^4*d^2 + 590*I*c^3*d^3 + 1240*c^2*d^4 - 385*I*c*d^5 + 13 6*d^6)*e^(8*I*f*x + 8*I*e) + 3*(35*c^6 + 142*I*c^5*d - 129*c^4*d^2 + 636*I *c^3*d^3 + 389*c^2*d^4 + 654*I*c*d^5 + 393*d^6)*e^(6*I*f*x + 6*I*e) + (65* c^6 + 254*I*c^5*d - 251*c^4*d^2 + 508*I*c^3*d^3 - 697*c^2*d^4 + 254*I*c*d^ 5 - 381*d^6)*e^(4*I*f*x + 4*I*e) + 4*(5*c^6 + 14*I*c^5*d + c^4*d^2 + 28*I* c^3*d^3 - 13*c^2*d^4 + 14*I*c*d^5 - 9*d^6)*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)) - 30*((-I*a^3*c^9 + a^3*c^8*d - 4*I*a^3*c^7*d^2 + 4*a^3*c^6*d^3 - 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 - 4*I*a^3*c^3*d^6 + 4*a^3* c^2*d^7 - I*a^3*c*d^8 + a^3*d^9)*f*e^(9*I*f*x + 9*I*e) + 2*(-I*a^3*c^9 + 3 *a^3*c^8*d + 8*a^3*c^6*d^3 + 6*I*a^3*c^5*d^4 + 6*a^3*c^4*d^5 + 8*I*a^3*c^3 *d^6 + 3*I*a^3*c*d^8 - a^3*d^9)*f*e^(7*I*f*x + 7*I*e) + (-I*a^3*c^9 + 5*a^ 3*c^8*d + 8*I*a^3*c^7*d^2 + 14*I*a^3*c^5*d^4 - 14*a^3*c^4*d^5 - 8*a^3*c^2* d^7 - 5*I*a^3*c*d^8 + a^3*d^9)*f*e^(5*I*f*x + 5*I*e))*sqrt(1/8*I/((-I*a^5* c^5 - 5*a^5*c^4*d + 10*I*a^5*c^3*d^2 + 10*a^5*c^2*d^3 - 5*I*a^5*c*d^4 - a^ 5*d^5)*f^2))*log(-4*(I*a^3*c^3 + 3*a^3*c^2*d - 3*I*a^3*c*d^2 - a^3*d^3)*f* sqrt(1/8*I/((-I*a^5*c^5 - 5*a^5*c^4*d + 10*I*a^5*c^3*d^2 + 10*a^5*c^2*d...
Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+I*a*tan(f*x+e))**(5/2)/(c+d*tan(f*x+e))**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm=" maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm=" giac")
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. \[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int(1/((a + a*tan(e + f*x)*1i)^(5/2)*(c + d*tan(e + f*x))^(5/2)),x)
Output:
int(1/((a + a*tan(e + f*x)*1i)^(5/2)*(c + d*tan(e + f*x))^(5/2)), x)
\[ \int \frac {1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {\int \frac {1}{\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{4} d^{2}+2 \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3} c d -2 \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3} d^{2} i +\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2} c^{2}-4 \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2} c d i -\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2} d^{2}-2 \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right ) c^{2} i -2 \sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right ) c d -\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {d \tan \left (f x +e \right )+c}\, c^{2}}d x}{\sqrt {a}\, a^{2}} \] Input:
int(1/(a+I*a*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(5/2),x)
Output:
( - int(1/(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x)* *4*d**2 + 2*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x) **3*c*d - 2*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x) **3*d**2*i + sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f*x )**2*c**2 - 4*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f* x)**2*c*d*i - sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f* x)**2*d**2 - 2*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f *x)*c**2*i - 2*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f *x)*c*d - sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*c**2),x))/(sqr t(a)*a**2)