\(\int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx\) [1311]

Optimal result

Integrand size = 29, antiderivative size = 417 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} (c-i d)^{3/2} f}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} (c+i d)^{3/2} f}-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \] Output:

-I*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c+d*tan(f*x 
+e))^(1/2))/(a-I*b)^(5/2)/(c-I*d)^(3/2)/f+I*arctanh((c+I*d)^(1/2)*(a+b*tan 
(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(a+I*b)^(5/2)/(c+I*d) 
^(3/2)/f-2/3*b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))^(3/2)/(c+d*tan(f* 
x+e))^(1/2)-4/3*b^2*(-5*a^2*d+3*a*b*c-2*b^2*d)/(a^2+b^2)^2/(-a*d+b*c)^2/f/ 
(a+b*tan(f*x+e))^(1/2)/(c+d*tan(f*x+e))^(1/2)+2/3*d*(3*a^4*d^3-6*a*b^3*c*( 
c^2+d^2)+b^4*d*(5*c^2+8*d^2)+a^2*b^2*d*(11*c^2+17*d^2))*(a+b*tan(f*x+e))^( 
1/2)/(a^2+b^2)^2/(-a*d+b*c)^3/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 6.37 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {2 \left (\frac {1}{2} b^2 \left (4 b^2 d-3 a (b c-a d)\right )-a \left (-2 a b^2 d+\frac {3}{2} b^2 (b c-a d)\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {3 (b c-a d)^3 \left (\frac {(a+i b)^2 (i c-d) \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+i b} \sqrt {-c+i d}}+\frac {(a-i b)^2 (i c+d) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} \sqrt {c+i d}}\right )}{4 (-b c+a d) \left (c^2+d^2\right ) f}-\frac {2 \left (-c \left (-\frac {3}{2} a b d (b c-a d)^2+b^2 c d \left (3 a b c-5 a^2 d-2 b^2 d\right )\right )+\frac {1}{4} d^2 \left (-6 a^3 b c d-6 a b^3 c d+3 a^4 d^2-b^4 \left (3 c^2-8 d^2\right )+a^2 b^2 \left (3 c^2+17 d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{(-b c+a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\right )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \] Input:

Integrate[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)),x]
 

Output:

(-2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + 
d*Tan[e + f*x]]) - (2*((-2*((b^2*(4*b^2*d - 3*a*(b*c - a*d)))/2 - a*(-2*a* 
b^2*d + (3*b^2*(b*c - a*d))/2)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan 
[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*((-3*(b*c - a*d)^3*(((a + I*b)^2 
*(I*c - d)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I* 
b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((a - I*b 
)^2*(I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I 
*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])))/(4*(-(b*c) 
 + a*d)*(c^2 + d^2)*f) - (2*(-(c*((-3*a*b*d*(b*c - a*d)^2)/2 + b^2*c*d*(3* 
a*b*c - 5*a^2*d - 2*b^2*d))) + (d^2*(-6*a^3*b*c*d - 6*a*b^3*c*d + 3*a^4*d^ 
2 - b^4*(3*c^2 - 8*d^2) + a^2*b^2*(3*c^2 + 17*d^2)))/4)*Sqrt[a + b*Tan[e + 
 f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])))/((a^2 + 
b^2)*(b*c - a*d))))/(3*(a^2 + b^2)*(b*c - a*d))
 

Rubi [A] (verified)

Time = 2.99 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.26, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 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.

\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 4052

\(\displaystyle -\frac {2 \int -\frac {-3 d a^2+3 b c a-4 b^2 d \tan ^2(e+f x)-4 b^2 d-3 b (b c-a d) \tan (e+f x)}{2 (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a-4 b^2 d \tan ^2(e+f x)-4 b^2 d-3 b (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a-4 b^2 d \tan (e+f x)^2-4 b^2 d-3 b (b c-a d) \tan (e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {2 \int \frac {-3 d^2 a^4+6 b c d a^3-b^2 \left (3 c^2+17 d^2\right ) a^2+6 b^3 c d a+6 b (b c-a d)^2 \tan (e+f x) a+4 b^2 d \left (-5 d a^2+3 b c a-2 b^2 d\right ) \tan ^2(e+f x)+b^4 \left (3 c^2-8 d^2\right )}{2 \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-b^2 \left (3 c^2+17 d^2\right ) a^2+6 b^3 c d a+6 b (b c-a d)^2 \tan (e+f x) a+4 b^2 d \left (-5 d a^2+3 b c a-2 b^2 d\right ) \tan ^2(e+f x)+b^4 \left (3 c^2-8 d^2\right )}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {-3 d^2 a^4+6 b c d a^3-b^2 \left (3 c^2+17 d^2\right ) a^2+6 b^3 c d a+6 b (b c-a d)^2 \tan (e+f x) a+4 b^2 d \left (-5 d a^2+3 b c a-2 b^2 d\right ) \tan (e+f x)^2+b^4 \left (3 c^2-8 d^2\right )}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {\frac {2 \int -\frac {3 \left ((b c-a d)^3 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^3 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {3 \int \frac {(b c-a d)^3 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^3 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {3 \int \frac {(b c-a d)^3 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^3 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{\left (a^2+b^2\right ) (b c-a d)}-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4099

\(\displaystyle -\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {3 \left (\frac {1}{2} (a-i b)^2 (c-i d) (b c-a d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (b c-a d)^3 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {3 \left (\frac {1}{2} (a-i b)^2 (c-i d) (b c-a d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (b c-a d)^3 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4098

\(\displaystyle -\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {3 \left (\frac {(a+i b)^2 (c+i d) (b c-a d)^3 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {(a-i b)^2 (c-i d) (b c-a d)^3 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {3 \left (\frac {(a-i b)^2 (c-i d) (b c-a d)^3 \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {(a+i b)^2 (c+i d) (b c-a d)^3 \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {3 \left (\frac {i (a-i b)^2 (c-i d) (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b} \sqrt {c+i d}}-\frac {i (a+i b)^2 (c+i d) (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b} \sqrt {c-i d}}\right )}{\left (c^2+d^2\right ) (b c-a d)}}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}\)

Input:

Int[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)),x]
 

Output:

(-2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + 
d*Tan[e + f*x]]) + ((-4*b^2*(3*a*b*c - 5*a^2*d - 2*b^2*d))/((a^2 + b^2)*(b 
*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - ((-3*(((- 
I)*(a + I*b)^2*(c + I*d)*(b*c - a*d)^3*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*T 
an[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*Sq 
rt[c - I*d]*f) + (I*(a - I*b)^2*(c - I*d)*(b*c - a*d)^3*ArcTanh[(Sqrt[c + 
I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/ 
(Sqrt[a + I*b]*Sqrt[c + I*d]*f)))/((b*c - a*d)*(c^2 + d^2)) - (2*d*(3*a^4* 
d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(5*c^2 + 8*d^2) + a^2*b^2*d*(11*c^2 + 
17*d^2))*Sqrt[a + b*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*T 
an[e + f*x]]))/((a^2 + b^2)*(b*c - a*d)))/(3*(a^2 + b^2)*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4052

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c 
+ d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 
/((m + 1)*(a^2 + b^2)*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + 
d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - 
 a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / 
; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] 
 && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ 
erQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4098

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[A^2/f   Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* 
x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
 

rule 4099

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(A + I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T 
an[e + f*x]), x], x] + Simp[(A - I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d 
*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A 
, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 
0]
 

rule 4132

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ 
[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [F(-1)]

Timed out.

\[\int \frac {1}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x)
 

Output:

int(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48847 vs. \(2 (355) = 710\).

Time = 57.54 (sec) , antiderivative size = 48847, normalized size of antiderivative = 117.14 \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fr 
icas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a+b*tan(f*x+e))**(5/2)/(c+d*tan(f*x+e))**(3/2),x)
 

Output:

Integral(1/((a + b*tan(e + f*x))**(5/2)*(c + d*tan(e + f*x))**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate(1/((b*tan(f*x + e) + a)^(5/2)*(d*tan(f*x + e) + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate(1/((b*tan(f*x + e) + a)^(5/2)*(d*tan(f*x + e) + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:

int(1/((a + b*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2)),x)
 

Output:

int(1/((a + b*tan(e + f*x))^(5/2)*(c + d*tan(e + f*x))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}}{\tan \left (f x +e \right )^{5} b^{3} d^{2}+3 \tan \left (f x +e \right )^{4} a \,b^{2} d^{2}+2 \tan \left (f x +e \right )^{4} b^{3} c d +3 \tan \left (f x +e \right )^{3} a^{2} b \,d^{2}+6 \tan \left (f x +e \right )^{3} a \,b^{2} c d +\tan \left (f x +e \right )^{3} b^{3} c^{2}+\tan \left (f x +e \right )^{2} a^{3} d^{2}+6 \tan \left (f x +e \right )^{2} a^{2} b c d +3 \tan \left (f x +e \right )^{2} a \,b^{2} c^{2}+2 \tan \left (f x +e \right ) a^{3} c d +3 \tan \left (f x +e \right ) a^{2} b \,c^{2}+a^{3} c^{2}}d x \] Input:

int(1/(a+b*tan(f*x+e))^(5/2)/(c+d*tan(f*x+e))^(3/2),x)
 

Output:

int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a))/(tan(e + f*x)**5*b 
**3*d**2 + 3*tan(e + f*x)**4*a*b**2*d**2 + 2*tan(e + f*x)**4*b**3*c*d + 3* 
tan(e + f*x)**3*a**2*b*d**2 + 6*tan(e + f*x)**3*a*b**2*c*d + tan(e + f*x)* 
*3*b**3*c**2 + tan(e + f*x)**2*a**3*d**2 + 6*tan(e + f*x)**2*a**2*b*c*d + 
3*tan(e + f*x)**2*a*b**2*c**2 + 2*tan(e + f*x)*a**3*c*d + 3*tan(e + f*x)*a 
**2*b*c**2 + a**3*c**2),x)