\(\int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx\) [1260]

Optimal result

Integrand size = 27, antiderivative size = 314 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{3/2} f}+\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{3/2} f}-\frac {b^{5/2} \left (4 a b c-7 a^2 d-3 b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \] Output:

-I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/(c-I*d)^(3/2)/f 
+I*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(a+I*b)^2/(c+I*d)^(3/2)/f 
-b^(5/2)*(-7*a^2*d+4*a*b*c-3*b^2*d)*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2) 
/(-a*d+b*c)^(1/2))/(a^2+b^2)^2/(-a*d+b*c)^(5/2)/f-d*(2*a^2*d^2+b^2*(c^2+3* 
d^2))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)-b^2/(a^2+b 
^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 5.68 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=-\frac {\frac {\frac {(a+i b)^2 (-i c+d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {(a-i b)^2 (i c+d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {b^{5/2} \left (-4 a b c+7 a^2 d+3 b^2 d\right ) \left (c^2+d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}}{\left (a^2+b^2\right ) (-b c+a d) \left (c^2+d^2\right )}+\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{(b c-a d) \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {b^2}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}}{\left (a^2+b^2\right ) (b c-a d) f} \] Input:

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

Output:

-(((((a + I*b)^2*((-I)*c + d)*(b*c - a*d)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x 
]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((a - I*b)^2*(I*c + d)*(b*c - a*d)^2*Ar 
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (b^(5/2)*(- 
4*a*b*c + 7*a^2*d + 3*b^2*d)*(c^2 + d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e 
 + f*x]])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/((a^2 + b^2)*(-(b*c) + a*d)*( 
c^2 + d^2)) + (d*(2*a^2*d^2 + b^2*(c^2 + 3*d^2)))/((b*c - a*d)*(c^2 + d^2) 
*Sqrt[c + d*Tan[e + f*x]]) + b^2/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + 
f*x]]))/((a^2 + b^2)*(b*c - a*d)*f))
 

Rubi [A] (warning: unable to verify)

Time = 2.85 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.21, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {\frac {\int \frac {2 \left ((b c-a d)^2 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^2 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \int \frac {(b c-a d)^2 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^2 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \int \frac {(b c-a d)^2 \left (c a^2-2 b d a-b^2 c\right )-(b c-a d)^2 \left (d a^2+2 b c a-b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c-i d) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c-i d) (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c+i d) (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a+i b)^2 (c+i d) (b c-a d)^2 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b)^2 (c-i d) (b c-a d)^2 \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a-i b)^2 (c-i d) (b c-a d)^2 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b)^2 (c+i d) (b c-a d)^2 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a-i b)^2 (c-i d) (b c-a d)^2 \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b)^2 (c+i d) (b c-a d)^2 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {2 b^3 \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {2 b^{5/2} \left (c^2+d^2\right ) \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) \sqrt {b c-a d}}+\frac {2 \left (\frac {(a+i b)^2 (c+i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b)^2 (c-i d) (b c-a d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{a^2+b^2}}{\left (c^2+d^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)}\)

Input:

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

Output:

-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f 
*x]])) + (((2*(((a + I*b)^2*(c + I*d)*(b*c - a*d)^2*ArcTan[Tan[e + f*x]/Sq 
rt[c - I*d]])/(Sqrt[c - I*d]*f) + ((a - I*b)^2*(c - I*d)*(b*c - a*d)^2*Arc 
Tan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f)))/(a^2 + b^2) - (2*b^(5 
/2)*(4*a*b*c - 7*a^2*d - 3*b^2*d)*(c^2 + d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d* 
Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]*f))/((b*c - 
a*d)*(c^2 + d^2)) - (2*d*(2*a^2*d^2 + b^2*(c^2 + 3*d^2)))/((b*c - a*d)*(c^ 
2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/(2*(a^2 + b^2)*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

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]
 

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 4020

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]
 

rule 4022

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]
 

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 4117

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

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])))
 

rule 4136

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) 
+ (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^ 
n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ 
(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ 
e + f*x]^2)/(a + b*Tan[e + f*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] & 
&  !GtQ[n, 0] &&  !LeQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(12888\) vs. \(2(280)=560\).

Time = 0.48 (sec) , antiderivative size = 12889, normalized size of antiderivative = 41.05

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(12889\)
default	\(\text {Expression too large to display}\)	\(12889\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[3,19,8]%%%}+%%%{8,[3,17,8]%%%}+%%%{28,[3,15,8]%%%}+ 
%%%{56,[3
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(2*sqrt(tan(e + f*x)*d + c) + int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)** 
4*a**2*b**2*d**4 + 4*tan(e + f*x)**4*a*b**3*c*d**3 + 4*tan(e + f*x)**4*b** 
4*c**2*d**2 + 2*tan(e + f*x)**3*a**3*b*d**4 + 10*tan(e + f*x)**3*a**2*b**2 
*c*d**3 + 16*tan(e + f*x)**3*a*b**3*c**2*d**2 + 8*tan(e + f*x)**3*b**4*c** 
3*d + tan(e + f*x)**2*a**4*d**4 + 8*tan(e + f*x)**2*a**3*b*c*d**3 + 21*tan 
(e + f*x)**2*a**2*b**2*c**2*d**2 + 20*tan(e + f*x)**2*a*b**3*c**3*d + 4*ta 
n(e + f*x)**2*b**4*c**4 + 2*tan(e + f*x)*a**4*c*d**3 + 10*tan(e + f*x)*a** 
3*b*c**2*d**2 + 16*tan(e + f*x)*a**2*b**2*c**3*d + 8*tan(e + f*x)*a*b**3*c 
**4 + a**4*c**2*d**2 + 4*a**3*b*c**3*d + 4*a**2*b**2*c**4),x)*tan(e + f*x) 
**2*a**3*b*d**4*f + 6*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*a**2*b 
**2*d**4 + 4*tan(e + f*x)**4*a*b**3*c*d**3 + 4*tan(e + f*x)**4*b**4*c**2*d 
**2 + 2*tan(e + f*x)**3*a**3*b*d**4 + 10*tan(e + f*x)**3*a**2*b**2*c*d**3 
+ 16*tan(e + f*x)**3*a*b**3*c**2*d**2 + 8*tan(e + f*x)**3*b**4*c**3*d + ta 
n(e + f*x)**2*a**4*d**4 + 8*tan(e + f*x)**2*a**3*b*c*d**3 + 21*tan(e + f*x 
)**2*a**2*b**2*c**2*d**2 + 20*tan(e + f*x)**2*a*b**3*c**3*d + 4*tan(e + f* 
x)**2*b**4*c**4 + 2*tan(e + f*x)*a**4*c*d**3 + 10*tan(e + f*x)*a**3*b*c**2 
*d**2 + 16*tan(e + f*x)*a**2*b**2*c**3*d + 8*tan(e + f*x)*a*b**3*c**4 + a* 
*4*c**2*d**2 + 4*a**3*b*c**3*d + 4*a**2*b**2*c**4),x)*tan(e + f*x)**2*a**2 
*b**2*c*d**3*f + 12*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*a**2*b** 
2*d**4 + 4*tan(e + f*x)**4*a*b**3*c*d**3 + 4*tan(e + f*x)**4*b**4*c**2*...