\(\int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\) [1265]

Optimal result

Integrand size = 27, antiderivative size = 272 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) (c-i d)^{5/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) (c+i d)^{5/2} f}-\frac {2 b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{5/2} f}+\frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 3.66 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\frac {\frac {3 \left (\frac {(-i a+b) (c+i d)^2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {(i a+b) (c-i d)^2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {2 b^{7/2} \left (c^2+d^2\right )^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (-b c+a d) \left (c^2+d^2\right )}-\frac {2 d^2}{(c+d \tan (e+f x))^{3/2}}-\frac {6 d^2 \left (-2 a c d+b \left (3 c^2+d^2\right )\right )}{(b c-a d) \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 (-b c+a d) \left (c^2+d^2\right ) f} \] Input:

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

Output:

((3*((((-I)*a + b)*(c + I*d)^2*(b*c - a*d)^2*ArcTanh[Sqrt[c + d*Tan[e + f* 
x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((I*a + b)*(c - I*d)^2*(b*c - a*d)^2*A 
rcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*b^(7/2) 
*(c^2 + d^2)^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)) - (2*d^2)/(c 
+ d*Tan[e + f*x])^(3/2) - (6*d^2*(-2*a*c*d + b*(3*c^2 + d^2)))/((b*c - a*d 
)*(c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]))/(3*(-(b*c) + a*d)*(c^2 + d^2)*f)
 

Rubi [A] (warning: unable to verify)

Time = 2.52 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.29, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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. \(13981\) vs. \(2(238)=476\).

Time = 0.42 (sec) , antiderivative size = 13982, normalized size of antiderivative = 51.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(13982\)
default	\(\text {Expression too large to display}\)	\(13982\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2),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. \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*tan(f*x+e))/(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:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 120.80 (sec) , antiderivative size = 356674, normalized size of antiderivative = 1311.30 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(log(((((((2*a^2*b^2*d^10*f^4 - b^4*d^10*f^4 - 4*a^2*b^2*c^10*f^4 - a^4*d^ 
10*f^4 + 20*a^4*c^2*d^8*f^4 - 110*a^4*c^4*d^6*f^4 + 100*a^4*c^6*d^4*f^4 - 
25*a^4*c^8*d^2*f^4 + 20*b^4*c^2*d^8*f^4 - 110*b^4*c^4*d^6*f^4 + 100*b^4*c^ 
6*d^4*f^4 - 25*b^4*c^8*d^2*f^4 - 140*a^2*b^2*c^2*d^8*f^4 + 620*a^2*b^2*c^4 
*d^6*f^4 - 640*a^2*b^2*c^6*d^4*f^4 + 130*a^2*b^2*c^8*d^2*f^4 + 20*a*b^3*c* 
d^9*f^4 + 20*a*b^3*c^9*d*f^4 - 20*a^3*b*c*d^9*f^4 - 20*a^3*b*c^9*d*f^4 - 2 
40*a*b^3*c^3*d^7*f^4 + 504*a*b^3*c^5*d^5*f^4 - 240*a*b^3*c^7*d^3*f^4 + 240 
*a^3*b*c^3*d^7*f^4 - 504*a^3*b*c^5*d^5*f^4 + 240*a^3*b*c^7*d^3*f^4)^(1/2) 
- a^2*c^5*f^2 + b^2*c^5*f^2 - 5*a^2*c*d^4*f^2 + 5*b^2*c*d^4*f^2 + 10*a^2*c 
^3*d^2*f^2 - 10*b^2*c^3*d^2*f^2 + 2*a*b*d^5*f^2 + 10*a*b*c^4*d*f^2 - 20*a* 
b*c^2*d^3*f^2)/(a^4*c^10*f^4 + a^4*d^10*f^4 + b^4*c^10*f^4 + b^4*d^10*f^4 
+ 2*a^2*b^2*c^10*f^4 + 2*a^2*b^2*d^10*f^4 + 5*a^4*c^2*d^8*f^4 + 10*a^4*c^4 
*d^6*f^4 + 10*a^4*c^6*d^4*f^4 + 5*a^4*c^8*d^2*f^4 + 5*b^4*c^2*d^8*f^4 + 10 
*b^4*c^4*d^6*f^4 + 10*b^4*c^6*d^4*f^4 + 5*b^4*c^8*d^2*f^4 + 10*a^2*b^2*c^2 
*d^8*f^4 + 20*a^2*b^2*c^4*d^6*f^4 + 20*a^2*b^2*c^6*d^4*f^4 + 10*a^2*b^2*c^ 
8*d^2*f^4))^(1/2)*(((((((2*a^2*b^2*d^10*f^4 - b^4*d^10*f^4 - 4*a^2*b^2*c^1 
0*f^4 - a^4*d^10*f^4 + 20*a^4*c^2*d^8*f^4 - 110*a^4*c^4*d^6*f^4 + 100*a^4* 
c^6*d^4*f^4 - 25*a^4*c^8*d^2*f^4 + 20*b^4*c^2*d^8*f^4 - 110*b^4*c^4*d^6*f^ 
4 + 100*b^4*c^6*d^4*f^4 - 25*b^4*c^8*d^2*f^4 - 140*a^2*b^2*c^2*d^8*f^4 + 6 
20*a^2*b^2*c^4*d^6*f^4 - 640*a^2*b^2*c^6*d^4*f^4 + 130*a^2*b^2*c^8*d^2*...
 

Reduce [F]

\[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(2*sqrt(tan(e + f*x)*d + c) + 6*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x) 
**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e 
 + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c**2*d + 3*tan(e + f*x)*a*c**2*d 
 + tan(e + f*x)*b*c**3 + a*c**3),x)*tan(e + f*x)**2*a*d**3*f + 12*int(sqrt 
(tan(e + f*x)*d + c)/(tan(e + f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3* 
tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2* 
b*c**2*d + 3*tan(e + f*x)*a*c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*tan( 
e + f*x)*a*c*d**2*f + 6*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*b*d* 
*3 + tan(e + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)* 
*2*a*c*d**2 + 3*tan(e + f*x)**2*b*c**2*d + 3*tan(e + f*x)*a*c**2*d + tan(e 
 + f*x)*b*c**3 + a*c**3),x)*a*c**2*d*f + 3*int((sqrt(tan(e + f*x)*d + c)*t 
an(e + f*x)**3)/(tan(e + f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e 
 + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c** 
2*d + 3*tan(e + f*x)*a*c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*tan(e + f 
*x)**2*b*d**3*f + 6*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e 
+ f*x)**4*b*d**3 + tan(e + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3 
*tan(e + f*x)**2*a*c*d**2 + 3*tan(e + f*x)**2*b*c**2*d + 3*tan(e + f*x)*a* 
c**2*d + tan(e + f*x)*b*c**3 + a*c**3),x)*tan(e + f*x)*b*c*d**2*f + 3*int( 
(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**4*b*d**3 + tan(e 
 + f*x)**3*a*d**3 + 3*tan(e + f*x)**3*b*c*d**2 + 3*tan(e + f*x)**2*a*c*...