\(\int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx\) [1279]

Optimal result

Integrand size = 27, antiderivative size = 328 \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=-\frac {4 b^{5/2} \left (7 a^2-3 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {3+4 \tan (e+f x)}}{\sqrt {4 a-3 b}}\right )}{(4 a-3 b)^{5/2} \left (a^2+b^2\right )^2 f}-\frac {2 \left (a^2-11 a b-b^2\right ) \arctan \left (\frac {a^2-11 a b-b^2-2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{\left (a^2-11 a b-b^2\right ) \sqrt {3+4 \tan (e+f x)}}\right )}{125 \left (a^2+b^2\right )^2 f}+\frac {\left (11 a^2+4 a b-11 b^2\right ) \text {arctanh}\left (\frac {2+\tan (e+f x)}{\sqrt {3+4 \tan (e+f x)}}\right )}{125 \left (a^2+b^2\right )^2 f}-\frac {4 \left (32 a^2+57 b^2\right )}{25 (4 a-3 b)^2 \left (a^2+b^2\right ) f \sqrt {3+4 \tan (e+f x)}}+\frac {b^2}{(4 a-3 b) \left (a^2+b^2\right ) f \sqrt {3+4 \tan (e+f x)} (a+b \tan (e+f x))} \] Output:

-4*b^(5/2)*(7*a^2-3*a*b+3*b^2)*arctan(b^(1/2)*(3+4*tan(f*x+e))^(1/2)/(4*a- 
3*b)^(1/2))/(4*a-3*b)^(5/2)/(a^2+b^2)^2/f-2/125*(a^2-11*a*b-b^2)*arctan((a 
^2-11*a*b-b^2-2*(a^2-11*a*b-b^2)*tan(f*x+e))/(a^2-11*a*b-b^2)/(3+4*tan(f*x 
+e))^(1/2))/(a^2+b^2)^2/f+1/125*(11*a^2+4*a*b-11*b^2)*arctanh((2+tan(f*x+e 
))/(3+4*tan(f*x+e))^(1/2))/(a^2+b^2)^2/f-4/25*(32*a^2+57*b^2)/(4*a-3*b)^2/ 
(a^2+b^2)/f/(3+4*tan(f*x+e))^(1/2)+b^2/(4*a-3*b)/(a^2+b^2)/f/(3+4*tan(f*x+ 
e))^(1/2)/(a+b*tan(f*x+e))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.53 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\frac {\frac {(-2+11 i) \sqrt {4 a-3 b} \left (4 a^2-(3+4 i) a b+3 i b^2\right )^2 \arctan \left (\left (\frac {1}{5}+\frac {2 i}{5}\right ) \sqrt {3+4 \tan (e+f x)}\right )+500 b^{5/2} \left (7 a^2-3 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {3+4 \tan (e+f x)}}{\sqrt {4 a-3 b}}\right )-(11-2 i) \sqrt {4 a-3 b} \left (4 a^2-(3-4 i) a b-3 i b^2\right )^2 \text {arctanh}\left (\left (\frac {2}{5}+\frac {i}{5}\right ) \sqrt {3+4 \tan (e+f x)}\right )}{(4 a-3 b)^{3/2} \left (a^2+b^2\right )}+\frac {20 \left (32 a^2+57 b^2\right )}{(4 a-3 b) \sqrt {3+4 \tan (e+f x)}}-\frac {125 b^2}{\sqrt {3+4 \tan (e+f x)} (a+b \tan (e+f x))}}{125 (-4 a+3 b) \left (a^2+b^2\right ) f} \] Input:

Integrate[1/((3 + 4*Tan[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2),x]
 

Output:

(((-2 + 11*I)*Sqrt[4*a - 3*b]*(4*a^2 - (3 + 4*I)*a*b + (3*I)*b^2)^2*ArcTan 
[(1/5 + (2*I)/5)*Sqrt[3 + 4*Tan[e + f*x]]] + 500*b^(5/2)*(7*a^2 - 3*a*b + 
3*b^2)*ArcTan[(Sqrt[b]*Sqrt[3 + 4*Tan[e + f*x]])/Sqrt[4*a - 3*b]] - (11 - 
2*I)*Sqrt[4*a - 3*b]*(4*a^2 - (3 - 4*I)*a*b - (3*I)*b^2)^2*ArcTanh[(2/5 + 
I/5)*Sqrt[3 + 4*Tan[e + f*x]]])/((4*a - 3*b)^(3/2)*(a^2 + b^2)) + (20*(32* 
a^2 + 57*b^2))/((4*a - 3*b)*Sqrt[3 + 4*Tan[e + f*x]]) - (125*b^2)/(Sqrt[3 
+ 4*Tan[e + f*x]]*(a + b*Tan[e + f*x])))/(125*(-4*a + 3*b)*(a^2 + b^2)*f)
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.43, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 4052, 3042, 4132, 27, 3042, 4136, 3042, 4019, 27, 3042, 4018, 216, 220, 4117, 73, 218}

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.

\(\displaystyle \int \frac {1}{(4 \tan (e+f x)+3)^{3/2} (a+b \tan (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(4 \tan (e+f x)+3)^{3/2} (a+b \tan (e+f x))^2}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {2 \int -\frac {48 a^3-200 b a^2+123 b^2 a-150 b^3-2 b \left (32 a^2+57 b^2\right ) \tan ^2(e+f x)-(4 a-3 b)^2 (4 a+3 b) \tan (e+f x)}{2 \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {48 a^3-200 b a^2+123 b^2 a-150 b^3-2 b \left (32 a^2+57 b^2\right ) \tan ^2(e+f x)-(4 a-3 b)^2 (4 a+3 b) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {48 a^3-200 b a^2+123 b^2 a-150 b^3-2 b \left (32 a^2+57 b^2\right ) \tan (e+f x)^2-(4 a-3 b)^2 (4 a+3 b) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {\frac {\int \frac {(a-3 b) (4 a-3 b)^2 (3 a+b)-2 (4 a-3 b)^2 (2 a-b) (a+2 b) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3}}dx}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan ^2(e+f x)+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {(a-3 b) (4 a-3 b)^2 (3 a+b)-2 (4 a-3 b)^2 (2 a-b) (a+2 b) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3}}dx}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 4019

\(\displaystyle \frac {\frac {\frac {\frac {1}{10} \int \frac {4 \left (2 \left (a^2-11 b a-b^2\right ) (4 a-3 b)^2+\left (a^2-11 b a-b^2\right ) \tan (e+f x) (4 a-3 b)^2\right )}{\sqrt {4 \tan (e+f x)+3}}dx-\frac {1}{10} \int -\frac {2 \left ((4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right )-2 (4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x)\right )}{\sqrt {4 \tan (e+f x)+3}}dx}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{5} \int \frac {(4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right )-2 (4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3}}dx+\frac {2}{5} \int \frac {2 \left (a^2-11 b a-b^2\right ) (4 a-3 b)^2+\left (a^2-11 b a-b^2\right ) \tan (e+f x) (4 a-3 b)^2}{\sqrt {4 \tan (e+f x)+3}}dx}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {1}{5} \int \frac {(4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right )-2 (4 a-3 b)^2 \left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x)}{\sqrt {4 \tan (e+f x)+3}}dx+\frac {2}{5} \int \frac {2 \left (a^2-11 b a-b^2\right ) (4 a-3 b)^2+\left (a^2-11 b a-b^2\right ) \tan (e+f x) (4 a-3 b)^2}{\sqrt {4 \tan (e+f x)+3}}dx}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 4018

\(\displaystyle \frac {\frac {\frac {-\frac {8 \left (11 a^2+4 a b-11 b^2\right )^2 (4 a-3 b)^4 \int \frac {1}{\frac {64 \left (2 \left (11 a^2+4 b a-11 b^2\right ) (4 a-3 b)^2+\left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x) (4 a-3 b)^2\right )^2}{4 \tan (e+f x)+3}-64 (4 a-3 b)^4 \left (11 a^2+4 b a-11 b^2\right )^2}d\frac {8 \left (2 \left (11 a^2+4 b a-11 b^2\right ) (4 a-3 b)^2+\left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x) (4 a-3 b)^2\right )}{\sqrt {4 \tan (e+f x)+3}}}{5 f}-\frac {4 \left (a^2-11 a b-b^2\right )^2 (4 a-3 b)^4 \int \frac {1}{4 \left (a^2-11 b a-b^2\right )^2 (4 a-3 b)^4+\frac {4 \left ((4 a-3 b)^2 \left (a^2-11 b a-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 b a-b^2\right ) \tan (e+f x)\right )^2}{4 \tan (e+f x)+3}}d\frac {2 \left ((4 a-3 b)^2 \left (a^2-11 b a-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 b a-b^2\right ) \tan (e+f x)\right )}{\sqrt {4 \tan (e+f x)+3}}}{5 f}}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {-\frac {8 \left (11 a^2+4 a b-11 b^2\right )^2 (4 a-3 b)^4 \int \frac {1}{\frac {64 \left (2 \left (11 a^2+4 b a-11 b^2\right ) (4 a-3 b)^2+\left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x) (4 a-3 b)^2\right )^2}{4 \tan (e+f x)+3}-64 (4 a-3 b)^4 \left (11 a^2+4 b a-11 b^2\right )^2}d\frac {8 \left (2 \left (11 a^2+4 b a-11 b^2\right ) (4 a-3 b)^2+\left (11 a^2+4 b a-11 b^2\right ) \tan (e+f x) (4 a-3 b)^2\right )}{\sqrt {4 \tan (e+f x)+3}}}{5 f}-\frac {2 \left (a^2-11 a b-b^2\right ) (4 a-3 b)^2 \arctan \left (\frac {(4 a-3 b)^2 \left (a^2-11 a b-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{(4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {\frac {\frac {\frac {(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \text {arctanh}\left (\frac {\left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2 \tan (e+f x)+2 \left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2}{(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}-\frac {2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \arctan \left (\frac {(4 a-3 b)^2 \left (a^2-11 a b-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{(4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {\tan (e+f x)^2+1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}dx}{a^2+b^2}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {\frac {\frac {\frac {(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \text {arctanh}\left (\frac {\left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2 \tan (e+f x)+2 \left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2}{(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}-\frac {2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \arctan \left (\frac {(4 a-3 b)^2 \left (a^2-11 a b-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{(4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}}{a^2+b^2}-\frac {50 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {1}{\sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}d\tan (e+f x)}{f \left (a^2+b^2\right )}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \text {arctanh}\left (\frac {\left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2 \tan (e+f x)+2 \left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2}{(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}-\frac {2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \arctan \left (\frac {(4 a-3 b)^2 \left (a^2-11 a b-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{(4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}}{a^2+b^2}-\frac {25 b^3 \left (7 a^2-3 a b+3 b^2\right ) \int \frac {1}{\frac {1}{4} (4 a-3 b)+\frac {1}{4} b (4 \tan (e+f x)+3)}d\sqrt {4 \tan (e+f x)+3}}{f \left (a^2+b^2\right )}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \text {arctanh}\left (\frac {\left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2 \tan (e+f x)+2 \left (11 a^2+4 a b-11 b^2\right ) (4 a-3 b)^2}{(4 a-3 b)^2 \left (11 a^2+4 a b-11 b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}-\frac {2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \arctan \left (\frac {(4 a-3 b)^2 \left (a^2-11 a b-b^2\right )-2 (4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \tan (e+f x)}{(4 a-3 b)^2 \left (a^2-11 a b-b^2\right ) \sqrt {4 \tan (e+f x)+3}}\right )}{5 f}}{a^2+b^2}-\frac {100 b^{5/2} \left (7 a^2-3 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {4 \tan (e+f x)+3}}{\sqrt {4 a-3 b}}\right )}{f \sqrt {4 a-3 b} \left (a^2+b^2\right )}}{25 (4 a-3 b)}-\frac {4 \left (32 a^2+57 b^2\right )}{25 f (4 a-3 b) \sqrt {4 \tan (e+f x)+3}}}{(4 a-3 b) \left (a^2+b^2\right )}+\frac {b^2}{f (4 a-3 b) \left (a^2+b^2\right ) \sqrt {4 \tan (e+f x)+3} (a+b \tan (e+f x))}\)

Input:

Int[1/((3 + 4*Tan[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2),x]
 

Output:

b^2/((4*a - 3*b)*(a^2 + b^2)*f*Sqrt[3 + 4*Tan[e + f*x]]*(a + b*Tan[e + f*x 
])) + (((-100*b^(5/2)*(7*a^2 - 3*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[3 + 4*T 
an[e + f*x]])/Sqrt[4*a - 3*b]])/(Sqrt[4*a - 3*b]*(a^2 + b^2)*f) + ((-2*(4* 
a - 3*b)^2*(a^2 - 11*a*b - b^2)*ArcTan[((4*a - 3*b)^2*(a^2 - 11*a*b - b^2) 
 - 2*(4*a - 3*b)^2*(a^2 - 11*a*b - b^2)*Tan[e + f*x])/((4*a - 3*b)^2*(a^2 
- 11*a*b - b^2)*Sqrt[3 + 4*Tan[e + f*x]])])/(5*f) + ((4*a - 3*b)^2*(11*a^2 
 + 4*a*b - 11*b^2)*ArcTanh[(2*(4*a - 3*b)^2*(11*a^2 + 4*a*b - 11*b^2) + (4 
*a - 3*b)^2*(11*a^2 + 4*a*b - 11*b^2)*Tan[e + f*x])/((4*a - 3*b)^2*(11*a^2 
 + 4*a*b - 11*b^2)*Sqrt[3 + 4*Tan[e + f*x]])])/(5*f))/(a^2 + b^2))/(25*(4* 
a - 3*b)) - (4*(32*a^2 + 57*b^2))/(25*(4*a - 3*b)*f*Sqrt[3 + 4*Tan[e + f*x 
]]))/((4*a - 3*b)*(a^2 + b^2))
 

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 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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 3042

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

rule 4018

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[-2*(d^2/f)   Subst[Int[1/(2*b*c*d - 4*a*d^2 
+ x^2), x], x, (b*c - 2*a*d - b*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] && NeQ[a^2 + b^2, 0 
] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
 

rule 4019

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[1/(2*q)   Int[( 
a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], 
 x], x] - Simp[1/(2*q)   Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f 
*x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && N 
eQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - 
 b*(c^2 - d^2), 0] && NiceSqrtQ[a^2 + b^2]
 

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 [A] (verified)

Time = 0.25 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\frac {64 \left (11 a^{2}+4 a b -11 b^{2}\right ) \ln \left (8+4 \tan \left (f x +e \right )+4 \sqrt {3+4 \tan \left (f x +e \right )}\right )+128 \left (2 a^{2}-22 a b -2 b^{2}\right ) \arctan \left (2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{16000 a^{4}+32000 a^{2} b^{2}+16000 b^{4}}-\frac {128}{25 \left (4 a -3 b \right )^{2} \sqrt {3+4 \tan \left (f x +e \right )}}-\frac {4 b^{3} \left (\frac {\left (\frac {a^{2}}{4}+\frac {b^{2}}{4}\right ) \sqrt {3+4 \tan \left (f x +e \right )}}{\frac {\left (3+4 \tan \left (f x +e \right )\right ) b}{4}+a -\frac {3 b}{4}}+\frac {4 \left (\frac {7}{4} a^{2}-\frac {3}{4} a b +\frac {3}{4} b^{2}\right ) \arctan \left (\frac {\sqrt {3+4 \tan \left (f x +e \right )}\, b}{\sqrt {b \left (4 a -3 b \right )}}\right )}{\sqrt {b \left (4 a -3 b \right )}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (4 a -3 b \right )^{2}}+\frac {64 \left (-11 a^{2}-4 a b +11 b^{2}\right ) \ln \left (8+4 \tan \left (f x +e \right )-4 \sqrt {3+4 \tan \left (f x +e \right )}\right )+128 \left (2 a^{2}-22 a b -2 b^{2}\right ) \arctan \left (-2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{16000 a^{4}+32000 a^{2} b^{2}+16000 b^{4}}}{f}\)	\(353\)
default	\(\frac {\frac {64 \left (11 a^{2}+4 a b -11 b^{2}\right ) \ln \left (8+4 \tan \left (f x +e \right )+4 \sqrt {3+4 \tan \left (f x +e \right )}\right )+128 \left (2 a^{2}-22 a b -2 b^{2}\right ) \arctan \left (2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{16000 a^{4}+32000 a^{2} b^{2}+16000 b^{4}}-\frac {128}{25 \left (4 a -3 b \right )^{2} \sqrt {3+4 \tan \left (f x +e \right )}}-\frac {4 b^{3} \left (\frac {\left (\frac {a^{2}}{4}+\frac {b^{2}}{4}\right ) \sqrt {3+4 \tan \left (f x +e \right )}}{\frac {\left (3+4 \tan \left (f x +e \right )\right ) b}{4}+a -\frac {3 b}{4}}+\frac {4 \left (\frac {7}{4} a^{2}-\frac {3}{4} a b +\frac {3}{4} b^{2}\right ) \arctan \left (\frac {\sqrt {3+4 \tan \left (f x +e \right )}\, b}{\sqrt {b \left (4 a -3 b \right )}}\right )}{\sqrt {b \left (4 a -3 b \right )}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (4 a -3 b \right )^{2}}+\frac {64 \left (-11 a^{2}-4 a b +11 b^{2}\right ) \ln \left (8+4 \tan \left (f x +e \right )-4 \sqrt {3+4 \tan \left (f x +e \right )}\right )+128 \left (2 a^{2}-22 a b -2 b^{2}\right ) \arctan \left (-2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{16000 a^{4}+32000 a^{2} b^{2}+16000 b^{4}}}{f}\)	\(353\)

Input:

int(1/(3+4*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^2,x,method=_RETURNVERBOSE)
 

Output:

1/f*(128/(16000*a^4+32000*a^2*b^2+16000*b^4)*(1/2*(11*a^2+4*a*b-11*b^2)*ln 
(8+4*tan(f*x+e)+4*(3+4*tan(f*x+e))^(1/2))+(2*a^2-22*a*b-2*b^2)*arctan(2+(3 
+4*tan(f*x+e))^(1/2)))-128/25/(4*a-3*b)^2/(3+4*tan(f*x+e))^(1/2)-4*b^3/(a^ 
4+2*a^2*b^2+b^4)/(4*a-3*b)^2*((1/4*a^2+1/4*b^2)*(3+4*tan(f*x+e))^(1/2)/(1/ 
4*(3+4*tan(f*x+e))*b+a-3/4*b)+4*(7/4*a^2-3/4*a*b+3/4*b^2)/(b*(4*a-3*b))^(1 
/2)*arctan((3+4*tan(f*x+e))^(1/2)*b/(b*(4*a-3*b))^(1/2)))+128/(16000*a^4+3 
2000*a^2*b^2+16000*b^4)*(1/2*(-11*a^2-4*a*b+11*b^2)*ln(8+4*tan(f*x+e)-4*(3 
+4*tan(f*x+e))^(1/2))+(2*a^2-22*a*b-2*b^2)*arctan(-2+(3+4*tan(f*x+e))^(1/2 
))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (304) = 608\).

Time = 2.37 (sec) , antiderivative size = 1970, normalized size of antiderivative = 6.01 \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(3+4*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^2,x, algorithm="fricas 
")
 

Output:

[1/250*(500*(21*a^3*b^2 - 9*a^2*b^3 + 9*a*b^4 + 4*(7*a^2*b^3 - 3*a*b^4 + 3 
*b^5)*tan(f*x + e)^2 + (28*a^3*b^2 + 9*a^2*b^3 + 3*a*b^4 + 9*b^5)*tan(f*x 
+ e))*sqrt(-b/(4*a - 3*b))*log(-((4*a - 3*b)*sqrt(-b/(4*a - 3*b))*sqrt(4*t 
an(f*x + e) + 3) - 2*b*tan(f*x + e) + 2*a - 3*b)/(b*tan(f*x + e) + a)) + 4 
*(48*a^5 - 600*a^4*b + 771*a^3*b^2 - 225*a^2*b^3 - 27*a*b^4 + 4*(16*a^4*b 
- 200*a^3*b^2 + 257*a^2*b^3 - 75*a*b^4 - 9*b^5)*tan(f*x + e)^2 + (64*a^5 - 
 752*a^4*b + 428*a^3*b^2 + 471*a^2*b^3 - 261*a*b^4 - 27*b^5)*tan(f*x + e)) 
*arctan(sqrt(4*tan(f*x + e) + 3) + 2) + 4*(48*a^5 - 600*a^4*b + 771*a^3*b^ 
2 - 225*a^2*b^3 - 27*a*b^4 + 4*(16*a^4*b - 200*a^3*b^2 + 257*a^2*b^3 - 75* 
a*b^4 - 9*b^5)*tan(f*x + e)^2 + (64*a^5 - 752*a^4*b + 428*a^3*b^2 + 471*a^ 
2*b^3 - 261*a*b^4 - 27*b^5)*tan(f*x + e))*arctan(sqrt(4*tan(f*x + e) + 3) 
- 2) + (528*a^5 - 600*a^4*b - 519*a^3*b^2 + 900*a^2*b^3 - 297*a*b^4 + 4*(1 
76*a^4*b - 200*a^3*b^2 - 173*a^2*b^3 + 300*a*b^4 - 99*b^5)*tan(f*x + e)^2 
+ (704*a^5 - 272*a^4*b - 1292*a^3*b^2 + 681*a^2*b^3 + 504*a*b^4 - 297*b^5) 
*tan(f*x + e))*log(sqrt(4*tan(f*x + e) + 3) + tan(f*x + e) + 2) - (528*a^5 
 - 600*a^4*b - 519*a^3*b^2 + 900*a^2*b^3 - 297*a*b^4 + 4*(176*a^4*b - 200* 
a^3*b^2 - 173*a^2*b^3 + 300*a*b^4 - 99*b^5)*tan(f*x + e)^2 + (704*a^5 - 27 
2*a^4*b - 1292*a^3*b^2 + 681*a^2*b^3 + 504*a*b^4 - 297*b^5)*tan(f*x + e))* 
log(-sqrt(4*tan(f*x + e) + 3) + tan(f*x + e) + 2) - 10*(128*a^5 + 256*a^3* 
b^2 + 75*a^2*b^3 + 128*a*b^4 + 75*b^5 + 4*(32*a^4*b + 89*a^2*b^3 + 57*b...
 

Sympy [F]

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

integrate(1/(3+4*tan(f*x+e))**(3/2)/(a+b*tan(f*x+e))**2,x)
 

Output:

Integral(1/((a + b*tan(e + f*x))**2*(4*tan(e + f*x) + 3)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(3+4*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^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(3*b-4*a>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(3+4*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^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 Type
 

Mupad [B] (verification not implemented)

Time = 11.16 (sec) , antiderivative size = 33043, normalized size of antiderivative = 100.74 \[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

int(1/((4*tan(e + f*x) + 3)^(3/2)*(a + b*tan(e + f*x))^2),x)
 

Output:

(atan(-(((((2254683933849600000000000000*b^39*f^6 - (((4*tan(e + f*x) + 3) 
^(1/2)*(37909766377497600000000000000*b^41*f^7 - 7201632396100608000000000 
00000*a*b^40*f^7 + 6682217574311424000000000000000*a^2*b^39*f^7 - 40299694 
124400230400000000000000*a^3*b^38*f^7 + 176247651190127385600000000000000* 
a^4*b^37*f^7 - 585590915528698982400000000000000*a^5*b^36*f^7 + 1482703892 
783430451200000000000000*a^6*b^35*f^7 - 2666888110746574848000000000000000 
*a^7*b^34*f^7 + 2128557857430065817600000000000000*a^8*b^33*f^7 + 67646071 
01070144921600000000000000*a^9*b^32*f^7 - 39276895021375929753600000000000 
000*a^10*b^31*f^7 + 122786670882567838924800000000000000*a^11*b^30*f^7 - 2 
96370830210915185920000000000000000*a^12*b^29*f^7 + 6027926643309771638784 
00000000000000*a^13*b^28*f^7 - 1071661221567962077593600000000000000*a^14* 
b^27*f^7 + 1697800170170703937536000000000000000*a^15*b^26*f^7 - 242447313 
9270790240128000000000000000*a^16*b^25*f^7 + 31429341767729881574400000000 
00000000*a^17*b^24*f^7 - 3714765740344859963904000000000000000*a^18*b^23*f 
^7 + 4012622794701839353856000000000000000*a^19*b^22*f^7 - 396394111638393 
0926208000000000000000*a^20*b^21*f^7 + 35780265203309952906240000000000000 
00*a^21*b^20*f^7 - 2943370702126384896000000000000000000*a^22*b^19*f^7 + 2 
196164963300913078272000000000000000*a^23*b^18*f^7 - 147471815400442154188 
8000000000000000*a^24*b^17*f^7 + 879875895902481363763200000000000000*a^25 
*b^16*f^7 - 456186114479830440345600000000000000*a^26*b^15*f^7 + 196634...
 

Reduce [F]

\[ \int \frac {1}{(3+4 \tan (e+f x))^{3/2} (a+b \tan (e+f x))^2} \, dx=\text {too large to display} \] Input:

int(1/(3+4*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^2,x)
 

Output:

(sqrt(4*tan(e + f*x) + 3) + 32*int(sqrt(4*tan(e + f*x) + 3)/(64*tan(e + f* 
x)**4*a**2*b**2 + 192*tan(e + f*x)**4*a*b**3 + 144*tan(e + f*x)**4*b**4 + 
128*tan(e + f*x)**3*a**3*b + 480*tan(e + f*x)**3*a**2*b**2 + 576*tan(e + f 
*x)**3*a*b**3 + 216*tan(e + f*x)**3*b**4 + 64*tan(e + f*x)**2*a**4 + 384*t 
an(e + f*x)**2*a**3*b + 756*tan(e + f*x)**2*a**2*b**2 + 540*tan(e + f*x)** 
2*a*b**3 + 81*tan(e + f*x)**2*b**4 + 96*tan(e + f*x)*a**4 + 360*tan(e + f* 
x)*a**3*b + 432*tan(e + f*x)*a**2*b**2 + 162*tan(e + f*x)*a*b**3 + 36*a**4 
 + 108*a**3*b + 81*a**2*b**2),x)*tan(e + f*x)**2*a**3*b*f + 144*int(sqrt(4 
*tan(e + f*x) + 3)/(64*tan(e + f*x)**4*a**2*b**2 + 192*tan(e + f*x)**4*a*b 
**3 + 144*tan(e + f*x)**4*b**4 + 128*tan(e + f*x)**3*a**3*b + 480*tan(e + 
f*x)**3*a**2*b**2 + 576*tan(e + f*x)**3*a*b**3 + 216*tan(e + f*x)**3*b**4 
+ 64*tan(e + f*x)**2*a**4 + 384*tan(e + f*x)**2*a**3*b + 756*tan(e + f*x)* 
*2*a**2*b**2 + 540*tan(e + f*x)**2*a*b**3 + 81*tan(e + f*x)**2*b**4 + 96*t 
an(e + f*x)*a**4 + 360*tan(e + f*x)*a**3*b + 432*tan(e + f*x)*a**2*b**2 + 
162*tan(e + f*x)*a*b**3 + 36*a**4 + 108*a**3*b + 81*a**2*b**2),x)*tan(e + 
f*x)**2*a**2*b**2*f + 216*int(sqrt(4*tan(e + f*x) + 3)/(64*tan(e + f*x)**4 
*a**2*b**2 + 192*tan(e + f*x)**4*a*b**3 + 144*tan(e + f*x)**4*b**4 + 128*t 
an(e + f*x)**3*a**3*b + 480*tan(e + f*x)**3*a**2*b**2 + 576*tan(e + f*x)** 
3*a*b**3 + 216*tan(e + f*x)**3*b**4 + 64*tan(e + f*x)**2*a**4 + 384*tan(e 
+ f*x)**2*a**3*b + 756*tan(e + f*x)**2*a**2*b**2 + 540*tan(e + f*x)**2*...