\(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx\) [1240]

Optimal result

Integrand size = 27, antiderivative size = 341 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=-\frac {(c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \] Output:

-(c-I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)^3/f+( 
c+I*d)^(3/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f+1/4 
*(24*a^3*b*c*d-40*a*b^3*c*d-3*a^4*d^2-2*a^2*b^2*(12*c^2-13*d^2)+b^4*(8*c^2 
-3*d^2))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/b^(1/2)/ 
(a^2+b^2)^3/(-a*d+b*c)^(1/2)/f-1/2*(-a*d+b*c)*(c+d*tan(f*x+e))^(1/2)/(a^2+ 
b^2)/f/(a+b*tan(f*x+e))^2-1/4*(-3*a^2*d+8*a*b*c+5*b^2*d)*(c+d*tan(f*x+e))^ 
(1/2)/(a^2+b^2)^2/f/(a+b*tan(f*x+e))
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2093\) vs. \(2(341)=682\).

Time = 6.33 (sec) , antiderivative size = 2093, normalized size of antiderivative = 6.14 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/2*(b^2*(c + d*Tan[e + f*x])^(5/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Ta 
n[e + f*x])^2) - (-((b*d*(c + d*Tan[e + f*x])^(3/2))/(f*(a + b*Tan[e + f*x 
]))) + (2*(-1/2*(b*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + b*Tan[e 
 + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^ 
2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d 
)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - 
 a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b 
*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4 
*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2) 
/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) - I*(a*(b*c - a*d 
)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2 
))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*(b*c - a 
*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + 
 (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d 
 - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*(( 
3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2 
)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - 
 (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - 
a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d 
- b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)...
 

Rubi [A] (warning: unable to verify)

Time = 2.92 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.14, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 4050, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {\int -\frac {-16 (a c+b d) \tan (e+f x) (b c-a d)^2-d \left (-3 d a^2+8 b c a+5 b^2 d\right ) \tan ^2(e+f x) (b c-a d)+\left (\left (8 c^2-5 d^2\right ) a^2+24 b c d a-b^2 \left (8 c^2-3 d^2\right )\right ) (b c-a d)}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-16 (a c+b d) \tan (e+f x) (b c-a d)^2-d \left (-3 d a^2+8 b c a+5 b^2 d\right ) \tan ^2(e+f x) (b c-a d)+\left (\left (8 c^2-5 d^2\right ) a^2+24 b c d a-b^2 \left (8 c^2-3 d^2\right )\right ) (b c-a d)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {-16 (a c+b d) \tan (e+f x) (b c-a d)^2-d \left (-3 d a^2+8 b c a+5 b^2 d\right ) \tan (e+f x)^2 (b c-a d)+\left (\left (8 c^2-5 d^2\right ) a^2+24 b c d a-b^2 \left (8 c^2-3 d^2\right )\right ) (b c-a d)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {\frac {\int \frac {8 \left ((b c-a d) \left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right )+(b c-a d) \left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {8 \int \frac {(b c-a d) \left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right )+(b c-a d) \left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {8 \int \frac {(b c-a d) \left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right )+(b c-a d) \left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 \left (a^2+b^2\right )}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} (a-i b)^3 (c+i d)^2 (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^3 (c-i d)^2 (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} (a-i b)^3 (c+i d)^2 (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^3 (c-i d)^2 (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i (a+i b)^3 (c-i d)^2 (b c-a d) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b)^3 (c+i d)^2 (b c-a d) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i (a-i b)^3 (c+i d)^2 (b c-a d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b)^3 (c-i d)^2 (b c-a d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {(a-i b)^3 (c+i d)^2 (b c-a d) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b)^3 (c-i d)^2 (b c-a d) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {(a-i b)^3 (c+i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^3 (c-i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {(b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {8 \left (\frac {(a-i b)^3 (c+i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^3 (c-i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {2 (b c-a d) \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {8 \left (\frac {(a-i b)^3 (c+i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^3 (c-i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {2 \sqrt {b c-a d} \left (3 a^4 d^2-24 a^3 b c d+24 a^2 b^2 c^2-26 a^2 b^2 d^2+40 a b^3 c d-8 b^4 c^2+3 b^4 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {8 \left (\frac {(a-i b)^3 (c+i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^3 (c-i d)^{3/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 \left (a^2+b^2\right )}\)

Input:

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

Output:

-1/2*((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + 
f*x])^2) + (((8*(((a + I*b)^3*(c - I*d)^(3/2)*(b*c - a*d)*ArcTan[Tan[e + f 
*x]/Sqrt[c - I*d]])/f + ((a - I*b)^3*(c + I*d)^(3/2)*(b*c - a*d)*ArcTan[Ta 
n[e + f*x]/Sqrt[c + I*d]])/f))/(a^2 + b^2) - (2*Sqrt[b*c - a*d]*(24*a^2*b^ 
2*c^2 - 8*b^4*c^2 - 24*a^3*b*c*d + 40*a*b^3*c*d + 3*a^4*d^2 - 26*a^2*b^2*d 
^2 + 3*b^4*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d] 
])/(Sqrt[b]*(a^2 + b^2)*f))/(2*(a^2 + b^2)*(b*c - a*d)) - ((8*a*b*c - 3*a^ 
2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x 
])))/(4*(a^2 + b^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 4050

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 
 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m 
 + 1)*(a^2 + b^2))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ 
(n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 
*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 
, x], 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] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 
2*m]
 

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. \(4732\) vs. \(2(303)=606\).

Time = 0.38 (sec) , antiderivative size = 4733, normalized size of antiderivative = 13.88

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4733\)
default	\(\text {Expression too large to display}\)	\(4733\)

Input:

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

Output:

1/f*d^2/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/ 
2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*b^3 
-1/f*d^2/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+ 
e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*b^ 
3+3/4/f*d^2/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2 
)/((a*d-b*c)*b)^(1/2))*b^4+1/4/f*d/(a^2+b^2)^3*ln((c+d*tan(f*x+e))^(1/2)*( 
2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^ 
(1/2)+2*c)^(1/2)*a^3-13/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*ta 
n(f*x+e))^(1/2)*a^4*b*c+2/f*d/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+d*tan( 
f*x+e))^(1/2)*a^3*b^2*c^2-5/2/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+ 
d*tan(f*x+e))^(1/2)*a^2*b^3*c+2/f*d/(a^2+b^2)^3/(tan(f*x+e)*b*d+a*d)^2*(c+ 
d*tan(f*x+e))^(1/2)*a*b^4*c^2+3/f/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2 
)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+ 
d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a^2*b*c-3/f/(a^2+b^2)^3/(2*(c^2+d^2 
)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e) 
)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a^2*b*c+3/4/f/d/(a 
^2+b^2)^3*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f* 
x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2+6/f*d/(a^2 
+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2 
)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b^2*c+1/4/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11055 vs. \(2 (297) = 594\).

Time = 112.16 (sec) , antiderivative size = 22129, normalized size of antiderivative = 64.89 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^3,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,[0,19,8]%%%}+%%%{8,[0,17,8]%%%}+%%%{28,[0,15,8]%%%}+ 
%%%{56,[0
 

Mupad [B] (verification not implemented)

Time = 34.63 (sec) , antiderivative size = 90146, normalized size of antiderivative = 264.36 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(((c + d*tan(e + f*x))^(1/2)*(5*a^3*d^3 - 3*a*b^2*d^3 + 3*b^3*c*d^2 + 8*a* 
b^2*c^2*d - 13*a^2*b*c*d^2))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (b*(c + d*tan(e 
 + f*x))^(3/2)*(5*b^2*d^2 - 3*a^2*d^2 + 8*a*b*c*d))/(4*(a^4 + b^4 + 2*a^2* 
b^2)))/(a^2*d^2*f - (2*b^2*c*f - 2*a*b*d*f)*(c + d*tan(e + f*x)) + b^2*c^2 
*f + b^2*f*(c + d*tan(e + f*x))^2 - 2*a*b*c*d*f) - atan(((((1036*a*b^15*d^ 
15*f^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^2 - 8988*a^3*b^13*d^15*f^2 
 + 6044*a^5*b^11*d^15*f^2 + 34388*a^7*b^9*d^15*f^2 + 10596*a^9*b^7*d^15*f^ 
2 - 6676*a^11*b^5*d^15*f^2 + 1012*a^13*b^3*d^15*f^2 + 932*b^16*c^3*d^12*f^ 
2 + 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*f^2 - 30836*a^2*b^14*c^3*d^1 
2*f^2 - 48000*a^2*b^14*c^5*d^10*f^2 + 5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^ 
3*b^13*c^2*d^13*f^2 + 57600*a^3*b^13*c^4*d^11*f^2 - 46464*a^3*b^13*c^6*d^9 
*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a^4*b^12*c^5*d^10*f^2 - 5696*a^ 
4*b^12*c^7*d^8*f^2 + 47004*a^5*b^11*c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11 
*f^2 - 30016*a^5*b^11*c^6*d^9*f^2 + 85404*a^6*b^10*c^3*d^12*f^2 + 169344*a 
^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*d^8*f^2 - 171180*a^7*b^9*c^2*d^1 
3*f^2 - 119808*a^7*b^9*c^4*d^11*f^2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8 
*b^8*c^3*d^12*f^2 - 49728*a^8*b^8*c^5*d^10*f^2 + 3776*a^8*b^8*c^7*d^8*f^2 
- 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9*b^7*c^4*d^11*f^2 + 36800*a^9*b^7* 
c^6*d^9*f^2 - 35356*a^10*b^6*c^3*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 
10368*a^10*b^6*c^7*d^8*f^2 + 55916*a^11*b^5*c^2*d^13*f^2 + 37632*a^11*b...
 

Reduce [F]

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

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

Output:

( - 4*sqrt(tan(e + f*x)*d + c)*a*c*d + 22*sqrt(tan(e + f*x)*d + c)*b*c**2 
- 6*sqrt(tan(e + f*x)*d + c)*b*d**2 + 2*int(sqrt(tan(e + f*x)*d + c)/(tan( 
e + f*x)**4*a*b**3*d**2 - 4*tan(e + f*x)**4*b**4*c*d + 3*tan(e + f*x)**3*a 
**2*b**2*d**2 - 11*tan(e + f*x)**3*a*b**3*c*d - 4*tan(e + f*x)**3*b**4*c** 
2 + 3*tan(e + f*x)**2*a**3*b*d**2 - 9*tan(e + f*x)**2*a**2*b**2*c*d - 12*t 
an(e + f*x)**2*a*b**3*c**2 + tan(e + f*x)*a**4*d**2 - tan(e + f*x)*a**3*b* 
c*d - 12*tan(e + f*x)*a**2*b**2*c**2 + a**4*c*d - 4*a**3*b*c**2),x)*tan(e 
+ f*x)**2*a**3*b**2*c*d**3*f - 24*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f* 
x)**4*a*b**3*d**2 - 4*tan(e + f*x)**4*b**4*c*d + 3*tan(e + f*x)**3*a**2*b* 
*2*d**2 - 11*tan(e + f*x)**3*a*b**3*c*d - 4*tan(e + f*x)**3*b**4*c**2 + 3* 
tan(e + f*x)**2*a**3*b*d**2 - 9*tan(e + f*x)**2*a**2*b**2*c*d - 12*tan(e + 
 f*x)**2*a*b**3*c**2 + tan(e + f*x)*a**4*d**2 - tan(e + f*x)*a**3*b*c*d - 
12*tan(e + f*x)*a**2*b**2*c**2 + a**4*c*d - 4*a**3*b*c**2),x)*tan(e + f*x) 
**2*a**2*b**3*c**2*d**2*f + 3*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)** 
4*a*b**3*d**2 - 4*tan(e + f*x)**4*b**4*c*d + 3*tan(e + f*x)**3*a**2*b**2*d 
**2 - 11*tan(e + f*x)**3*a*b**3*c*d - 4*tan(e + f*x)**3*b**4*c**2 + 3*tan( 
e + f*x)**2*a**3*b*d**2 - 9*tan(e + f*x)**2*a**2*b**2*c*d - 12*tan(e + f*x 
)**2*a*b**3*c**2 + tan(e + f*x)*a**4*d**2 - tan(e + f*x)*a**3*b*c*d - 12*t 
an(e + f*x)*a**2*b**2*c**2 + a**4*c*d - 4*a**3*b*c**2),x)*tan(e + f*x)**2* 
a**2*b**3*d**4*f + 96*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**4*a*b...