\(\int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx\) [1150]

Optimal result

Integrand size = 32, antiderivative size = 378 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {\sqrt [4]{-1} a^{3/2} \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}} \] Output:

-1/8*(-1)^(1/4)*a^(3/2)*(5*I*c^3+45*c^2*d-55*I*c*d^2-23*d^3)*arctanh((-1)^ 
(3/4)*d^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c+d*tan(f*x+e))^(1/2))/d^( 
1/2)/f-2*I*2^(1/2)*a^(3/2)*(c-I*d)^(5/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan( 
f*x+e))^(1/2)/(c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))/f+1/8*a*(c-3*I*d)*(5 
*I*c+3*d)*(a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/f+1/12*a*(5*I*c+ 
7*d)*(a+I*a*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/f+1/3*a^2*(c+I*d)*(c+ 
d*tan(f*x+e))^(5/2)/d/f/(a+I*a*tan(f*x+e))^(1/2)-1/3*a^2*(c+d*tan(f*x+e))^ 
(7/2)/d/f/(a+I*a*tan(f*x+e))^(1/2)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1078\) vs. \(2(378)=756\).

Time = 6.63 (sec) , antiderivative size = 1078, normalized size of antiderivative = 2.85 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\frac {d (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {-\frac {3 a (3 c-i d) d (a+i a \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {\frac {a \left (-\frac {3}{4} a^3 d \left (11 c^2-14 i c d-7 d^2\right )-\frac {3}{4} i a^3 \left (8 c^3-13 i c^2 d-10 c d^2+i d^3\right )\right ) \left (-\frac {2 \sqrt {2} \arctan \left (\frac {\sqrt {-a c+i a d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a c+i a d}}-\frac {2 (-1)^{3/4} \sqrt {c+i d} \sqrt {\frac {1}{\frac {c}{c+i d}+\frac {i d}{c+i d}}} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}} \arcsin \left (\frac {\sqrt [4]{-1} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+i d} \sqrt {\frac {c}{c+i d}+\frac {i d}{c+i d}}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{\sqrt {a} \sqrt {d} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {3 (i a c-a d)^2 \left (11 c^2-14 i c d-7 d^2\right ) \sqrt {\frac {i a}{-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}} \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )^2 \sqrt {\frac {i a (c+d \tan (e+f x))}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}} \left (\frac {2 i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}-\frac {2 \sqrt [4]{-1} \sqrt {a} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a c-a d} \sqrt {-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}} \sqrt {1+\frac {i a d (a+i a \tan (e+f x))}{(i a c-a d) \left (-\frac {a^2 c}{i a c-a d}-\frac {i a^2 d}{i a c-a d}\right )}}}\right )}{8 d f \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}}{2 a}}{3 a} \] Input:

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

Output:

(d*(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - ((-3*a 
*(3*c - I*d)*d*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(4*f 
) + ((a*((-3*a^3*d*(11*c^2 - (14*I)*c*d - 7*d^2))/4 - ((3*I)/4)*a^3*(8*c^3 
 - (13*I)*c^2*d - 10*c*d^2 + I*d^3))*((-2*Sqrt[2]*ArcTan[(Sqrt[-(a*c) + I* 
a*d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])])/Sq 
rt[-(a*c) + I*a*d] - (2*(-1)^(3/4)*Sqrt[c + I*d]*Sqrt[(c/(c + I*d) + (I*d) 
/(c + I*d))^(-1)]*Sqrt[c/(c + I*d) + (I*d)/(c + I*d)]*ArcSin[((-1)^(1/4)*S 
qrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + I*d]*Sqrt[c/(c + I*d) 
 + (I*d)/(c + I*d)])]*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(Sqrt[a]*Sqrt[ 
d]*Sqrt[c + d*Tan[e + f*x]])))/f + (3*(I*a*c - a*d)^2*(11*c^2 - (14*I)*c*d 
 - 7*d^2)*Sqrt[(I*a)/(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))] 
*(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d))^2*Sqrt[(I*a*(c + d*T 
an[e + f*x]))/(I*a*c - a*d)]*Sqrt[1 + (I*a*d*(a + I*a*Tan[e + f*x]))/((I*a 
*c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)))]*(((2*I)*a 
*d*(a + I*a*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(I*a*c - a*d)) - (I*a 
^2*d)/(I*a*c - a*d))) - (2*(-1)^(1/4)*Sqrt[a]*Sqrt[d]*ArcSinh[((-1)^(1/4)* 
Sqrt[a]*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2 
*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a*d)])]*Sqrt[a + I*a*Tan[e + f*x]] 
)/(Sqrt[I*a*c - a*d]*Sqrt[-((a^2*c)/(I*a*c - a*d)) - (I*a^2*d)/(I*a*c - a* 
d)]*Sqrt[1 + (I*a*d*(a + I*a*Tan[e + f*x]))/((I*a*c - a*d)*(-((a^2*c)/(...
 

Rubi [A] (verified)

Time = 2.70 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {3042, 4039, 27, 3042, 4078, 3042, 4080, 27, 3042, 4080, 27, 3042, 4084, 3042, 4027, 221, 4082, 66, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4039

\(\displaystyle \frac {a \int -\frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{2 \sqrt {i \tan (e+f x) a+a}}dx}{3 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \int \frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{\sqrt {i \tan (e+f x) a+a}}dx}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(a (i c-13 d)+a (c-11 i d) \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{\sqrt {i \tan (e+f x) a+a}}dx}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4078

\(\displaystyle -\frac {a \left (-\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left ((7 c-5 i d) d a^2+d (5 i c+7 d) \tan (e+f x) a^2\right )dx}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {\int \sqrt {i \tan (e+f x) a+a} (c+d \tan (e+f x))^{3/2} \left ((7 c-5 i d) d a^2+d (5 i c+7 d) \tan (e+f x) a^2\right )dx}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4080

\(\displaystyle -\frac {a \left (-\frac {\frac {\int \frac {3}{2} \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{2 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \int \sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)} \left (d \left (11 c^2-14 i d c-7 d^2\right ) a^3+(c-3 i d) d (5 i c+3 d) \tan (e+f x) a^3\right )dx}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4080

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left ((3 c-i d) d \left (9 c^2-14 i d c-9 d^2\right ) a^4+d \left (5 i c^3+45 d c^2-55 i d^2 c-23 d^3\right ) \tan (e+f x) a^4\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4084

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {32 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {32 a^4 d (c-i d)^3 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4027

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {64 i a^6 d (c-i d)^3 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-a^3 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4082

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-\frac {a^5 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {a \left (-\frac {\frac {3 \left (\frac {-\frac {2 a^5 d \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}+\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a \left (-\frac {\frac {a^2 d (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {3 \left (\frac {-\frac {2 (-1)^{3/4} a^{9/2} \sqrt {d} \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {32 i \sqrt {2} a^{9/2} d (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a}+\frac {a^3 d (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )}{4 a}}{a^2}-\frac {2 a (c+i d) (c+d \tan (e+f x))^{5/2}}{f \sqrt {a+i a \tan (e+f x)}}\right )}{6 d}\)

Input:

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

Output:

-1/3*(a^2*(c + d*Tan[e + f*x])^(7/2))/(d*f*Sqrt[a + I*a*Tan[e + f*x]]) - ( 
a*((-2*a*(c + I*d)*(c + d*Tan[e + f*x])^(5/2))/(f*Sqrt[a + I*a*Tan[e + f*x 
]]) - ((a^2*d*((5*I)*c + 7*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f* 
x])^(3/2))/(2*f) + (3*(((-2*(-1)^(3/4)*a^(9/2)*Sqrt[d]*(5*c^3 - (45*I)*c^2 
*d - 55*c*d^2 + (23*I)*d^3)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e 
 + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f - ((32*I)*Sqrt[2]*a^(9/2) 
*(c - I*d)^(5/2)*d*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqr 
t[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f)/(2*a) + (a^3*(c - (3*I)*d)*d*( 
(5*I)*c + 3*d)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f))/(4 
*a))/a^2))/(6*d)
 

Defintions of rubi rules used

rule 27

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

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 4027

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

rule 4039

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 - 2)*((c + 
 d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[a/(d*(m + n - 1)) 
Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + 
a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x 
] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 
 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] 
 && (IntegerQ[m] || IntegersQ[2*m, 2*n])
 

rule 4078

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

rule 4080

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

rule 4082

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

rule 4084

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1502 vs. \(2 (305 ) = 610\).

Time = 0.42 (sec) , antiderivative size = 1503, normalized size of antiderivative = 3.98

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1503\)
default	\(\text {Expression too large to display}\)	\(1503\)

Input:

int((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/96/f*(a*(1+I*tan(f*x+e)))^(1/2)*(c+d*tan(f*x+e))^(1/2)*a*(52*I*(I*a*d)^( 
1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2 
)*c*d*tan(f*x+e)-69*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e 
)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/( 
I*a*d)^(1/2))*a*d^3+16*I*2^(1/2)*d^2*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e) 
)*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+e)^2+48*I*(I*a*d)^(1/2)*ln 
((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/ 
2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*a*c-54*I*( 
I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e 
)))^(1/2)*d^2-48*I*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I 
*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a 
*d)^(1/2))*a*d-15*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I* 
a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a* 
d)^(1/2))*a*c^3+165*2^(1/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+ 
I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I* 
a*d)^(1/2))*a*c*d^2+28*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*ta 
n(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d^2*tan(f*x+e)+48*I*(I*a*d)^(1/2)*ln((3* 
a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*( 
a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*a*d+135*I*2^(1 
/2)*(-a*(I*d-c))^(1/2)*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1493 vs. \(2 (290) = 580\).

Time = 0.18 (sec) , antiderivative size = 1493, normalized size of antiderivative = 3.95 \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/48*(48*sqrt(2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqr 
t(-(a^3*c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c* 
d^4 - I*a^3*d^5)/f^2)*log((I*sqrt(2)*f*sqrt(-(a^3*c^5 - 5*I*a^3*c^4*d - 10 
*a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c*d^4 - I*a^3*d^5)/f^2)*e^(I*f*x + 
 I*e) + sqrt(2)*(a*c^2 - 2*I*a*c*d - a*d^2 + (a*c^2 - 2*I*a*c*d - a*d^2)*e 
^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I 
*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a 
*c^2 - 2*I*a*c*d - a*d^2)) - 48*sqrt(2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2* 
I*f*x + 2*I*e) + f)*sqrt(-(a^3*c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I 
*a^3*c^2*d^3 + 5*a^3*c*d^4 - I*a^3*d^5)/f^2)*log((-I*sqrt(2)*f*sqrt(-(a^3* 
c^5 - 5*I*a^3*c^4*d - 10*a^3*c^3*d^2 + 10*I*a^3*c^2*d^3 + 5*a^3*c*d^4 - I* 
a^3*d^5)/f^2)*e^(I*f*x + I*e) + sqrt(2)*(a*c^2 - 2*I*a*c*d - a*d^2 + (a*c^ 
2 - 2*I*a*c*d - a*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2 
*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 
1)))*e^(-I*f*x - I*e)/(a*c^2 - 2*I*a*c*d - a*d^2)) + 2*sqrt(2)*((33*I*a*c^ 
2 + 94*a*c*d - 49*I*a*d^2)*e^(5*I*f*x + 5*I*e) - 2*(-33*I*a*c^2 - 68*a*c*d 
 + 19*I*a*d^2)*e^(3*I*f*x + 3*I*e) - 3*(-11*I*a*c^2 - 14*a*c*d + 7*I*a*d^2 
)*e^(I*f*x + I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I* 
f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) + 3*(f*e^(4*I*f*x + 4 
*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*sqrt((-25*I*a^3*c^6 - 450*a^3*c^5*...
 

Sympy [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

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

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*d-c>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

Exception generated. \[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeError: Bad Argument TypeError: Bad Argument TypeRecursive ass 
umption s
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f 
*x)**3,x)*d**2*i + 2*int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c) 
*tan(e + f*x)**2,x)*c*d*i + int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x) 
*d + c)*tan(e + f*x)**2,x)*d**2 + int(sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e 
+ f*x)*d + c)*tan(e + f*x),x)*c**2*i + 2*int(sqrt(tan(e + f*x)*i + 1)*sqrt 
(tan(e + f*x)*d + c)*tan(e + f*x),x)*c*d + int(sqrt(tan(e + f*x)*i + 1)*sq 
rt(tan(e + f*x)*d + c),x)*c**2)