\(\int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx\) [57]

Optimal result

Integrand size = 20, antiderivative size = 839 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b) (a+i b)^2 \left (i a+b-(i a-b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a+i b)^2 d}-\frac {b (c+d x)^4}{(a+i b)^2 (i a+b) d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f}-\frac {2 i b^2 (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a+i b)^2 (i a+b) f^2}-\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f^3}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 (a+i b)^2 (i a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \] Output:

-3*I*b^2*d^2*(d*x+c)*polylog(2,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^ 
2)^2/f^3-2*b^2*(d*x+c)^3/(a-I*b)/(a+I*b)^2/(I*a+b-(I*a-b)*exp(2*I*e+2*I*f* 
x))/f+1/4*(d*x+c)^4/(a+I*b)^2/d-b*(d*x+c)^4/(a+I*b)^2/(I*a+b)/d-b^2*(d*x+c 
)^4/(a^2+b^2)^2/d+3*b^2*d*(d*x+c)^2*ln(1-(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b 
))/(a^2+b^2)^2/f^2-2*b*(d*x+c)^3*ln(1-(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/ 
(a-I*b)/(a+I*b)^2/f-2*I*b^2*(d*x+c)^3/(a^2+b^2)^2/f-3*I*b^2*d^2*(d*x+c)*po 
lylog(3,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^3-3*b*d*(d*x+c)^ 
2*polylog(2,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a+I*b)^2/(I*a+b)/f^2-3*b^ 
2*d*(d*x+c)^2*polylog(2,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^ 
2+3/2*b^2*d^3*polylog(3,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^2+b^2)^2/f^ 
4-3*b*d^2*(d*x+c)*polylog(3,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a-I*b)/(a 
+I*b)^2/f^3-2*I*b^2*(d*x+c)^3*ln(1-(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a^ 
2+b^2)^2/f+3/2*b*d^3*polylog(4,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/(a+I*b) 
^2/(I*a+b)/f^4+3/2*b^2*d^3*polylog(4,(a+I*b)*exp(2*I*e+2*I*f*x)/(a-I*b))/( 
a^2+b^2)^2/f^4
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1733\) vs. \(2(839)=1678\).

Time = 9.27 (sec) , antiderivative size = 1733, normalized size of antiderivative = 2.07 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(b*((4*c^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-3*b*d + 2*a*c* 
f)*x)/(a^2 + b^2) - (4*b*(c + d*x)^3)/(a + I*b) + (2*a*f*(c + d*x)^4)/((a 
+ I*b)*d) + (12*c*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(-(b*d) 
 + a*c*f)*x*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I*b 
)*((-I)*a + b)*f) - (6*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))* 
(b*d - 2*a*c*f)*x^2*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/( 
(a - I*b)*((-I)*a + b)*f) + (4*a*d^3*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^(( 
2*I)*e)))*x^3*Log[1 + (-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a - I 
*b)*((-I)*a + b)) + (2*c^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))* 
(-3*b*d + 2*a*c*f)*Log[a - I*b - (a + I*b)*E^((2*I)*(e + f*x))])/((a - I*b 
)*((-I)*a + b)*f) - (6*c*d*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))* 
(-(b*d) + a*c*f)*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/(( 
a^2 + b^2)*f^2) + (3*d^2*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(b 
*d - 2*a*c*f)*(2*f*x*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] 
 - I*PolyLog[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)* 
f^3) - (3*a*d^3*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(2*f^2*x^2* 
PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - (2*I)*f*x*PolyLog[ 
3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - PolyLog[4, (a - I*b)/((a + 
 I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3)))/(2*(a - I*b)*(a + I*b)*( 
(-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e)))*f) + (3*x^2*(-(a*c^2*d...
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 839, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b (c+d x)^4}{(a+i b)^2 (i a+b) d}+\frac {(c+d x)^4}{4 (a+i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}-\frac {2 b \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)^3}{(a-i b) (a+i b)^2 f}-\frac {2 i b^2 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b) (a+i b)^2 \left (i a-(i a-b) e^{2 i e+2 i f x}+b\right ) f}-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {3 b^2 d \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}-\frac {3 b d \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)^2}{(a+i b)^2 (i a+b) f^2}-\frac {3 b^2 d \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}-\frac {3 i b^2 d^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^2 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)}{(a-i b) (a+i b)^2 f^3}-\frac {3 i b^2 d^2 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 (a+i b)^2 (i a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}\)

Input:

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

Output:

((-2*I)*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f) - (2*b^2*(c + d*x)^3)/((a - I*b 
)*(a + I*b)^2*(I*a + b - (I*a - b)*E^((2*I)*e + (2*I)*f*x))*f) + (c + d*x) 
^4/(4*(a + I*b)^2*d) - (b*(c + d*x)^4)/((a + I*b)^2*(I*a + b)*d) - (b^2*(c 
 + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 - ((a + I*b)*E^( 
(2*I)*e + (2*I)*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) - (2*b*(c + d*x)^3*L 
og[1 - ((a + I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/((a - I*b)*(a + I*b 
)^2*f) - ((2*I)*b^2*(c + d*x)^3*Log[1 - ((a + I*b)*E^((2*I)*e + (2*I)*f*x) 
)/(a - I*b)])/((a^2 + b^2)^2*f) - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, ((a 
+ I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3) - (3*b*d*( 
c + d*x)^2*PolyLog[2, ((a + I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/((a 
+ I*b)^2*(I*a + b)*f^2) - (3*b^2*d*(c + d*x)^2*PolyLog[2, ((a + I*b)*E^((2 
*I)*e + (2*I)*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) + (3*b^2*d^3*PolyLog[3 
, ((a + I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/(2*(a^2 + b^2)^2*f^4) - 
(3*b*d^2*(c + d*x)*PolyLog[3, ((a + I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b 
)])/((a - I*b)*(a + I*b)^2*f^3) - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[3, ((a 
+ I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3) + (3*b*d^3 
*PolyLog[4, ((a + I*b)*E^((2*I)*e + (2*I)*f*x))/(a - I*b)])/(2*(a + I*b)^2 
*(I*a + b)*f^4) + (3*b^2*d^3*PolyLog[4, ((a + I*b)*E^((2*I)*e + (2*I)*f*x) 
)/(a - I*b)])/(2*(a^2 + b^2)^2*f^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5930 vs. \(2 (754 ) = 1508\).

Time = 1.11 (sec) , antiderivative size = 5931, normalized size of antiderivative = 7.07

method	result	size
risch	\(\text {Expression too large to display}\)	\(5931\)

Input:

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

Output:

result too large to display
 

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. 3363 vs. \(2 (687) = 1374\).

Time = 0.17 (sec) , antiderivative size = 3363, normalized size of antiderivative = 4.01 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4641 vs. \(2 (687) = 1374\).

Time = 2.16 (sec) , antiderivative size = 4641, normalized size of antiderivative = 5.53 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(36*(b^2/((a^4 + a^2*b^2)*f*tan(f*x + e) + (a^3*b + a*b^3)*f) + 2*a*b 
*log(a*tan(f*x + e) + b)/((a^4 + 2*a^2*b^2 + b^4)*f) - a*b*log(tan(f*x + e 
)^2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*f) - (a^2 - b^2)*(f*x + e)/((a^4 + 2*a^2 
*b^2 + b^4)*f))*c^2*d*e - 12*(2*a*b*log(a*tan(f*x + e) + b)/(a^4 + 2*a^2*b 
^2 + b^4) - a*b*log(tan(f*x + e)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^2 - b 
^2)*(f*x + e)/(a^4 + 2*a^2*b^2 + b^4) + b^2/(a^3*b + a*b^3 + (a^4 + a^2*b^ 
2)*tan(f*x + e)))*c^3 - (3*(a^3 + I*a^2*b + a*b^2 + I*b^3)*(f*x + e)^4*d^3 
 + 24*(-I*a*b^2 - b^3)*d^3*e^3 + 72*(I*a*b^2 + b^3)*c*d^2*e^2*f - 12*((a^3 
 + I*a^2*b + a*b^2 + I*b^3)*d^3*e - (a^3 + I*a^2*b + a*b^2 + I*b^3)*c*d^2* 
f)*(f*x + e)^3 + 18*((a^3 + I*a^2*b + a*b^2 + I*b^3)*d^3*e^2 - 2*(a^3 + I* 
a^2*b + a*b^2 + I*b^3)*c*d^2*e*f + (a^3 + I*a^2*b + a*b^2 + I*b^3)*c^2*d*f 
^2)*(f*x + e)^2 - 12*((a^3 + I*a^2*b + a*b^2 + I*b^3)*d^3*e^3 - 3*(a^3 + I 
*a^2*b + a*b^2 + I*b^3)*c*d^2*e^2*f)*(f*x + e) + 12*(2*(I*a^2*b + a*b^2)*d 
^3*e^3 + 3*(I*a*b^2 + b^3)*d^3*e^2 + 3*(I*a*b^2 + b^3)*c^2*d*f^2 + 6*((-I* 
a^2*b - a*b^2)*c*d^2*e^2 + (-I*a*b^2 - b^3)*c*d^2*e)*f + (2*(-I*a^2*b + a* 
b^2)*d^3*e^3 + 3*(-I*a*b^2 + b^3)*d^3*e^2 + 3*(-I*a*b^2 + b^3)*c^2*d*f^2 + 
 6*((I*a^2*b - a*b^2)*c*d^2*e^2 + (I*a*b^2 - b^3)*c*d^2*e)*f)*cos(2*f*x + 
2*e) + (2*(a^2*b + I*a*b^2)*d^3*e^3 + 3*(a*b^2 + I*b^3)*d^3*e^2 + 3*(a*b^2 
 + I*b^3)*c^2*d*f^2 - 6*((a^2*b + I*a*b^2)*c*d^2*e^2 + (a*b^2 + I*b^3)*c*d 
^2*e)*f)*sin(2*f*x + 2*e))*arctan2(b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

( - 216*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b* 
*2))*cos(e + f*x)*a**6*b*d**3*i + 144*sqrt(a**2 + b**2)*atan((tan((e + f*x 
)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a**5*b**2*c*d**2*f*i - 48* 
sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos 
(e + f*x)*a**4*b**3*c**2*d*f**2*i - 336*sqrt(a**2 + b**2)*atan((tan((e + f 
*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a**4*b**3*d**3*i + 192*s 
qrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*cos( 
e + f*x)*a**3*b**4*c*d**2*f*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2 
)*b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a**2*b**5*c**2*d*f**2*i - 120 
*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*co 
s(e + f*x)*a**2*b**5*d**3*i + 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)* 
b*i - a*i)/sqrt(a**2 + b**2))*cos(e + f*x)*a*b**6*c*d**2*f*i - 216*sqrt(a* 
*2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + f* 
x)*a**7*d**3*i + 144*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/s 
qrt(a**2 + b**2))*sin(e + f*x)*a**6*b*c*d**2*f*i - 48*sqrt(a**2 + b**2)*at 
an((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + f*x)*a**5*b**2* 
c**2*d*f**2*i - 336*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sq 
rt(a**2 + b**2))*sin(e + f*x)*a**5*b**2*d**3*i + 192*sqrt(a**2 + b**2)*ata 
n((tan((e + f*x)/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(e + f*x)*a**4*b**3*c 
*d**2*f*i - 48*sqrt(a**2 + b**2)*atan((tan((e + f*x)/2)*b*i - a*i)/sqrt...