\(\int \frac {e^{2 \arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [280]

Optimal result

Integrand size = 21, antiderivative size = 123 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9 e^{2 \arctan (a x)}}{160 a c^4}+\frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \arctan (a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \arctan (a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )} \] Output:

9/160*exp(2*arctan(a*x))/a/c^4+1/20*exp(2*arctan(a*x))*(3*a*x+1)/a/c^4/(a^ 
2*x^2+1)^3+3/40*exp(2*arctan(a*x))*(2*a*x+1)/a/c^4/(a^2*x^2+1)^2+9/80*exp( 
2*arctan(a*x))*(a*x+1)/a/c^4/(a^2*x^2+1)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {8 c e^{2 \arctan (a x)} (1+3 a x)+3 \left (c+a^2 c x^2\right ) \left (4 e^{2 \arctan (a x)} (1+2 a x)+3 (1-i a x)^i (1+i a x)^{-i} (-i+a x) (i+a x) \left (3+2 a x+a^2 x^2\right )\right )}{160 a c^2 \left (c+a^2 c x^2\right )^3} \] Input:

Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]
 

Output:

(8*c*E^(2*ArcTan[a*x])*(1 + 3*a*x) + 3*(c + a^2*c*x^2)*(4*E^(2*ArcTan[a*x] 
)*(1 + 2*a*x) + (3*(1 - I*a*x)^I*(-I + a*x)*(I + a*x)*(3 + 2*a*x + a^2*x^2 
))/(1 + I*a*x)^I))/(160*a*c^2*(c + a^2*c*x^2)^3)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5593, 27, 5593, 5593, 5594}

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.

\(\displaystyle \int \frac {e^{2 \arctan (a x)}}{\left (a^2 c x^2+c\right )^4} \, dx\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \int \frac {e^{2 \arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx}{4 c}+\frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {e^{2 \arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{4 c^4}+\frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \left (\frac {3}{5} \int \frac {e^{2 \arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx+\frac {(2 a x+1) e^{2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}\right )}{4 c^4}+\frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {3 \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {e^{2 \arctan (a x)}}{a^2 x^2+1}dx+\frac {(a x+1) e^{2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}\right )+\frac {(2 a x+1) e^{2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}\right )}{4 c^4}+\frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5594

\(\displaystyle \frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}+\frac {3 \left (\frac {(2 a x+1) e^{2 \arctan (a x)}}{10 a \left (a^2 x^2+1\right )^2}+\frac {3}{5} \left (\frac {(a x+1) e^{2 \arctan (a x)}}{4 a \left (a^2 x^2+1\right )}+\frac {e^{2 \arctan (a x)}}{8 a}\right )\right )}{4 c^4}\)

Input:

Int[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]
 

Output:

(E^(2*ArcTan[a*x])*(1 + 3*a*x))/(20*a*c^4*(1 + a^2*x^2)^3) + (3*((E^(2*Arc 
Tan[a*x])*(1 + 2*a*x))/(10*a*(1 + a^2*x^2)^2) + (3*(E^(2*ArcTan[a*x])/(8*a 
) + (E^(2*ArcTan[a*x])*(1 + a*x))/(4*a*(1 + a^2*x^2))))/5))/(4*c^4)
 

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 5593

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[(n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 
4*(p + 1)^2))), x] + Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2)))   I 
nt[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] 
&& EqQ[d, a^2*c] && LtQ[p, -1] &&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 
 0] && IntegerQ[2*p]
 

rule 5594

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E 
^(n*ArcTan[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
 

Maple [A] (verified)

Time = 36.80 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59

method	result	size
gosper	\(\frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (9 x^{6} a^{6}+18 a^{5} x^{5}+45 a^{4} x^{4}+60 a^{3} x^{3}+75 a^{2} x^{2}+66 a x +47\right )}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(73\)
orering	\(\frac {\left (9 x^{6} a^{6}+18 a^{5} x^{5}+45 a^{4} x^{4}+60 a^{3} x^{3}+75 a^{2} x^{2}+66 a x +47\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{160 a \left (a^{2} c \,x^{2}+c \right )^{4}}\)	\(80\)
parallelrisch	\(\frac {9 a^{6} {\mathrm e}^{2 \arctan \left (a x \right )} x^{6}+18 a^{5} {\mathrm e}^{2 \arctan \left (a x \right )} x^{5}+45 a^{4} {\mathrm e}^{2 \arctan \left (a x \right )} x^{4}+60 a^{3} {\mathrm e}^{2 \arctan \left (a x \right )} x^{3}+75 a^{2} {\mathrm e}^{2 \arctan \left (a x \right )} x^{2}+66 \,{\mathrm e}^{2 \arctan \left (a x \right )} a x +47 \,{\mathrm e}^{2 \arctan \left (a x \right )}}{160 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(116\)

Input:

int(exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/160*exp(2*arctan(a*x))*(9*a^6*x^6+18*a^5*x^5+45*a^4*x^4+60*a^3*x^3+75*a^ 
2*x^2+66*a*x+47)/(a^2*x^2+1)^3/c^4/a
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (9 \, a^{6} x^{6} + 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} + 66 \, a x + 47\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")
 

Output:

1/160*(9*a^6*x^6 + 18*a^5*x^5 + 45*a^4*x^4 + 60*a^3*x^3 + 75*a^2*x^2 + 66* 
a*x + 47)*e^(2*arctan(a*x))/(a^7*c^4*x^6 + 3*a^5*c^4*x^4 + 3*a^3*c^4*x^2 + 
 a*c^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (112) = 224\).

Time = 8.58 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.33 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\begin {cases} \frac {9 a^{6} x^{6} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {18 a^{5} x^{5} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {45 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {60 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {75 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {66 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {47 e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \text {otherwise} \end {cases} \] Input:

integrate(exp(2*atan(a*x))/(a**2*c*x**2+c)**4,x)
 

Output:

Piecewise((9*a**6*x**6*exp(2*atan(a*x))/(160*a**7*c**4*x**6 + 480*a**5*c** 
4*x**4 + 480*a**3*c**4*x**2 + 160*a*c**4) + 18*a**5*x**5*exp(2*atan(a*x))/ 
(160*a**7*c**4*x**6 + 480*a**5*c**4*x**4 + 480*a**3*c**4*x**2 + 160*a*c**4 
) + 45*a**4*x**4*exp(2*atan(a*x))/(160*a**7*c**4*x**6 + 480*a**5*c**4*x**4 
 + 480*a**3*c**4*x**2 + 160*a*c**4) + 60*a**3*x**3*exp(2*atan(a*x))/(160*a 
**7*c**4*x**6 + 480*a**5*c**4*x**4 + 480*a**3*c**4*x**2 + 160*a*c**4) + 75 
*a**2*x**2*exp(2*atan(a*x))/(160*a**7*c**4*x**6 + 480*a**5*c**4*x**4 + 480 
*a**3*c**4*x**2 + 160*a*c**4) + 66*a*x*exp(2*atan(a*x))/(160*a**7*c**4*x** 
6 + 480*a**5*c**4*x**4 + 480*a**3*c**4*x**2 + 160*a*c**4) + 47*exp(2*atan( 
a*x))/(160*a**7*c**4*x**6 + 480*a**5*c**4*x**4 + 480*a**3*c**4*x**2 + 160* 
a*c**4), Ne(a, 0)), (x/c**4, True))
 

Maxima [F]

\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \] Input:

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")
 

Output:

integrate(e^(2*arctan(a*x))/(a^2*c*x^2 + c)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (111) = 222\).

Time = 0.16 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.63 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {47 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} - 132 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{11} + 18 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 180 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} + 225 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} - 456 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} - 4 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 456 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 225 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} - 180 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 18 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + 132 \, e^{\left (2 \, \arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 47 \, e^{\left (2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} + 6 \, a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 15 \, a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} + 20 \, a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 15 \, a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} + 6 \, a c^{4} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + a c^{4}\right )}} \] Input:

integrate(exp(2*arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")
 

Output:

1/160*(47*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^12 - 132*e^(2*arctan(a*x) 
)*tan(1/2*arctan(a*x))^11 + 18*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^10 + 
 180*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^9 + 225*e^(2*arctan(a*x))*tan( 
1/2*arctan(a*x))^8 - 456*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^7 - 4*e^(2 
*arctan(a*x))*tan(1/2*arctan(a*x))^6 + 456*e^(2*arctan(a*x))*tan(1/2*arcta 
n(a*x))^5 + 225*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^4 - 180*e^(2*arctan 
(a*x))*tan(1/2*arctan(a*x))^3 + 18*e^(2*arctan(a*x))*tan(1/2*arctan(a*x))^ 
2 + 132*e^(2*arctan(a*x))*tan(1/2*arctan(a*x)) + 47*e^(2*arctan(a*x)))/(a* 
c^4*tan(1/2*arctan(a*x))^12 + 6*a*c^4*tan(1/2*arctan(a*x))^10 + 15*a*c^4*t 
an(1/2*arctan(a*x))^8 + 20*a*c^4*tan(1/2*arctan(a*x))^6 + 15*a*c^4*tan(1/2 
*arctan(a*x))^4 + 6*a*c^4*tan(1/2*arctan(a*x))^2 + a*c^4)
 

Mupad [B] (verification not implemented)

Time = 23.63 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x+1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x+1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \] Input:

int(exp(2*atan(a*x))/(c + a^2*c*x^2)^4,x)
 

Output:

(9*exp(2*atan(a*x)))/(160*a*c^4) + (9*exp(2*atan(a*x))*(a*x + 1))/(80*a*c^ 
4*(a^2*x^2 + 1)) + (3*exp(2*atan(a*x))*(2*a*x + 1))/(40*a*c^4*(a^2*x^2 + 1 
)^2) + (exp(2*atan(a*x))*(3*a*x + 1))/(20*a*c^4*(a^2*x^2 + 1)^3)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {e^{2 \mathit {atan} \left (a x \right )} \left (9 a^{6} x^{6}+18 a^{5} x^{5}+45 a^{4} x^{4}+60 a^{3} x^{3}+75 a^{2} x^{2}+66 a x +47\right )}{160 a \,c^{4} \left (a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1\right )} \] Input:

int(exp(2*atan(a*x))/(a^2*c*x^2+c)^4,x)
 

Output:

(e**(2*atan(a*x))*(9*a**6*x**6 + 18*a**5*x**5 + 45*a**4*x**4 + 60*a**3*x** 
3 + 75*a**2*x**2 + 66*a*x + 47))/(160*a*c**4*(a**6*x**6 + 3*a**4*x**4 + 3* 
a**2*x**2 + 1))