\(\int \frac {e^{\arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [265]

Optimal result

Integrand size = 19, antiderivative size = 116 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {144 e^{\arctan (a x)}}{629 a c^4}+\frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\arctan (a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {72 e^{\arctan (a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \] Output:

144/629*exp(arctan(a*x))/a/c^4+1/37*exp(arctan(a*x))*(6*a*x+1)/a/c^4/(a^2* 
x^2+1)^3+30/629*exp(arctan(a*x))*(4*a*x+1)/a/c^4/(a^2*x^2+1)^2+72/629*exp( 
arctan(a*x))*(2*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 = 123, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {17 c e^{\arctan (a x)} (1+6 a x)+6 \left (c+a^2 c x^2\right ) \left (5 e^{\arctan (a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (-i+a x) (i+a x) \left (3+2 a x+2 a^2 x^2\right )\right )}{629 a c^2 \left (c+a^2 c x^2\right )^3} \] Input:

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

Output:

(17*c*E^ArcTan[a*x]*(1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^ArcTan[a*x]*(1 + 
4*a*x) + (12*(1 - I*a*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2 
))/(1 + I*a*x)^(I/2)))/(629*a*c^2*(c + a^2*c*x^2)^3)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, 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.

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

\(\Big \downarrow \) 5593

\(\displaystyle \frac {30 \int \frac {e^{\arctan (a x)}}{c^3 \left (a^2 x^2+1\right )^3}dx}{37 c}+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {30 \int \frac {e^{\arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx}{37 c^4}+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {30 \left (\frac {12}{17} \int \frac {e^{\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx+\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )}{37 c^4}+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {30 \left (\frac {12}{17} \left (\frac {2}{5} \int \frac {e^{\arctan (a x)}}{a^2 x^2+1}dx+\frac {(2 a x+1) e^{\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}\right )+\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )}{37 c^4}+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}\)

\(\Big \downarrow \) 5594

\(\displaystyle \frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}+\frac {30 \left (\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}+\frac {12}{17} \left (\frac {(2 a x+1) e^{\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}+\frac {2 e^{\arctan (a x)}}{5 a}\right )\right )}{37 c^4}\)

Input:

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

Output:

(E^ArcTan[a*x]*(1 + 6*a*x))/(37*a*c^4*(1 + a^2*x^2)^3) + (30*((E^ArcTan[a* 
x]*(1 + 4*a*x))/(17*a*(1 + a^2*x^2)^2) + (12*((2*E^ArcTan[a*x])/(5*a) + (E 
^ArcTan[a*x]*(1 + 2*a*x))/(5*a*(1 + a^2*x^2))))/17))/(37*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 = 36.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61

method	result	size
gosper	\(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (144 x^{6} a^{6}+144 a^{5} x^{5}+504 a^{4} x^{4}+408 a^{3} x^{3}+606 a^{2} x^{2}+366 a x +263\right )}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(71\)
orering	\(\frac {\left (144 x^{6} a^{6}+144 a^{5} x^{5}+504 a^{4} x^{4}+408 a^{3} x^{3}+606 a^{2} x^{2}+366 a x +263\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{\arctan \left (a x \right )}}{629 a \left (a^{2} c \,x^{2}+c \right )^{4}}\)	\(78\)
parallelrisch	\(\frac {144 a^{6} {\mathrm e}^{\arctan \left (a x \right )} x^{6}+144 a^{5} {\mathrm e}^{\arctan \left (a x \right )} x^{5}+504 a^{4} {\mathrm e}^{\arctan \left (a x \right )} x^{4}+408 a^{3} {\mathrm e}^{\arctan \left (a x \right )} x^{3}+606 a^{2} {\mathrm e}^{\arctan \left (a x \right )} x^{2}+366 \,{\mathrm e}^{\arctan \left (a x \right )} a x +263 \,{\mathrm e}^{\arctan \left (a x \right )}}{629 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(102\)

Input:

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

Output:

1/629*exp(arctan(a*x))*(144*a^6*x^6+144*a^5*x^5+504*a^4*x^4+408*a^3*x^3+60 
6*a^2*x^2+366*a*x+263)/(a^2*x^2+1)^3/c^4/a
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (144 \, a^{6} x^{6} + 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} + 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} + 366 \, a x + 263\right )} e^{\left (\arctan \left (a x\right )\right )}}{629 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \] Input:

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

Output:

1/629*(144*a^6*x^6 + 144*a^5*x^5 + 504*a^4*x^4 + 408*a^3*x^3 + 606*a^2*x^2 
 + 366*a*x + 263)*e^(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. 398 vs. \(2 (107) = 214\).

Time = 8.31 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.43 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\begin {cases} \frac {144 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {144 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {504 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {408 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {606 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {366 a x e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {263 e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((144*a**6*x**6*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c* 
*4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c**4) + 144*a**5*x**5*exp(atan(a*x)) 
/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c 
**4) + 504*a**4*x**4*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x 
**4 + 1887*a**3*c**4*x**2 + 629*a*c**4) + 408*a**3*x**3*exp(atan(a*x))/(62 
9*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c**4) 
 + 606*a**2*x**2*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 
+ 1887*a**3*c**4*x**2 + 629*a*c**4) + 366*a*x*exp(atan(a*x))/(629*a**7*c** 
4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c**4) + 263*exp 
(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x** 
2 + 629*a*c**4), Ne(a, 0)), (x/c**4, True))
 

Maxima [F]

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

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

Output:

integrate(e^(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. 297 vs. \(2 (104) = 208\).

Time = 0.14 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {263 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{12} - 732 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{11} + 846 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{10} + 396 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{9} + 2313 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{8} - 2136 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{7} + 2372 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{6} + 2136 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{5} + 2313 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{4} - 396 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{3} + 846 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right )^{2} + 732 \, e^{\left (\arctan \left (a x\right )\right )} \tan \left (\frac {1}{2} \, \arctan \left (a x\right )\right ) + 263 \, e^{\left (\arctan \left (a x\right )\right )}}{629 \, {\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(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")
 

Output:

1/629*(263*e^(arctan(a*x))*tan(1/2*arctan(a*x))^12 - 732*e^(arctan(a*x))*t 
an(1/2*arctan(a*x))^11 + 846*e^(arctan(a*x))*tan(1/2*arctan(a*x))^10 + 396 
*e^(arctan(a*x))*tan(1/2*arctan(a*x))^9 + 2313*e^(arctan(a*x))*tan(1/2*arc 
tan(a*x))^8 - 2136*e^(arctan(a*x))*tan(1/2*arctan(a*x))^7 + 2372*e^(arctan 
(a*x))*tan(1/2*arctan(a*x))^6 + 2136*e^(arctan(a*x))*tan(1/2*arctan(a*x))^ 
5 + 2313*e^(arctan(a*x))*tan(1/2*arctan(a*x))^4 - 396*e^(arctan(a*x))*tan( 
1/2*arctan(a*x))^3 + 846*e^(arctan(a*x))*tan(1/2*arctan(a*x))^2 + 732*e^(a 
rctan(a*x))*tan(1/2*arctan(a*x)) + 263*e^(arctan(a*x)))/(a*c^4*tan(1/2*arc 
tan(a*x))^12 + 6*a*c^4*tan(1/2*arctan(a*x))^10 + 15*a*c^4*tan(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 = 23.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {144\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {72\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {30\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x+1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \] Input:

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

Output:

(144*exp(atan(a*x)))/(629*a*c^4) + (72*exp(atan(a*x))*(2*a*x + 1))/(629*a* 
c^4*(a^2*x^2 + 1)) + (30*exp(atan(a*x))*(4*a*x + 1))/(629*a*c^4*(a^2*x^2 + 
 1)^2) + (exp(atan(a*x))*(6*a*x + 1))/(37*a*c^4*(a^2*x^2 + 1)^3)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {e^{\mathit {atan} \left (a x \right )} \left (144 a^{6} x^{6}+144 a^{5} x^{5}+504 a^{4} x^{4}+408 a^{3} x^{3}+606 a^{2} x^{2}+366 a x +263\right )}{629 a \,c^{4} \left (a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(e**atan(a*x)*(144*a**6*x**6 + 144*a**5*x**5 + 504*a**4*x**4 + 408*a**3*x* 
*3 + 606*a**2*x**2 + 366*a*x + 263))/(629*a*c**4*(a**6*x**6 + 3*a**4*x**4 
+ 3*a**2*x**2 + 1))