\(\int \frac {e^{\arctan (a x)}}{(c+a^2 c x^2)^3} \, dx\) [250]

Optimal result

Integrand size = 19, antiderivative size = 83 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {24 e^{\arctan (a x)}}{85 a c^3}+\frac {e^{\arctan (a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\arctan (a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \]

[Out]

24/85*exp(arctan(a*x))/a/c^3+1/17*exp(arctan(a*x))*(4*a*x+1)/a/c^3/(a^2*x^2+1)^2+12/85*exp(arctan(a*x))*(2*a*x
+1)/a/c^3/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5178, 5179} \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {12 (2 a x+1) e^{\arctan (a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac {(4 a x+1) e^{\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac {24 e^{\arctan (a x)}}{85 a c^3} \]

[In]

Int[E^ArcTan[a*x]/(c + a^2*c*x^2)^3,x]

[Out]

(24*E^ArcTan[a*x])/(85*a*c^3) + (E^ArcTan[a*x]*(1 + 4*a*x))/(17*a*c^3*(1 + a^2*x^2)^2) + (12*E^ArcTan[a*x]*(1
+ 2*a*x))/(85*a*c^3*(1 + a^2*x^2))

Rule 5178

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(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] + Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))
), Int[(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 5179

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

Rubi steps \begin{align*} \text {integral}& = \frac {e^{\arctan (a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c} \\ & = \frac {e^{\arctan (a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\arctan (a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac {24 \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx}{85 c^2} \\ & = \frac {24 e^{\arctan (a x)}}{85 a c^3}+\frac {e^{\arctan (a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\arctan (a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {5 e^{\arctan (a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} \left (1+a^2 x^2\right ) \left (3+2 a x+2 a^2 x^2\right )}{85 a c^3 \left (1+a^2 x^2\right )^2} \]

[In]

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^3,x]

[Out]

(5*E^ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a*x)^(I/2)*(1 + a^2*x^2)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2))
/(85*a*c^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 10.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66

method	result	size
gosper	\(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (24 a^{4} x^{4}+24 a^{3} x^{3}+60 a^{2} x^{2}+44 a x +41\right )}{85 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\)	\(55\)
parallelrisch	\(\frac {24 a^{4} {\mathrm e}^{\arctan \left (a x \right )} x^{4}+24 a^{3} x^{3} {\mathrm e}^{\arctan \left (a x \right )}+60 x^{2} {\mathrm e}^{\arctan \left (a x \right )} a^{2}+44 \,{\mathrm e}^{\arctan \left (a x \right )} a x +41 \,{\mathrm e}^{\arctan \left (a x \right )}}{85 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\)	\(76\)

[In]

int(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/85*exp(arctan(a*x))*(24*a^4*x^4+24*a^3*x^3+60*a^2*x^2+44*a*x+41)/(a^2*x^2+1)^2/c^3/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {{\left (24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 44 \, a x + 41\right )} e^{\left (\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/85*(24*a^4*x^4 + 24*a^3*x^3 + 60*a^2*x^2 + 44*a*x + 41)*e^(arctan(a*x))/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3
)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (75) = 150\).

Time = 2.87 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} \frac {24 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {24 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {60 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {44 a x e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {41 e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate(exp(atan(a*x))/(a**2*c*x**2+c)**3,x)

[Out]

Piecewise((24*a**4*x**4*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 24*a**3*x**3*exp
(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 60*a**2*x**2*exp(atan(a*x))/(85*a**5*c**3*x
**4 + 170*a**3*c**3*x**2 + 85*a*c**3) + 44*a*x*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c
**3) + 41*exp(atan(a*x))/(85*a**5*c**3*x**4 + 170*a**3*c**3*x**2 + 85*a*c**3), Ne(a, 0)), (x/c**3, True))

Maxima [F]

\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^3, x)

Giac [F]

\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {24\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {12\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]

[In]

int(exp(atan(a*x))/(c + a^2*c*x^2)^3,x)

[Out]

(24*exp(atan(a*x)))/(85*a*c^3) + (12*exp(atan(a*x))*(2*a*x + 1))/(85*a*c^3*(a^2*x^2 + 1)) + (exp(atan(a*x))*(4
*a*x + 1))/(17*a*c^3*(a^2*x^2 + 1)^2)