\(\int \frac {e^{2 \arctan (a x)}}{(c+a^2 c x^2)^3} \, dx\) [265]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 e^{2 \arctan (a x)}}{40 a c^3}+\frac {e^{2 \arctan (a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 e^{2 \arctan (a x)} (1+a x)}{20 a c^3 \left (1+a^2 x^2\right )} \]

[Out]

3/40*exp(2*arctan(a*x))/a/c^3+1/10*exp(2*arctan(a*x))*(2*a*x+1)/a/c^3/(a^2*x^2+1)^2+3/20*exp(2*arctan(a*x))*(a
*x+1)/a/c^3/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 (a x+1) e^{2 \arctan (a x)}}{20 a c^3 \left (a^2 x^2+1\right )}+\frac {(2 a x+1) e^{2 \arctan (a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 e^{2 \arctan (a x)}}{40 a c^3} \]

[In]

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

[Out]

(3*E^(2*ArcTan[a*x]))/(40*a*c^3) + (E^(2*ArcTan[a*x])*(1 + 2*a*x))/(10*a*c^3*(1 + a^2*x^2)^2) + (3*E^(2*ArcTan
[a*x])*(1 + a*x))/(20*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^{2 \arctan (a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c} \\ & = \frac {e^{2 \arctan (a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 e^{2 \arctan (a x)} (1+a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx}{20 c^2} \\ & = \frac {3 e^{2 \arctan (a x)}}{40 a c^3}+\frac {e^{2 \arctan (a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 e^{2 \arctan (a x)} (1+a x)}{20 a c^3 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {4 e^{2 \arctan (a x)} (1+2 a x)+3 (1-i a x)^i (1+i a x)^{-i} \left (1+a^2 x^2\right ) \left (3+2 a x+a^2 x^2\right )}{40 a c^3 \left (1+a^2 x^2\right )^2} \]

[In]

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

[Out]

(4*E^(2*ArcTan[a*x])*(1 + 2*a*x) + (3*(1 - I*a*x)^I*(1 + a^2*x^2)*(3 + 2*a*x + a^2*x^2))/(1 + I*a*x)^I)/(40*a*
c^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 10.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (3 a^{4} x^{4}+6 a^{3} x^{3}+12 a^{2} x^{2}+14 a x +13\right )}{40 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\)	\(57\)
parallelrisch	\(\frac {3 a^{4} {\mathrm e}^{2 \arctan \left (a x \right )} x^{4}+6 a^{3} x^{3} {\mathrm e}^{2 \arctan \left (a x \right )}+12 x^{2} {\mathrm e}^{2 \arctan \left (a x \right )} a^{2}+14 \,{\mathrm e}^{2 \arctan \left (a x \right )} a x +13 \,{\mathrm e}^{2 \arctan \left (a x \right )}}{40 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\)	\(86\)

[In]

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

[Out]

1/40*exp(2*arctan(a*x))*(3*a^4*x^4+6*a^3*x^3+12*a^2*x^2+14*a*x+13)/(a^2*x^2+1)^2/c^3/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {{\left (3 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 14 \, a x + 13\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

[In]

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

[Out]

1/40*(3*a^4*x^4 + 6*a^3*x^3 + 12*a^2*x^2 + 14*a*x + 13)*e^(2*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. 231 vs. \(2 (78) = 156\).

Time = 2.87 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.62 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} \frac {3 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {6 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {12 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {14 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {13 e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*a**4*x**4*exp(2*atan(a*x))/(40*a**5*c**3*x**4 + 80*a**3*c**3*x**2 + 40*a*c**3) + 6*a**3*x**3*exp(
2*atan(a*x))/(40*a**5*c**3*x**4 + 80*a**3*c**3*x**2 + 40*a*c**3) + 12*a**2*x**2*exp(2*atan(a*x))/(40*a**5*c**3
*x**4 + 80*a**3*c**3*x**2 + 40*a*c**3) + 14*a*x*exp(2*atan(a*x))/(40*a**5*c**3*x**4 + 80*a**3*c**3*x**2 + 40*a
*c**3) + 13*exp(2*atan(a*x))/(40*a**5*c**3*x**4 + 80*a**3*c**3*x**2 + 40*a*c**3), Ne(a, 0)), (x/c**3, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.90 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x+1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}+\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]

[In]

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

[Out]

(3*exp(2*atan(a*x)))/(40*a*c^3) + (3*exp(2*atan(a*x))*(a*x + 1))/(20*a*c^3*(a^2*x^2 + 1)) + (exp(2*atan(a*x))*
(2*a*x + 1))/(10*a*c^3*(a^2*x^2 + 1)^2)