\(\int \frac {e^{-2 \arctan (a x)}}{(c+a^2 c x^2)^{7/2}} \, dx\) [300]

Optimal result

Integrand size = 23, antiderivative size = 115 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {e^{-2 \arctan (a x)} (2-5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}-\frac {20 e^{-2 \arctan (a x)} (2-3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}-\frac {24 e^{-2 \arctan (a x)} (2-a x)}{377 a c^3 \sqrt {c+a^2 c x^2}} \]

[Out]

1/29*(5*a*x-2)/a/c/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(5/2)-20/377*(-3*a*x+2)/a/c^2/exp(2*arctan(a*x))/(a^2*c*x^
2+c)^(3/2)-24/377*(-a*x+2)/a/c^3/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5178, 5177} \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {24 (2-a x) e^{-2 \arctan (a x)}}{377 a c^3 \sqrt {a^2 c x^2+c}}-\frac {20 (2-3 a x) e^{-2 \arctan (a x)}}{377 a c^2 \left (a^2 c x^2+c\right )^{3/2}}-\frac {(2-5 a x) e^{-2 \arctan (a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}} \]

[In]

Int[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(7/2)),x]

[Out]

-1/29*(2 - 5*a*x)/(a*c*E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(5/2)) - (20*(2 - 3*a*x))/(377*a*c^2*E^(2*ArcTan[a*x]
)*(c + a^2*c*x^2)^(3/2)) - (24*(2 - a*x))/(377*a*c^3*E^(2*ArcTan[a*x])*Sqrt[c + a^2*c*x^2])

Rule 5177

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n + a*x)*(E^(n*ArcTan[a*x])/(
a*c*(n^2 + 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] &&  !IntegerQ[I*n]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-2 \arctan (a x)} (2-5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{29 c} \\ & = -\frac {e^{-2 \arctan (a x)} (2-5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}-\frac {20 e^{-2 \arctan (a x)} (2-3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {120 \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{377 c^2} \\ & = -\frac {e^{-2 \arctan (a x)} (2-5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}-\frac {20 e^{-2 \arctan (a x)} (2-3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}-\frac {24 e^{-2 \arctan (a x)} (2-a x)}{377 a c^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{-2 \arctan (a x)} \left (-114+149 a x-136 a^2 x^2+108 a^3 x^3-48 a^4 x^4+24 a^5 x^5\right )}{377 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(7/2)),x]

[Out]

(-114 + 149*a*x - 136*a^2*x^2 + 108*a^3*x^3 - 48*a^4*x^4 + 24*a^5*x^5)/(377*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x
^2)^2*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {\left (a^{2} x^{2}+1\right ) \left (24 a^{5} x^{5}-48 a^{4} x^{4}+108 a^{3} x^{3}-136 a^{2} x^{2}+149 a x -114\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{377 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\)	\(74\)

[In]

int(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/377*(a^2*x^2+1)*(24*a^5*x^5-48*a^4*x^4+108*a^3*x^3-136*a^2*x^2+149*a*x-114)/a/exp(2*arctan(a*x))/(a^2*c*x^2+
c)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (24 \, a^{5} x^{5} - 48 \, a^{4} x^{4} + 108 \, a^{3} x^{3} - 136 \, a^{2} x^{2} + 149 \, a x - 114\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{377 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

1/377*(24*a^5*x^5 - 48*a^4*x^4 + 108*a^3*x^3 - 136*a^2*x^2 + 149*a*x - 114)*sqrt(a^2*c*x^2 + c)*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 [F(-1)]

Timed out. \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/exp(2*atan(a*x))/(a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(e^(-2*arctan(a*x))/(a^2*c*x^2 + c)^(7/2), x)

Giac [F]

\[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/exp(2*arctan(a*x))/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {114}{377\,a^5\,c^3}-\frac {24\,x^5}{377\,c^3}-\frac {149\,x}{377\,a^4\,c^3}+\frac {48\,x^4}{377\,a\,c^3}-\frac {108\,x^3}{377\,a^2\,c^3}+\frac {136\,x^2}{377\,a^3\,c^3}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^4}+x^4\,\sqrt {c\,a^2\,x^2+c}+\frac {2\,x^2\,\sqrt {c\,a^2\,x^2+c}}{a^2}} \]

[In]

int(exp(-2*atan(a*x))/(c + a^2*c*x^2)^(7/2),x)

[Out]

-(exp(-2*atan(a*x))*(114/(377*a^5*c^3) - (24*x^5)/(377*c^3) - (149*x)/(377*a^4*c^3) + (48*x^4)/(377*a*c^3) - (
108*x^3)/(377*a^2*c^3) + (136*x^2)/(377*a^3*c^3)))/((c + a^2*c*x^2)^(1/2)/a^4 + x^4*(c + a^2*c*x^2)^(1/2) + (2
*x^2*(c + a^2*c*x^2)^(1/2))/a^2)