\(\int \frac {e^{\arctan (a x)}}{(c+a^2 c x^2)^{7/2}} \, dx\) [258]

Optimal result

Integrand size = 21, antiderivative size = 108 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{\arctan (a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {e^{\arctan (a x)} (1+3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {3 e^{\arctan (a x)} (1+a x)}{13 a c^3 \sqrt {c+a^2 c x^2}} \]

[Out]

1/26*exp(arctan(a*x))*(5*a*x+1)/a/c/(a^2*c*x^2+c)^(5/2)+1/13*exp(arctan(a*x))*(3*a*x+1)/a/c^2/(a^2*c*x^2+c)^(3
/2)+3/13*exp(arctan(a*x))*(a*x+1)/a/c^3/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 108, 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, 5177} \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {3 (a x+1) e^{\arctan (a x)}}{13 a c^3 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+1) e^{\arctan (a x)}}{13 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {(5 a x+1) e^{\arctan (a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}} \]

[In]

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

[Out]

(E^ArcTan[a*x]*(1 + 5*a*x))/(26*a*c*(c + a^2*c*x^2)^(5/2)) + (E^ArcTan[a*x]*(1 + 3*a*x))/(13*a*c^2*(c + a^2*c*
x^2)^(3/2)) + (3*E^ArcTan[a*x]*(1 + a*x))/(13*a*c^3*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^{\arctan (a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {10 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{13 c} \\ & = \frac {e^{\arctan (a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {e^{\arctan (a x)} (1+3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{13 c^2} \\ & = \frac {e^{\arctan (a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {e^{\arctan (a x)} (1+3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {3 e^{\arctan (a x)} (1+a x)}{13 a c^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {e^{\arctan (a x)} \left (9+17 a x+14 a^2 x^2+18 a^3 x^3+6 a^4 x^4+6 a^5 x^5\right )}{26 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2}} \]

[In]

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

[Out]

(E^ArcTan[a*x]*(9 + 17*a*x + 14*a^2*x^2 + 18*a^3*x^3 + 6*a^4*x^4 + 6*a^5*x^5))/(26*a*c^3*(1 + a^2*x^2)^2*Sqrt[
c + a^2*c*x^2])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{5} x^{5}+6 a^{4} x^{4}+18 a^{3} x^{3}+14 a^{2} x^{2}+17 a x +9\right ) {\mathrm e}^{\arctan \left (a x \right )}}{26 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\)	\(70\)

[In]

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

[Out]

1/26*(a^2*x^2+1)*(6*a^5*x^5+6*a^4*x^4+18*a^3*x^3+14*a^2*x^2+17*a*x+9)*exp(arctan(a*x))/a/(a^2*c*x^2+c)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (6 \, a^{5} x^{5} + 6 \, a^{4} x^{4} + 18 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 17 \, a x + 9\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (\arctan \left (a x\right )\right )}}{26 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]

[In]

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

[Out]

1/26*(6*a^5*x^5 + 6*a^4*x^4 + 18*a^3*x^3 + 14*a^2*x^2 + 17*a*x + 9)*sqrt(a^2*c*x^2 + c)*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 [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {9}{26\,a^5\,c^3}+\frac {3\,x^5}{13\,c^3}+\frac {17\,x}{26\,a^4\,c^3}+\frac {3\,x^4}{13\,a\,c^3}+\frac {9\,x^3}{13\,a^2\,c^3}+\frac {7\,x^2}{13\,a^3\,c^3}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^4}+x^4\,\sqrt {c\,a^2\,x^2+c}+\frac {2\,x^2\,\sqrt {c\,a^2\,x^2+c}}{a^2}} \]

[In]

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

[Out]

(exp(atan(a*x))*(9/(26*a^5*c^3) + (3*x^5)/(13*c^3) + (17*x)/(26*a^4*c^3) + (3*x^4)/(13*a*c^3) + (9*x^3)/(13*a^
2*c^3) + (7*x^2)/(13*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)