3.252 \(\int \frac{e^{\tan ^{-1}(a x)}}{(c+a^2 c x^2)^5} \, dx\)

Optimal. Leaf size=149 \[ \frac{4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac{336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac{56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac{(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]

[Out]

(8064*E^ArcTan[a*x])/(40885*a*c^5) + (E^ArcTan[a*x]*(1 + 8*a*x))/(65*a*c^5*(1 + a^2*x^2)^4) + (56*E^ArcTan[a*x
]*(1 + 6*a*x))/(2405*a*c^5*(1 + a^2*x^2)^3) + (336*E^ArcTan[a*x]*(1 + 4*a*x))/(8177*a*c^5*(1 + a^2*x^2)^2) + (
4032*E^ArcTan[a*x]*(1 + 2*a*x))/(40885*a*c^5*(1 + a^2*x^2))

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Rubi [A]  time = 0.140465, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac{4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac{336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac{56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac{(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTan[a*x]/(c + a^2*c*x^2)^5,x]

[Out]

(8064*E^ArcTan[a*x])/(40885*a*c^5) + (E^ArcTan[a*x]*(1 + 8*a*x))/(65*a*c^5*(1 + a^2*x^2)^4) + (56*E^ArcTan[a*x
]*(1 + 6*a*x))/(2405*a*c^5*(1 + a^2*x^2)^3) + (336*E^ArcTan[a*x]*(1 + 4*a*x))/(8177*a*c^5*(1 + a^2*x^2)^2) + (
4032*E^ArcTan[a*x]*(1 + 2*a*x))/(40885*a*c^5*(1 + a^2*x^2))

Rule 5070

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 5071

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*} \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^5} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx}{65 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{481 c^2}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{8177 c^3}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}+\frac{8064 \int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{40885 c^4}\\ &=\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5}+\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.257846, size = 153, normalized size = 1.03 \[ \frac{629 (8 a x+1) e^{\tan ^{-1}(a x)}+\frac{56 \left (a^2 c x^2+c\right ) \left (17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left (a^2 c x^2+c\right ) \left (5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x-i) (a x+i) \left (2 a^2 x^2+2 a x+3\right )\right )\right )}{c^2}}{40885 a c \left (a^2 c x^2+c\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^5,x]

[Out]

(629*E^ArcTan[a*x]*(1 + 8*a*x) + (56*(c + a^2*c*x^2)*(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
))))/c^2)/(40885*a*c*(c + a^2*c*x^2)^4)

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Maple [A]  time = 0.037, size = 87, normalized size = 0.6 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }} \left ( 8064\,{a}^{8}{x}^{8}+8064\,{a}^{7}{x}^{7}+36288\,{a}^{6}{x}^{6}+30912\,{a}^{5}{x}^{5}+62160\,{a}^{4}{x}^{4}+43344\,{a}^{3}{x}^{3}+48664\,{a}^{2}{x}^{2}+25528\,ax+15357 \right ) }{40885\, \left ({a}^{2}{x}^{2}+1 \right ) ^{4}{c}^{5}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x)

[Out]

1/40885*exp(arctan(a*x))*(8064*a^8*x^8+8064*a^7*x^7+36288*a^6*x^6+30912*a^5*x^5+62160*a^4*x^4+43344*a^3*x^3+48
664*a^2*x^2+25528*a*x+15357)/(a^2*x^2+1)^4/c^5/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="maxima")

[Out]

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^5, x)

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Fricas [A]  time = 1.97205, size = 304, normalized size = 2.04 \begin{align*} \frac{{\left (8064 \, a^{8} x^{8} + 8064 \, a^{7} x^{7} + 36288 \, a^{6} x^{6} + 30912 \, a^{5} x^{5} + 62160 \, a^{4} x^{4} + 43344 \, a^{3} x^{3} + 48664 \, a^{2} x^{2} + 25528 \, a x + 15357\right )} e^{\left (\arctan \left (a x\right )\right )}}{40885 \,{\left (a^{9} c^{5} x^{8} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} + 4 \, a^{3} c^{5} x^{2} + a c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="fricas")

[Out]

1/40885*(8064*a^8*x^8 + 8064*a^7*x^7 + 36288*a^6*x^6 + 30912*a^5*x^5 + 62160*a^4*x^4 + 43344*a^3*x^3 + 48664*a
^2*x^2 + 25528*a*x + 15357)*e^(arctan(a*x))/(a^9*c^5*x^8 + 4*a^7*c^5*x^6 + 6*a^5*c^5*x^4 + 4*a^3*c^5*x^2 + a*c
^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(atan(a*x))/(a**2*c*x**2+c)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="giac")

[Out]

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^5, x)