3.250 \(\int \frac{e^{\tan ^{-1}(a x)}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=83 \[ \frac{12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac{(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]

[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))

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Rubi [A]  time = 0.0777701, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac{12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac{(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]

Antiderivative was successfully verified.

[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 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 )^3} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac{24 \int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{85 c^2}\\ &=\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3}+\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}\\ \end{align*}

Mathematica [C]  time = 0.248842, size = 114, normalized size = 1.37 \[ \frac{\frac{5 (4 a x+1) e^{\tan ^{-1}(a x)}}{\left (a^2 x^2+1\right )^2}+\frac{24 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x+(1-i))}{a x-i}+(12-24 i) (1-i a x)^{-1+\frac{i}{2}} (1+i a x)^{-1-\frac{i}{2}}}{85 a c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((12 - 24*I)/((1 - I*a*x)^(1 - I/2)*(1 + I*a*x)^(1 + I/2)) + (24*(1 - I*a*x)^(I/2)*((1 - I) + a*x))/((1 + I*a*
x)^(I/2)*(-I + a*x)) + (5*E^ArcTan[a*x]*(1 + 4*a*x))/(1 + a^2*x^2)^2)/(85*a*c^3)

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Maple [A]  time = 0.036, size = 55, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }} \left ( 24\,{a}^{4}{x}^{4}+24\,{a}^{3}{x}^{3}+60\,{a}^{2}{x}^{2}+44\,ax+41 \right ) }{85\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}a{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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/a/c^3

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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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02264, size = 153, normalized size = 1.84 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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