3.4 \(\int \frac{e^{\cot ^{-1}(x)}}{(a+a x^2)^3} \, dx\)

Optimal. Leaf size=58 \[ -\frac{(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac{12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3} \]

[Out]

(-24*E^ArcCot[x])/(85*a^3) - (E^ArcCot[x]*(1 - 4*x))/(17*a^3*(1 + x^2)^2) - (12*E^ArcCot[x]*(1 - 2*x))/(85*a^3
*(1 + x^2))

________________________________________________________________________________________

Rubi [A]  time = 0.0668886, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac{(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left (x^2+1\right )^2}-\frac{12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )}-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2)^3,x]

[Out]

(-24*E^ArcCot[x])/(85*a^3) - (E^ArcCot[x]*(1 - 4*x))/(17*a^3*(1 + x^2)^2) - (12*E^ArcCot[x]*(1 - 2*x))/(85*a^3
*(1 + x^2))

Rule 5115

Int[E^(ArcCot[(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*ArcCot[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*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&& NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] &&  !(IntegerQ[p] && IntegerQ[(I*n)/2]) &&  !( !IntegerQ[p] && I
ntegerQ[(I*n - 1)/2])

Rule 5113

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> -Simp[E^(n*ArcCot[a*x])/(a*c*n), x] /; Fr
eeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps

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

Mathematica [A]  time = 0.144909, size = 38, normalized size = 0.66 \[ -\frac{\left (24 x^4-24 x^3+60 x^2-44 x+41\right ) e^{\cot ^{-1}(x)}}{85 a^3 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^ArcCot[x]*(41 - 44*x + 60*x^2 - 24*x^3 + 24*x^4))/(85*a^3*(1 + x^2)^2)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 36, normalized size = 0.6 \begin{align*} -{\frac{{{\rm e}^{{\rm arccot} \left (x\right )}} \left ( 24\,{x}^{4}-24\,{x}^{3}+60\,{x}^{2}-44\,x+41 \right ) }{85\, \left ({x}^{2}+1 \right ) ^{2}{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/85*exp(arccot(x))*(24*x^4-24*x^3+60*x^2-44*x+41)/(x^2+1)^2/a^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.49881, size = 116, normalized size = 2. \begin{align*} -\frac{{\left (24 \, x^{4} - 24 \, x^{3} + 60 \, x^{2} - 44 \, x + 41\right )} e^{\operatorname{arccot}\left (x\right )}}{85 \,{\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/85*(24*x^4 - 24*x^3 + 60*x^2 - 44*x + 41)*e^arccot(x)/(a^3*x^4 + 2*a^3*x^2 + a^3)

________________________________________________________________________________________

Sympy [B]  time = 16.5687, size = 155, normalized size = 2.67 \begin{align*} - \frac{24 x^{4} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac{24 x^{3} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac{60 x^{2} e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} + \frac{44 x e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} - \frac{41 e^{\operatorname{acot}{\left (x \right )}}}{85 a^{3} x^{4} + 170 a^{3} x^{2} + 85 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*x**4*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3) + 24*x**3*exp(acot(x))/(85*a**3*x**4 + 170*a**3
*x**2 + 85*a**3) - 60*x**2*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3) + 44*x*exp(acot(x))/(85*a**3*
x**4 + 170*a**3*x**2 + 85*a**3) - 41*exp(acot(x))/(85*a**3*x**4 + 170*a**3*x**2 + 85*a**3)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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