3.7 \(\int \frac{e^{\cot ^{-1}(x)}}{(a+a x^2)^{7/2}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac{(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]

[Out]

-(E^ArcCot[x]*(1 - 5*x))/(26*a*(a + a*x^2)^(5/2)) - (E^ArcCot[x]*(1 - 3*x))/(13*a^2*(a + a*x^2)^(3/2)) - (3*E^
ArcCot[x]*(1 - x))/(13*a^3*Sqrt[a + a*x^2])

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Rubi [A]  time = 0.090235, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5115, 5114} \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac{(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCot[x]/(a + a*x^2)^(7/2),x]

[Out]

-(E^ArcCot[x]*(1 - 5*x))/(26*a*(a + a*x^2)^(5/2)) - (E^ArcCot[x]*(1 - 3*x))/(13*a^2*(a + a*x^2)^(3/2)) - (3*E^
ArcCot[x]*(1 - x))/(13*a^3*Sqrt[a + a*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 5114

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[((n - a*x)*E^(n*ArcCot[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 - 1)/2]

Rubi steps

\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{7/2}} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}+\frac{10 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{5/2}} \, dx}{13 a}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}+\frac{6 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx}{13 a^2}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}-\frac{3 e^{\cot ^{-1}(x)} (1-x)}{13 a^3 \sqrt{a+a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.208714, size = 95, normalized size = 1.16 \[ \frac{e^{\cot ^{-1}(x)} \left (-39 \sqrt{\frac{1}{x^2}+1} x \cos \left (3 \cot ^{-1}(x)\right )+5 \sqrt{\frac{1}{x^2}+1} x \cos \left (5 \cot ^{-1}(x)\right )+13 \sqrt{\frac{1}{x^2}+1} x \sin \left (3 \cot ^{-1}(x)\right )-\sqrt{\frac{1}{x^2}+1} x \sin \left (5 \cot ^{-1}(x)\right )+130 x-130\right )}{416 a^3 \sqrt{a \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^ArcCot[x]*(-130 + 130*x - 39*Sqrt[1 + x^(-2)]*x*Cos[3*ArcCot[x]] + 5*Sqrt[1 + x^(-2)]*x*Cos[5*ArcCot[x]] +
13*Sqrt[1 + x^(-2)]*x*Sin[3*ArcCot[x]] - Sqrt[1 + x^(-2)]*x*Sin[5*ArcCot[x]]))/(416*a^3*Sqrt[a*(1 + x^2)])

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Maple [A]  time = 0.042, size = 45, normalized size = 0.6 \begin{align*}{\frac{ \left ({x}^{2}+1 \right ) \left ( 6\,{x}^{5}-6\,{x}^{4}+18\,{x}^{3}-14\,{x}^{2}+17\,x-9 \right ){{\rm e}^{{\rm arccot} \left (x\right )}}}{26} \left ( a{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.4986, size = 161, normalized size = 1.96 \begin{align*} \frac{{\left (6 \, x^{5} - 6 \, x^{4} + 18 \, x^{3} - 14 \, x^{2} + 17 \, x - 9\right )} \sqrt{a x^{2} + a} e^{\operatorname{arccot}\left (x\right )}}{26 \,{\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26*(6*x^5 - 6*x^4 + 18*x^3 - 14*x^2 + 17*x - 9)*sqrt(a*x^2 + a)*e^arccot(x)/(a^4*x^6 + 3*a^4*x^4 + 3*a^4*x^2
+ a^4)

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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(acot(x))/(a*x**2+a)**(7/2),x)

[Out]

Timed out

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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 )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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