3.10 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^9} \, dx\)

Optimal. Leaf size=141 \[ -\frac{2 (-e)^{7/2} \sqrt{d+e x^2}}{35 d^4 x}-\frac{(-e)^{5/2} \sqrt{d+e x^2}}{35 d^3 x^3}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8} \]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(56*d*x^7) - (3*(-e)^(3/2)*Sqrt[d + e*x^2])/(140*d^2*x^5) - ((-e)^(5/2)*Sqrt[d + e
*x^2])/(35*d^3*x^3) - (2*(-e)^(7/2)*Sqrt[d + e*x^2])/(35*d^4*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(8*x^8)

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Rubi [A]  time = 0.0495252, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5151, 271, 264} \[ -\frac{2 (-e)^{7/2} \sqrt{d+e x^2}}{35 d^4 x}-\frac{(-e)^{5/2} \sqrt{d+e x^2}}{35 d^3 x^3}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^9,x]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(56*d*x^7) - (3*(-e)^(3/2)*Sqrt[d + e*x^2])/(140*d^2*x^5) - ((-e)^(5/2)*Sqrt[d + e
*x^2])/(35*d^3*x^3) - (2*(-e)^(7/2)*Sqrt[d + e*x^2])/(35*d^4*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(8*x^8)

Rule 5151

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcTa
n[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^9} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8}+\frac{1}{8} \sqrt{-e} \int \frac{1}{x^8 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8}+\frac{\left (3 (-e)^{3/2}\right ) \int \frac{1}{x^6 \sqrt{d+e x^2}} \, dx}{28 d}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8}+\frac{\left (3 (-e)^{5/2}\right ) \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx}{35 d^2}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{(-e)^{5/2} \sqrt{d+e x^2}}{35 d^3 x^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8}+\frac{\left (2 (-e)^{7/2}\right ) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{35 d^3}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{56 d x^7}-\frac{3 (-e)^{3/2} \sqrt{d+e x^2}}{140 d^2 x^5}-\frac{(-e)^{5/2} \sqrt{d+e x^2}}{35 d^3 x^3}-\frac{2 (-e)^{7/2} \sqrt{d+e x^2}}{35 d^4 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0682556, size = 89, normalized size = 0.63 \[ \frac{\sqrt{-e} x \sqrt{d+e x^2} \left (6 d^2 e x^2-5 d^3-8 d e^2 x^4+16 e^3 x^6\right )-35 d^4 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{280 d^4 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^9,x]

[Out]

(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-5*d^3 + 6*d^2*e*x^2 - 8*d*e^2*x^4 + 16*e^3*x^6) - 35*d^4*ArcTan[(Sqrt[-e]*x)/Sqr
t[d + e*x^2]])/(280*d^4*x^8)

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Maple [A]  time = 0.043, size = 167, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,{x}^{8}}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{e}{40\,{d}^{2}{x}^{5}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{e}^{2}}{30\,{d}^{3}{x}^{3}}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{{e}^{3}}{15\,{d}^{4}x}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{1}{56\,{d}^{2}{x}^{7}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{e}{70\,{d}^{3}{x}^{5}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}}{105\,{d}^{4}{x}^{3}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^9,x)

[Out]

-1/8*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^8+1/40*(-e)^(1/2)*e/d^2/x^5*(e*x^2+d)^(1/2)-1/30*(-e)^(1/2)*e^2/d^
3/x^3*(e*x^2+d)^(1/2)+1/15*(-e)^(1/2)*e^3/d^4/x*(e*x^2+d)^(1/2)-1/56*(-e)^(1/2)/d^2/x^7*(e*x^2+d)^(3/2)+1/70*(
-e)^(1/2)/d^3*e/x^5*(e*x^2+d)^(3/2)-1/105*(-e)^(1/2)/d^4*e^2/x^3*(e*x^2+d)^(3/2)

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Maxima [A]  time = 1.0016, size = 178, normalized size = 1.26 \begin{align*} \frac{{\left (8 \, e^{3} x^{6} + 4 \, d e^{2} x^{4} - d^{2} e x^{2} + 3 \, d^{3}\right )} \sqrt{-e} e}{120 \, \sqrt{e x^{2} + d} d^{4} x^{5}} - \frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{8 \, x^{8}} - \frac{{\left (8 \, e^{3} x^{6} - 4 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + 15 \, d^{3}\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{840 \, d^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^9,x, algorithm="maxima")

[Out]

1/120*(8*e^3*x^6 + 4*d*e^2*x^4 - d^2*e*x^2 + 3*d^3)*sqrt(-e)*e/(sqrt(e*x^2 + d)*d^4*x^5) - 1/8*arctan(sqrt(-e)
*x/sqrt(e*x^2 + d))/x^8 - 1/840*(8*e^3*x^6 - 4*d*e^2*x^4 + 3*d^2*e*x^2 + 15*d^3)*sqrt(e*x^2 + d)*sqrt(-e)/(d^4
*x^7)

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Fricas [A]  time = 3.18562, size = 189, normalized size = 1.34 \begin{align*} -\frac{35 \, d^{4} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (16 \, e^{3} x^{7} - 8 \, d e^{2} x^{5} + 6 \, d^{2} e x^{3} - 5 \, d^{3} x\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{280 \, d^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^9,x, algorithm="fricas")

[Out]

-1/280*(35*d^4*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (16*e^3*x^7 - 8*d*e^2*x^5 + 6*d^2*e*x^3 - 5*d^3*x)*sqrt(e*
x^2 + d)*sqrt(-e))/(d^4*x^8)

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Sympy [B]  time = 31.1258, size = 575, normalized size = 4.08 \begin{align*} - \frac{5 d^{6} e^{\frac{19}{2}} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{8 \left (35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}\right )} - \frac{9 d^{5} e^{\frac{21}{2}} x^{2} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{8 \left (35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}\right )} - \frac{5 d^{4} e^{\frac{23}{2}} x^{4} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{8 \left (35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}\right )} + \frac{5 d^{3} e^{\frac{25}{2}} x^{6} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{8 \left (35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}\right )} + \frac{15 d^{2} e^{\frac{27}{2}} x^{8} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{4 \left (35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}\right )} + \frac{5 d e^{\frac{29}{2}} x^{10} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} + \frac{2 e^{\frac{31}{2}} x^{12} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2))/x**9,x)

[Out]

-5*d**6*e**(19/2)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(8*(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x
**10 + 35*d**4*e**12*x**12)) - 9*d**5*e**(21/2)*x**2*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(8*(35*d**7*e**9*x**6 + 105
*d**6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x**12)) - 5*d**4*e**(23/2)*x**4*sqrt(-e)*sqrt(d/(e*x**
2) + 1)/(8*(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x**12)) + 5*d**3*e*
*(25/2)*x**6*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(8*(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x**10
+ 35*d**4*e**12*x**12)) + 15*d**2*e**(27/2)*x**8*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(4*(35*d**7*e**9*x**6 + 105*d**
6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x**12)) + 5*d*e**(29/2)*x**10*sqrt(-e)*sqrt(d/(e*x**2) + 1
)/(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x**12) + 2*e**(31/2)*x**12*s
qrt(-e)*sqrt(d/(e*x**2) + 1)/(35*d**7*e**9*x**6 + 105*d**6*e**10*x**8 + 105*d**5*e**11*x**10 + 35*d**4*e**12*x
**12) - atan(x*sqrt(-e)/sqrt(d + e*x**2))/(8*x**8)

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Giac [B]  time = 1.21304, size = 487, normalized size = 3.45 \begin{align*} -\frac{x^{7}{\left (\frac{245 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{4} e^{\left (-4\right )}}{x^{4}} + \frac{1225 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{6} e^{\left (-8\right )}}{x^{6}} + \frac{49 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{2}}{x^{2}} + 5 \, e^{4}\right )} e^{14}}{35840 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{7} d^{4}} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{8 \, x^{8}} + \frac{{\left (\frac{1225 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} d^{24} e^{30}}{x} + \frac{245 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{3} d^{24} e^{26}}{x^{3}} + \frac{49 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{5} d^{24} e^{22}}{x^{5}} + \frac{5 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{7} d^{24} e^{18}}{x^{7}}\right )} e^{\left (-28\right )}}{35840 \, d^{28}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^9,x, algorithm="giac")

[Out]

-1/35840*x^7*(245*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^4*e^(-4)/x^4 + 1225*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-
d*e)*e)^6*e^(-8)/x^6 + 49*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^2/x^2 + 5*e^4)*e^14/((sqrt(-x^2*e^2 - d*e)*e
- sqrt(-d*e)*e)^7*d^4) - 1/8*arctan(x*sqrt(-e)/sqrt(x^2*e + d))/x^8 + 1/35840*(1225*(sqrt(-x^2*e^2 - d*e)*e -
sqrt(-d*e)*e)*d^24*e^30/x + 245*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^3*d^24*e^26/x^3 + 49*(sqrt(-x^2*e^2 -
d*e)*e - sqrt(-d*e)*e)^5*d^24*e^22/x^5 + 5*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^7*d^24*e^18/x^7)*e^(-28)/d
^28