3.9 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^7} \, dx\)

Optimal. Leaf size=113 \[ -\frac{4 (-e)^{5/2} \sqrt{d+e x^2}}{45 d^3 x}-\frac{2 (-e)^{3/2} \sqrt{d+e x^2}}{45 d^2 x^3}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6} \]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(30*d*x^5) - (2*(-e)^(3/2)*Sqrt[d + e*x^2])/(45*d^2*x^3) - (4*(-e)^(5/2)*Sqrt[d +
e*x^2])/(45*d^3*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(6*x^6)

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Rubi [A]  time = 0.0385326, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5151, 271, 264} \[ -\frac{4 (-e)^{5/2} \sqrt{d+e x^2}}{45 d^3 x}-\frac{2 (-e)^{3/2} \sqrt{d+e x^2}}{45 d^2 x^3}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^7,x]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(30*d*x^5) - (2*(-e)^(3/2)*Sqrt[d + e*x^2])/(45*d^2*x^3) - (4*(-e)^(5/2)*Sqrt[d +
e*x^2])/(45*d^3*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(6*x^6)

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^7} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6}+\frac{1}{6} \sqrt{-e} \int \frac{1}{x^6 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6}+\frac{\left (2 (-e)^{3/2}\right ) \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx}{15 d}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}-\frac{2 (-e)^{3/2} \sqrt{d+e x^2}}{45 d^2 x^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6}+\frac{\left (4 (-e)^{5/2}\right ) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{45 d^2}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{30 d x^5}-\frac{2 (-e)^{3/2} \sqrt{d+e x^2}}{45 d^2 x^3}-\frac{4 (-e)^{5/2} \sqrt{d+e x^2}}{45 d^3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0571997, size = 78, normalized size = 0.69 \[ \frac{\sqrt{-e} x \sqrt{d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )-15 d^3 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{90 d^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^7,x]

[Out]

(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-3*d^2 + 4*d*e*x^2 - 8*e^2*x^4) - 15*d^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(9
0*d^3*x^6)

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Maple [A]  time = 0.042, size = 117, normalized size = 1. \begin{align*} -{\frac{1}{6\,{x}^{6}}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{e}{18\,{d}^{2}{x}^{3}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{e}^{2}}{9\,{d}^{3}x}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{1}{30\,{d}^{2}{x}^{5}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{e}{45\,{d}^{3}{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^7,x)

[Out]

-1/6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^6+1/18*(-e)^(1/2)*e/d^2/x^3*(e*x^2+d)^(1/2)-1/9*(-e)^(1/2)*e^2/d^3
/x*(e*x^2+d)^(1/2)-1/30*(-e)^(1/2)/d^2/x^5*(e*x^2+d)^(3/2)+1/45*(-e)^(1/2)/d^3*e/x^3*(e*x^2+d)^(3/2)

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Maxima [A]  time = 1.01803, size = 147, normalized size = 1.3 \begin{align*} -\frac{{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} \sqrt{-e} e}{18 \, \sqrt{e x^{2} + d} d^{3} x^{3}} - \frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{6 \, x^{6}} + \frac{{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{90 \, d^{3} x^{5}} \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^7,x, algorithm="maxima")

[Out]

-1/18*(2*e^2*x^4 + d*e*x^2 - d^2)*sqrt(-e)*e/(sqrt(e*x^2 + d)*d^3*x^3) - 1/6*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)
)/x^6 + 1/90*(2*e^2*x^4 - d*e*x^2 - 3*d^2)*sqrt(e*x^2 + d)*sqrt(-e)/(d^3*x^5)

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Fricas [A]  time = 2.68875, size = 165, normalized size = 1.46 \begin{align*} -\frac{15 \, d^{3} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (8 \, e^{2} x^{5} - 4 \, d e x^{3} + 3 \, d^{2} x\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{90 \, d^{3} x^{6}} \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^7,x, algorithm="fricas")

[Out]

-1/90*(15*d^3*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (8*e^2*x^5 - 4*d*e*x^3 + 3*d^2*x)*sqrt(e*x^2 + d)*sqrt(-e))
/(d^3*x^6)

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Sympy [B]  time = 12.4261, size = 352, normalized size = 3.12 \begin{align*} - \frac{d^{4} e^{\frac{9}{2}} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{2 \left (15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}\right )} - \frac{d^{3} e^{\frac{11}{2}} x^{2} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{3 \left (15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}\right )} - \frac{d^{2} e^{\frac{13}{2}} x^{4} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{2 \left (15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}\right )} - \frac{2 d e^{\frac{15}{2}} x^{6} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}} - \frac{4 e^{\frac{17}{2}} x^{8} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{3 \left (15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}\right )} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{6 x^{6}} \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**7,x)

[Out]

-d**4*e**(9/2)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(2*(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d**3*e**6*x**8)) -
d**3*e**(11/2)*x**2*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(3*(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d**3*e**6*x*
*8)) - d**2*e**(13/2)*x**4*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(2*(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d**3*e
**6*x**8)) - 2*d*e**(15/2)*x**6*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d**3
*e**6*x**8) - 4*e**(17/2)*x**8*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(3*(15*d**5*e**4*x**4 + 30*d**4*e**5*x**6 + 15*d*
*3*e**6*x**8)) - atan(x*sqrt(-e)/sqrt(d + e*x**2))/(6*x**6)

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Giac [B]  time = 1.2122, size = 381, normalized size = 3.37 \begin{align*} \frac{x^{5}{\left (\frac{25 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} + \frac{150 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{4} e^{\left (-5\right )}}{x^{4}} + 3 \, e^{3}\right )} e^{10}}{2880 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{5} d^{3}} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{6 \, x^{6}} - \frac{{\left (\frac{150 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} d^{12} e^{16}}{x} + \frac{25 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{3} d^{12} e^{12}}{x^{3}} + \frac{3 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{5} d^{12} e^{8}}{x^{5}}\right )} e^{\left (-15\right )}}{2880 \, d^{15}} \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^7,x, algorithm="giac")

[Out]

1/2880*x^5*(25*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^2*e^(-1)/x^2 + 150*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)
*e)^4*e^(-5)/x^4 + 3*e^3)*e^10/((sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^5*d^3) - 1/6*arctan(x*sqrt(-e)/sqrt(x^
2*e + d))/x^6 - 1/2880*(150*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)*d^12*e^16/x + 25*(sqrt(-x^2*e^2 - d*e)*e -
sqrt(-d*e)*e)^3*d^12*e^12/x^3 + 3*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^5*d^12*e^8/x^5)*e^(-15)/d^15