3.8 \(\int \frac {\tan ^{-1}(\frac {\sqrt {-e} x}{\sqrt {d+e x^2}})}{x^5} \, dx\)

Optimal. Leaf size=85 \[ -\frac {(-e)^{3/2} \sqrt {d+e x^2}}{6 d^2 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{12 d x^3} \]

[Out]

-1/4*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^4-1/6*(-e)^(3/2)*(e*x^2+d)^(1/2)/d^2/x-1/12*(-e)^(1/2)*(e*x^2+d)^(
1/2)/d/x^3

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Rubi [A]  time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5151, 271, 264} \[ -\frac {(-e)^{3/2} \sqrt {d+e x^2}}{6 d^2 x}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{12 d x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5,x]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(12*d*x^3) - ((-e)^(3/2)*Sqrt[d + e*x^2])/(6*d^2*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d +
e*x^2]]/(4*x^4)

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]

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 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]

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}+\frac {1}{4} \sqrt {-e} \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{12 d x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}+\frac {(-e)^{3/2} \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{6 d}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{12 d x^3}-\frac {(-e)^{3/2} \sqrt {d+e x^2}}{6 d^2 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 67, normalized size = 0.79 \[ \frac {\sqrt {-e} x \sqrt {d+e x^2} \left (2 e x^2-d\right )-3 d^2 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{12 d^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5,x]

[Out]

(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-d + 2*e*x^2) - 3*d^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(12*d^2*x^4)

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fricas [A]  time = 0.63, size = 58, normalized size = 0.68 \[ -\frac {3 \, d^{2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, e x^{3} - d x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{12 \, d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="fricas")

[Out]

-1/12*(3*d^2*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (2*e*x^3 - d*x)*sqrt(e*x^2 + d)*sqrt(-e))/(d^2*x^4)

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giac [B]  time = 0.26, size = 198, normalized size = 2.33 \[ -\frac {x^{3} {\left (\frac {9 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{2} e^{\left (-2\right )}}{x^{2}} + e^{2}\right )} e^{6}}{96 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{3} d^{2}} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{4 \, x^{4}} + \frac {{\left (\frac {9 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} d^{4} e^{6}}{x} + \frac {{\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{3} d^{4} e^{2}}{x^{3}}\right )} e^{\left (-6\right )}}{96 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="giac")

[Out]

-1/96*x^3*(9*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^2*e^(-2)/x^2 + e^2)*e^6/((sqrt(-x^2*e^2 - d*e)*e - sqrt(-
d*e)*e)^3*d^2) - 1/4*arctan(x*sqrt(-e)/sqrt(x^2*e + d))/x^4 + 1/96*(9*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)*
d^4*e^6/x + (sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^3*d^4*e^2/x^3)*e^(-6)/d^6

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maple [A]  time = 0.04, size = 69, normalized size = 0.81 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{4 x^{4}}+\frac {\sqrt {-e}\, e \sqrt {e \,x^{2}+d}}{4 d^{2} x}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{12 d^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x)

[Out]

-1/4*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^4+1/4*(-e)^(1/2)*e/d^2/x*(e*x^2+d)^(1/2)-1/12*(-e)^(1/2)/d^2/x^3*(
e*x^2+d)^(3/2)

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maxima [A]  time = 0.33, size = 68, normalized size = 0.80 \[ \frac {\sqrt {e x^{2} + d} \sqrt {-e} e}{4 \, d^{2} x} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} \sqrt {-e}}{12 \, d^{2} x^{3}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="maxima")

[Out]

1/4*sqrt(e*x^2 + d)*sqrt(-e)*e/(d^2*x) - 1/12*(e*x^2 + d)^(3/2)*sqrt(-e)/(d^2*x^3) - 1/4*arctan(sqrt(-e)*x/sqr
t(e*x^2 + d))/x^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^5,x)

[Out]

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^5, x)

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sympy [A]  time = 5.13, size = 83, normalized size = 0.98 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{4 x^{4}} - \frac {\sqrt {e} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{12 d x^{2}} + \frac {e^{\frac {3}{2}} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{6 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2))/x**5,x)

[Out]

-atan(x*sqrt(-e)/sqrt(d + e*x**2))/(4*x**4) - sqrt(e)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(12*d*x**2) + e**(3/2)*sqr
t(-e)*sqrt(d/(e*x**2) + 1)/(6*d**2)

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