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

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

[Out]

-1/8*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^8-3/140*(-e)^(3/2)*(e*x^2+d)^(1/2)/d^2/x^5-1/35*(-e)^(5/2)*(e*x^2+
d)^(1/2)/d^3/x^3-2/35*(-e)^(7/2)*(e*x^2+d)^(1/2)/d^4/x-1/56*(-e)^(1/2)*(e*x^2+d)^(1/2)/d/x^7

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Rubi [A]  time = 0.05, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 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^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*}

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Mathematica [A]  time = 0.07, size = 89, normalized size = 0.63 \[ \frac {\sqrt {-e} x \sqrt {d+e x^2} \left (-5 d^3+6 d^2 e x^2-8 d e^2 x^4+16 e^3 x^6\right )-35 d^4 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{280 d^4 x^8} \]

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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fricas [A]  time = 0.79, size = 80, normalized size = 0.57 \[ -\frac {35 \, d^{4} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (16 \, e^{3} x^{7} - 8 \, d e^{2} x^{5} + 6 \, d^{2} e x^{3} - 5 \, d^{3} x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{280 \, d^{4} x^{8}} \]

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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giac [B]  time = 0.26, size = 361, normalized size = 2.56 \[ -\frac {x^{7} {\left (\frac {245 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{4} e^{\left (-4\right )}}{x^{4}} + \frac {1225 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{6} e^{\left (-8\right )}}{x^{6}} + \frac {49 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{2}}{x^{2}} + 5 \, e^{4}\right )} e^{14}}{35840 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{7} d^{4}} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{8 \, x^{8}} + \frac {{\left (\frac {1225 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} d^{24} e^{30}}{x} + \frac {245 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{3} d^{24} e^{26}}{x^{3}} + \frac {49 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{5} d^{24} e^{22}}{x^{5}} + \frac {5 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{7} d^{24} e^{18}}{x^{7}}\right )} e^{\left (-28\right )}}{35840 \, d^{28}} \]

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

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maple [A]  time = 0.05, size = 167, normalized size = 1.18 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{8 x^{8}}+\frac {\sqrt {-e}\, e \sqrt {e \,x^{2}+d}}{40 d^{2} x^{5}}-\frac {\sqrt {-e}\, e^{2} \sqrt {e \,x^{2}+d}}{30 d^{3} x^{3}}+\frac {\sqrt {-e}\, e^{3} \sqrt {e \,x^{2}+d}}{15 d^{4} x}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{56 d^{2} x^{7}}+\frac {\sqrt {-e}\, e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{70 d^{3} x^{5}}-\frac {\sqrt {-e}\, e^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{105 d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^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 = 0.34, size = 132, normalized size = 0.94 \[ \frac {{\left (8 \, e^{3} x^{6} + 4 \, d e^{2} x^{4} - d^{2} e x^{2} + 3 \, d^{3}\right )} \sqrt {-e} e}{120 \, \sqrt {e x^{2} + d} d^{4} x^{5}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{8 \, x^{8}} - \frac {{\left (8 \, e^{3} x^{6} - 4 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + 15 \, d^{3}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{840 \, d^{4} x^{7}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 9.15, size = 575, normalized size = 4.08 \[ - \frac {5 d^{6} e^{\frac {19}{2}} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} - \frac {9 d^{5} e^{\frac {21}{2}} x^{2} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} - \frac {5 d^{4} e^{\frac {23}{2}} x^{4} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} + \frac {5 d^{3} e^{\frac {25}{2}} x^{6} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{280 d^{7} e^{9} x^{6} + 840 d^{6} e^{10} x^{8} + 840 d^{5} e^{11} x^{10} + 280 d^{4} e^{12} x^{12}} + \frac {15 d^{2} e^{\frac {27}{2}} x^{8} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{140 d^{7} e^{9} x^{6} + 420 d^{6} e^{10} x^{8} + 420 d^{5} e^{11} x^{10} + 140 d^{4} e^{12} x^{12}} + \frac {5 d e^{\frac {29}{2}} x^{10} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} + \frac {2 e^{\frac {31}{2}} x^{12} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{35 d^{7} e^{9} x^{6} + 105 d^{6} e^{10} x^{8} + 105 d^{5} e^{11} x^{10} + 35 d^{4} e^{12} x^{12}} - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{8 x^{8}} \]

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)/(280*d**7*e**9*x**6 + 840*d**6*e**10*x**8 + 840*d**5*e**11*x**
10 + 280*d**4*e**12*x**12) - 9*d**5*e**(21/2)*x**2*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840*d**
6*e**10*x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) - 5*d**4*e**(23/2)*x**4*sqrt(-e)*sqrt(d/(e*x**2) +
1)/(280*d**7*e**9*x**6 + 840*d**6*e**10*x**8 + 840*d**5*e**11*x**10 + 280*d**4*e**12*x**12) + 5*d**3*e**(25/2
)*x**6*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(280*d**7*e**9*x**6 + 840*d**6*e**10*x**8 + 840*d**5*e**11*x**10 + 280*d*
*4*e**12*x**12) + 15*d**2*e**(27/2)*x**8*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(140*d**7*e**9*x**6 + 420*d**6*e**10*x*
*8 + 420*d**5*e**11*x**10 + 140*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*sqrt(-e)*sq
rt(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) - at
an(x*sqrt(-e)/sqrt(d + e*x**2))/(8*x**8)

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