3.2.45 \(\int \frac {1}{\sqrt {a+b x^2} \text {ArcTan}(\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}})^3} \, dx\) [145]

Optimal. Leaf size=68 \[ -\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2} \]

[Out]

-1/2*(-a*e^2/b-e^2*x^2)^(1/2)/e/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^2/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5265, 5263} \begin {gather*} -\frac {\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3),x]

[Out]

-1/2*Sqrt[-((a*e^2)/b) - e^2*x^2]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2)

Rule 5263

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

Rule 5265

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]^(m_.)/Sqrt[(d_.) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a
+ b*x^2]/Sqrt[d + e*x^2], Int[ArcTan[c*(x/Sqrt[a + b*x^2])]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e
, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^3} \, dx &=\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2} \int \frac {1}{\sqrt {-\frac {a e^2}{b}-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^3} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2}}{2 e \sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 62, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3),x]

[Out]

-1/2*Sqrt[-((e^2*(a + b*x^2))/b)]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]]^2)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\arctan \left (\frac {e x}{\sqrt {-\frac {a \,e^{2}}{b}-e^{2} x^{2}}}\right )^{3} \sqrt {x^{2} b +a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x)

[Out]

int(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found  sqrt((-_SAGE_VAR_b*_SAGE_VAR_x^2)-_SAGE_VAR_a)

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Fricas [A]
time = 2.58, size = 62, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-\frac {{\left (b x^{2} + a\right )} e^{2}}{b}} e^{\left (-1\right )}}{2 \, \sqrt {b x^{2} + a} \arctan \left (\frac {b x \sqrt {-\frac {{\left (b x^{2} + a\right )} e^{2}}{b}} e^{\left (-1\right )}}{b x^{2} + a}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-(b*x^2 + a)*e^2/b)*e^(-1)/(sqrt(b*x^2 + a)*arctan(b*x*sqrt(-(b*x^2 + a)*e^2/b)*e^(-1)/(b*x^2 + a))^
2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x^{2}} \operatorname {atan}^{3}{\left (\frac {e x}{\sqrt {- \frac {a e^{2}}{b} - e^{2} x^{2}}} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atan(e*x/(-a*e**2/b-e**2*x**2)**(1/2))**3/(b*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x**2)*atan(e*x/sqrt(-a*e**2/b - e**2*x**2))**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + a)*arctan(e*x/sqrt(-e^2*x^2 - a*e^2/b))^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {atan}\left (\frac {e\,x}{\sqrt {-e^2\,x^2-\frac {a\,e^2}{b}}}\right )}^3\,\sqrt {b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(atan((e*x)/(- e^2*x^2 - (a*e^2)/b)^(1/2))^3*(a + b*x^2)^(1/2)),x)

[Out]

int(1/(atan((e*x)/(- e^2*x^2 - (a*e^2)/b)^(1/2))^3*(a + b*x^2)^(1/2)), x)

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