3.2082 \(\int \frac {1}{\sqrt {-d+e x} \sqrt {d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {661, 205} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-d + e*x]*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 661

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 53, normalized size = 1.00 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-d + e*x]*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

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fricas [A]  time = 0.84, size = 147, normalized size = 2.77 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{d}} \log \left (-\frac {e^{2} x^{2} + 2 \, d e x - 2 \, \sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d} d \sqrt {-\frac {1}{d}} - 3 \, d^{2}}{e^{2} x^{2} - 2 \, d e x + d^{2}}\right )}{2 \, e}, \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d} \sqrt {d}}{e^{2} x^{2} - d^{2}}\right )}{\sqrt {d} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*sqrt(-1/d)*log(-(e^2*x^2 + 2*d*e*x - 2*sqrt(2)*sqrt(-e^2*x^2 + d^2)*sqrt(e*x - d)*d*sqrt(-1/d) -
3*d^2)/(e^2*x^2 - 2*d*e*x + d^2))/e, sqrt(2)*arctan(sqrt(2)*sqrt(-e^2*x^2 + d^2)*sqrt(e*x - d)*sqrt(d)/(e^2*x^
2 - d^2))/(sqrt(d)*e)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*sqrt(e*x - d)), x)

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maple [A]  time = 0.06, size = 63, normalized size = 1.19 \[ \frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-e x -d}\, \sqrt {2}}{2 \sqrt {d}}\right )}{\sqrt {e x -d}\, \sqrt {-e x -d}\, \sqrt {d}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*sqrt(e*x - d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {d^2-e^2\,x^2}\,\sqrt {e\,x-d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \sqrt {- d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*sqrt(-d + e*x)), x)

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