3.22.16 \(\int \frac {-b^2+a x^2}{(b^2+a x^2) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\)

Optimal. Leaf size=153 \[ \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}}-\frac {4 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 1.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 120, normalized size = 0.78 \begin {gather*} \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b^2 + a*x^2)/((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]

[Out]

(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - (4*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt
[a] + (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} - b^{2}}{{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^2 - b^2)/((a*x^2 + b^2)*sqrt(b + sqrt(a*x^2 + b^2))), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b^{2}}{\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} - b^{2}}{{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^2 - b^2)/((a*x^2 + b^2)*sqrt(b + sqrt(a*x^2 + b^2))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*x**2 - b**2)/(sqrt(b + sqrt(a*x**2 + b**2))*(a*x**2 + b**2)), x)

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