3.51 \(\int \frac{1}{A^4-A^2 B^2+\left (-A^2+B^2\right ) x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]

[Out]

ArcTanh[x/A]/(A*(A^2 - B^2))

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Rubi [A]  time = 0.0202066, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A^4 - A^2*B^2 + (-A^2 + B^2)*x^2)^(-1),x]

[Out]

ArcTanh[x/A]/(A*(A^2 - B^2))

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Rubi in Sympy [A]  time = 1.3105, size = 12, normalized size = 0.57 \[ \frac{\operatorname{atanh}{\left (\frac{x}{A} \right )}}{A \left (A^{2} - B^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(A**4-A**2*B**2+(-A**2+B**2)*x**2),x)

[Out]

atanh(x/A)/(A*(A**2 - B**2))

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Mathematica [A]  time = 0.00642334, size = 21, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(A^4 - A^2*B^2 + (-A^2 + B^2)*x^2)^(-1),x]

[Out]

ArcTanh[x/A]/(A*(A^2 - B^2))

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Maple [B]  time = 0.009, size = 44, normalized size = 2.1 \[ -{\frac{\ln \left ( A-x \right ) }{ \left ( 2\,{A}^{2}-2\,{B}^{2} \right ) A}}+{\frac{\ln \left ( A+x \right ) }{ \left ( 2\,{A}^{2}-2\,{B}^{2} \right ) A}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(A^4-A^2*B^2+(-A^2+B^2)*x^2),x)

[Out]

-1/2/(A^2-B^2)/A*ln(A-x)+1/2/(A^2-B^2)/A*ln(A+x)

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Maxima [A]  time = 1.34928, size = 53, normalized size = 2.52 \[ \frac{\log \left (A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} - \frac{\log \left (-A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(A^4 - A^2*B^2 - (A^2 - B^2)*x^2),x, algorithm="maxima")

[Out]

1/2*log(A + x)/(A^3 - A*B^2) - 1/2*log(-A + x)/(A^3 - A*B^2)

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Fricas [A]  time = 0.198048, size = 36, normalized size = 1.71 \[ \frac{\log \left (A + x\right ) - \log \left (-A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(A^4 - A^2*B^2 - (A^2 - B^2)*x^2),x, algorithm="fricas")

[Out]

1/2*(log(A + x) - log(-A + x))/(A^3 - A*B^2)

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Sympy [A]  time = 0.399846, size = 70, normalized size = 3.33 \[ - \frac{\log{\left (- \frac{A^{3}}{\left (A - B\right ) \left (A + B\right )} + \frac{A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} + \frac{\log{\left (\frac{A^{3}}{\left (A - B\right ) \left (A + B\right )} - \frac{A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(A**4-A**2*B**2+(-A**2+B**2)*x**2),x)

[Out]

-log(-A**3/((A - B)*(A + B)) + A*B**2/((A - B)*(A + B)) + x)/(2*A*(A - B)*(A + B
)) + log(A**3/((A - B)*(A + B)) - A*B**2/((A - B)*(A + B)) + x)/(2*A*(A - B)*(A
+ B))

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GIAC/XCAS [A]  time = 0.209492, size = 55, normalized size = 2.62 \[ \frac{{\rm ln}\left ({\left | A + x \right |}\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} - \frac{{\rm ln}\left ({\left | -A + x \right |}\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(A^4 - A^2*B^2 - (A^2 - B^2)*x^2),x, algorithm="giac")

[Out]

1/2*ln(abs(A + x))/(A^3 - A*B^2) - 1/2*ln(abs(-A + x))/(A^3 - A*B^2)