Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
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Rubi [A] time = 0.0801177, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
[In] Int[1/((c + d*x^2)*Sqrt[e + f*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 11.5189, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atanh}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{c} \sqrt{e + f x^{2}}} \right )}}{\sqrt{c} \sqrt{c f - d e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x**2+c)/(f*x**2+e)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0387733, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + d*x^2)*Sqrt[e + f*x^2]),x]
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Maple [B] time = 0.015, size = 306, normalized size = 6.2 \[ -{\frac{1}{2}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}+{\frac{1}{2}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x^2+c)/(f*x^2+e)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264382, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left ({\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2}\right )} \sqrt{-c d e + c^{2} f} + 4 \,{\left ({\left (c d^{2} e^{2} - 3 \, c^{2} d e f + 2 \, c^{3} f^{2}\right )} x^{3} -{\left (c^{2} d e^{2} - c^{3} e f\right )} x\right )} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, \sqrt{-c d e + c^{2} f}}, \frac{\arctan \left (\frac{{\left (d e - 2 \, c f\right )} x^{2} - c e}{2 \, \sqrt{c d e - c^{2} f} \sqrt{f x^{2} + e} x}\right )}{2 \, \sqrt{c d e - c^{2} f}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c + d x^{2}\right ) \sqrt{e + f x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x**2+c)/(f*x**2+e)**(1/2),x)
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GIAC/XCAS [A] time = 0.251006, size = 100, normalized size = 2.04 \[ -\frac{\sqrt{f} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")
[Out]