3.61 \(\int \frac{1}{\left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}} \]

[Out]

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

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

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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]

atanh(x*sqrt(c*f - d*e)/(sqrt(c)*sqrt(e + f*x**2)))/(sqrt(c)*sqrt(c*f - d*e))

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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]

[Out]

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

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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)

[Out]

-1/2/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d+2*f*(-c*d)^(1/2)/d*(x-
(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*f+2*f*(-c*d)^(1/2)/
d*(x-(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x-(-c*d)^(1/2)/d))+1/2/(-c*d)^(1/2)/(-
(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*(-
(c*f-d*e)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*f-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)
-(c*f-d*e)/d)^(1/2))/(x+(-c*d)^(1/2)/d))

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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]

Exception raised: ValueError

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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]

[1/4*log((((d^2*e^2 - 8*c*d*e*f + 8*c^2*f^2)*x^4 + c^2*e^2 - 2*(3*c*d*e^2 - 4*c^
2*e*f)*x^2)*sqrt(-c*d*e + c^2*f) + 4*((c*d^2*e^2 - 3*c^2*d*e*f + 2*c^3*f^2)*x^3
- (c^2*d*e^2 - c^3*e*f)*x)*sqrt(f*x^2 + e))/(d^2*x^4 + 2*c*d*x^2 + c^2))/sqrt(-c
*d*e + c^2*f), 1/2*arctan(1/2*((d*e - 2*c*f)*x^2 - c*e)/(sqrt(c*d*e - c^2*f)*sqr
t(f*x^2 + e)*x))/sqrt(c*d*e - c^2*f)]

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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)

[Out]

Integral(1/((c + d*x**2)*sqrt(e + f*x**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]

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