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

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

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Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {377, 205} \begin {gather*} \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}} \end {gather*}

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])

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.11, size = 103, normalized size = 2.10 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {d \sqrt {f} x^2}{\sqrt {c} \sqrt {d e-c f}}-\frac {d x \sqrt {e+f x^2}}{\sqrt {c} \sqrt {d e-c f}}+\frac {\sqrt {c} \sqrt {f}}{\sqrt {d e-c f}}\right )}{\sqrt {c} \sqrt {d e-c f}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((c + d*x^2)*Sqrt[e + f*x^2]),x]

[Out]

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

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fricas [B]  time = 1.56, size = 241, normalized size = 4.92 \begin {gather*} \left [-\frac {\sqrt {-c d e + c^{2} f} \log \left (\frac {{\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} - 4 \, {\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt {-c d e + c^{2} f} \sqrt {f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left (c d e - c^{2} f\right )}}, \frac {\arctan \left (\frac {\sqrt {c d e - c^{2} f} {\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt {f x^{2} + e}}{2 \, {\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} + {\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right )}{2 \, \sqrt {c d e - c^{2} f}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.36, size = 74, normalized size = 1.51 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x^2+c)/(f*x^2+e)^(1/2),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)

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maple [B]  time = 0.01, size = 306, normalized size = 6.24 \begin {gather*} -\frac {\ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {\ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}} \end {gather*}

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x^{2} + c\right )} \sqrt {f x^{2} + e}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \left \{\begin {array}{cl} \frac {\mathrm {atan}\left (\frac {x\,\sqrt {d\,e-c\,f}}{\sqrt {c}\,\sqrt {f\,x^2+e}}\right )}{\sqrt {-c\,\left (c\,f-d\,e\right )}} & \text {\ if\ \ }0<d\,e-c\,f\\ \frac {\ln \left (\frac {\sqrt {c\,\left (f\,x^2+e\right )}+x\,\sqrt {c\,f-d\,e}}{\sqrt {c\,\left (f\,x^2+e\right )}-x\,\sqrt {c\,f-d\,e}}\right )}{2\,\sqrt {c\,\left (c\,f-d\,e\right )}} & \text {\ if\ \ }d\,e-c\,f<0\\ \int \frac {1}{\left (d\,x^2+c\right )\,\sqrt {f\,x^2+e}} \,d x & \text {\ if\ \ }d\,e-c\,f\notin \mathbb {R}\vee c\,f=d\,e \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(0 < - c*f + d*e, atan((x*(- c*f + d*e)^(1/2))/(c^(1/2)*(e + f*x^2)^(1/2)))/(-c*(c*f - d*e))^(1/2), -
c*f + d*e < 0, log(((c*(e + f*x^2))^(1/2) + x*(c*f - d*e)^(1/2))/((c*(e + f*x^2))^(1/2) - x*(c*f - d*e)^(1/2)
))/(2*(c*(c*f - d*e))^(1/2)), ~in(- c*f + d*e, 'real') | c*f == d*e, int(1/((c + d*x^2)*(e + f*x^2)^(1/2)), x)
)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c + d x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \end {gather*}

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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