3.3.19 \(\int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {875, 208} \begin {gather*} -2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

-2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 875

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e^2, Subst[I
nt[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 67, normalized size = 1.72 \begin {gather*} -\frac {2 \sqrt {c} \sqrt {\frac {a x}{c}+\frac {1}{c}} \sqrt {c-a c x} \tanh ^{-1}\left (\sqrt {c} \sqrt {\frac {a x}{c}+\frac {1}{c}}\right )}{\sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a*c*x]/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

(-2*Sqrt[c]*Sqrt[c^(-1) + (a*x)/c]*Sqrt[c - a*c*x]*ArcTanh[Sqrt[c]*Sqrt[c^(-1) + (a*x)/c]])/Sqrt[1 - a^2*x^2]

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IntegrateAlgebraic [A]  time = 0.34, size = 54, normalized size = 1.38 \begin {gather*} -\frac {2 \sqrt {c-a c x} \tan ^{-1}\left (\frac {\sqrt {a x-1}}{\sqrt {-(a x-1)^2-2 (a x-1)}}\right )}{\sqrt {a x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[c - a*c*x]/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

(-2*Sqrt[c - a*c*x]*ArcTan[Sqrt[-1 + a*x]/Sqrt[-2*(-1 + a*x) - (-1 + a*x)^2]])/Sqrt[-1 + a*x]

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fricas [A]  time = 0.41, size = 110, normalized size = 2.82 \begin {gather*} \left [\sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ), -2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)), -2*sqrt(
-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c))]

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giac [A]  time = 0.17, size = 57, normalized size = 1.46 \begin {gather*} -\frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c}\right )}}{{\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^3*(arctan(sqrt(2)*sqrt(c)/sqrt(-c))/(sqrt(-c)*c) - arctan(sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c))/abs(c)

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maple [A]  time = 0.02, size = 58, normalized size = 1.49 \begin {gather*} \frac {2 \sqrt {-\left (a x -1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {c}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right )}{\left (a x -1\right ) \sqrt {\left (a x +1\right ) c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-(a*x-1)*c)^(1/2)*(-a^2*x^2+1)^(1/2)/(a*x-1)/((a*x+1)*c)^(1/2)*c^(1/2)*arctanh(((a*x+1)*c)^(1/2)/c^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-a c x + c}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*c*x + c)/(sqrt(-a^2*x^2 + 1)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {c-a\,c\,x}}{x\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1))/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)

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