3.651 \(\int \frac {\sqrt {c+d x}}{1-x^2} \, dx\)

Optimal. Leaf size=58 \[ \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {700, 1130, 206} \[ \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(1 - x^2),x]

[Out]

-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c + d]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 700

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x}}{1-x^2} \, dx &=(2 d) \operatorname {Subst}\left (\int \frac {x^2}{-c^2+d^2+2 c x^2-x^4} \, dx,x,\sqrt {c+d x}\right )\\ &=-\left ((c-d) \operatorname {Subst}\left (\int \frac {1}{c-d-x^2} \, dx,x,\sqrt {c+d x}\right )\right )+(c+d) \operatorname {Subst}\left (\int \frac {1}{c+d-x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right )+\sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 58, normalized size = 1.00 \[ \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(1 - x^2),x]

[Out]

-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c + d]]

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fricas [A]  time = 1.00, size = 295, normalized size = 5.09 \[ \left [\frac {1}{2} \, \sqrt {c - d} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c - d} + 2 \, c - d}{x + 1}\right ) + \frac {1}{2} \, \sqrt {c + d} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt {-c + d} \arctan \left (-\frac {\sqrt {d x + c} \sqrt {-c + d}}{c - d}\right ) + \frac {1}{2} \, \sqrt {c + d} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c - d}}{c + d}\right ) + \frac {1}{2} \, \sqrt {c - d} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c - d} + 2 \, c - d}{x + 1}\right ), -\sqrt {-c + d} \arctan \left (-\frac {\sqrt {d x + c} \sqrt {-c + d}}{c - d}\right ) - \sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c - d}}{c + d}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(c - d)*log((d*x - 2*sqrt(d*x + c)*sqrt(c - d) + 2*c - d)/(x + 1)) + 1/2*sqrt(c + d)*log((d*x + 2*sqr
t(d*x + c)*sqrt(c + d) + 2*c + d)/(x - 1)), -sqrt(-c + d)*arctan(-sqrt(d*x + c)*sqrt(-c + d)/(c - d)) + 1/2*sq
rt(c + d)*log((d*x + 2*sqrt(d*x + c)*sqrt(c + d) + 2*c + d)/(x - 1)), -sqrt(-c - d)*arctan(sqrt(d*x + c)*sqrt(
-c - d)/(c + d)) + 1/2*sqrt(c - d)*log((d*x - 2*sqrt(d*x + c)*sqrt(c - d) + 2*c - d)/(x + 1)), -sqrt(-c + d)*a
rctan(-sqrt(d*x + c)*sqrt(-c + d)/(c - d)) - sqrt(-c - d)*arctan(sqrt(d*x + c)*sqrt(-c - d)/(c + d))]

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giac [A]  time = 0.20, size = 54, normalized size = 0.93 \[ -\sqrt {-c + d} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c + d}}\right ) + \sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c - d}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-c + d)*arctan(sqrt(d*x + c)/sqrt(-c + d)) + sqrt(-c - d)*arctan(sqrt(d*x + c)/sqrt(-c - d))

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maple [A]  time = 0.09, size = 47, normalized size = 0.81 \[ \sqrt {c +d}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c +d}}\right )-\sqrt {-c +d}\, \arctan \left (\frac {\sqrt {d x +c}}{\sqrt {-c +d}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c-4*d>0)', see `assume?` for
more details)Is 4*c-4*d positive or negative?

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mupad [B]  time = 0.48, size = 46, normalized size = 0.79 \[ \mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c+d}}\right )\,\sqrt {c+d}-\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c-d}}\right )\,\sqrt {c-d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(c + d*x)^(1/2)/(x^2 - 1),x)

[Out]

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

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sympy [A]  time = 5.25, size = 61, normalized size = 1.05 \[ \frac {2 \left (\frac {d \left (c - d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + d}} \right )}}{2 \sqrt {- c + d}} - \frac {d \left (c + d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c - d}} \right )}}{2 \sqrt {- c - d}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*(c - d)*atan(sqrt(c + d*x)/sqrt(-c + d))/(2*sqrt(-c + d)) - d*(c + d)*atan(sqrt(c + d*x)/sqrt(-c - d))/(2
*sqrt(-c - d)))/d

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