3.1248 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}-\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6142, 653, 217, 203} \[ -\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}-\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 6142

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(c + d*x^2)^(p
+ n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && I
LtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}-\int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}-\operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=-\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}-\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 100, normalized size = 1.64 \[ \frac {2 \sqrt {1-a^2 x^2} \left (\sqrt {a x+1} (a x-1)+\sqrt {1-a x} (a x+1) \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-a x} (a x+1) \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 - a^2*x^2]*((-1 + a*x)*Sqrt[1 + a*x] + Sqrt[1 - a*x]*(1 + a*x)*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sq
rt[1 - a*x]*(1 + a*x)*Sqrt[c - a^2*c*x^2])

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fricas [A]  time = 0.74, size = 150, normalized size = 2.46 \[ \left [-\frac {{\left (a x + 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x + a c\right )}}, \frac {{\left (a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 4*sqrt(-a^2*c*x^2 + c))
/(a^2*c*x + a*c), ((a*x + 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*c*
x^2 + c))/(a^2*c*x + a*c)]

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giac [B]  time = 0.21, size = 107, normalized size = 1.75 \[ -\frac {2 \, {\left (\frac {{\left (c \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {-c} \sqrt {c}\right )} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{c^{\frac {3}{2}}} - \frac {\frac {\arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{c}}{\mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}\right )}}{{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((c*arctan(sqrt(-c)/sqrt(c)) - sqrt(-c)*sqrt(c))*sgn(1/(a*x + 1))*sgn(a)/c^(3/2) - (arctan(sqrt(-c + 2*c/(a
*x + 1))/sqrt(c))/sqrt(c) - sqrt(-c + 2*c/(a*x + 1))/c)/(sgn(1/(a*x + 1))*sgn(a)))/abs(a)

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maple [A]  time = 0.04, size = 74, normalized size = 1.21 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}{a^{2} c \left (x +\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 40, normalized size = 0.66 \[ -\frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c} - \frac {\arcsin \left (a x\right )}{a \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a^2*c*x^2 + c)/(a^2*c*x + a*c) - arcsin(a*x)/(a*sqrt(c))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {a^2\,x^2-1}{\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{a x \sqrt {- a^{2} c x^{2} + c} + \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{a x \sqrt {- a^{2} c x^{2} + c} + \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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