3.8.19 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\) [719]

Optimal. Leaf size=77 \[ \frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6295, 6285, 45} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6285

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

Rule 6295

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 50, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (\frac {x}{a}-\frac {\log (1+a x)}{a^2}\right )}{\sqrt {c-\frac {c}{a^2 x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.83, size = 51, normalized size = 0.66

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-a x +\ln \left (a x +1\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.28, size = 21, normalized size = 0.27 \begin {gather*} \frac {i \, x}{\sqrt {c}} - \frac {i \, \log \left (a x + 1\right )}{a \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x/sqrt(c) - I*log(a*x + 1)/(a*sqrt(c))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (69) = 138\).
time = 0.38, size = 367, normalized size = 4.77 \begin {gather*} \left [\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x - {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right )}{2 \, {\left (a^{3} c x^{2} - a c\right )}}, \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right )}{a^{3} c x^{2} - a c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - (a^2*x^2 - 1)*sqrt(-c)*log((a^6*c*x^6 + 4
*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x - (a^5*x^5 + 4*a^4*x^4 + 6*a^3*x^3 + 4*a^2*x^2)*sqrt(-a^2*x^2
+ 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)))/(a^3*c*x^2 - a*c), (
sqrt(-a^2*x^2 + 1)*a^2*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - (a^2*x^2 - 1)*sqrt(c)*arctan((a^2*x^2 + 2*a*x + 2
)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^3*c*x^3 + 2*a^2*c*x^2 - a*c*x - 2*c)))/(a^3*c*
x^2 - a*c)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________