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

Optimal. Leaf size=111 \[ \frac {(1-a x)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 (a x+1) (1-a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {a x+1} \sqrt {1-a x} \sin ^{-1}(a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.24, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6159, 6129, 78, 50, 41, 216} \[ \frac {(1-a x)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 (a x+1) (1-a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {a x+1} \sqrt {1-a x} \sin ^{-1}(a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 78

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

Rule 216

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

Rule 6129

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

Rule 6159

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 69, normalized size = 0.62 \[ \frac {-a^2 x^2+2 \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )-2 a x+3}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3 - 2*a*x - a^2*x^2 + 2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)

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fricas [A]  time = 0.60, size = 214, normalized size = 1.93 \[ \left [\frac {{\left (a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}, -\frac {2 \, {\left (a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 97, normalized size = 0.87 \[ -\frac {{\left (a x + 1\right )} a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} + \frac {4 \, a^{2} c \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}}}{a^{3} c \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*x + 1)*a^2*sqrt(c - 2*c/(a*x + 1)) + 4*a^2*c*arctan(sqrt(c - 2*c/(a*x + 1))/sqrt(-c))/sqrt(-c) + 2*a^2*sq
rt(c - 2*c/(a*x + 1)))/(a^3*c*sgn(-1/(a*x + 1) + 1))

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maple [A]  time = 0.04, size = 177, normalized size = 1.59 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \left (\sqrt {c}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2}-2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x a c +2 a \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a \sqrt {c}-2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) c \right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,c^{\frac {3}{2}} a \left (a x +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*(a^2*x^2-1)/a^2)^(1/2)*(c^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x*a^2-2*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*
x*a*c+2*a*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2)*a*c^(1/2)-2*ln(x*c^(1/2)+(c*(a^2*x^2
-1)/a^2)^(1/2))*c)/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x/c^(3/2)/a/(a*x+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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