3.777 \(\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx\)

Optimal. Leaf size=140 \[ -a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 x}-\frac {(1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {a^3 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]

[Out]

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

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Rubi [A]  time = 0.39, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6159, 6129, 96, 94, 92, 208} \[ -a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 x}-\frac {(1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {a^3 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^3),x]

[Out]

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

Rule 92

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

Rule 94

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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 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}}}{x^3} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^4} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^4 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}{3 x^2}-\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^3 \sqrt {1+a x}} \, dx}{3 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{x^2 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}{3 x^2}-\frac {\left (a^3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac {\left (a^4 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-a^2 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{3 x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac {a^3 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 86, normalized size = 0.61 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (5 a^2 x^2-3 a x+1\right )+3 a^3 x^3 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{3 x^2 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^3),x]

[Out]

-1/3*(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1 - 3*a*x + 5*a^2*x^2) + 3*a^3*x^3*ArcTan[1/Sqrt[-1 + a^2*x^2
]]))/(x^2*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 0.72, size = 200, normalized size = 1.43 \[ \left [\frac {3 \, a^{2} \sqrt {-c} x^{2} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, -\frac {3 \, a^{2} \sqrt {c} x^{2} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.78, size = 231, normalized size = 1.65 \[ \frac {2}{3} \, {\left (3 \, a \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a c \mathrm {sgn}\relax (x) + 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) + 12 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{3} \mathrm {sgn}\relax (x) + 5 \, c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{3}}\right )} {\left | a \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm="giac")

[Out]

2/3*(3*a*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x) - (3*(sqrt(a^2*c)*x - sqrt(a^2*
c*x^2 - c))^5*a*c*sgn(x) + 3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*c^(3/2)*abs(a)*sgn(x) + 12*(sqrt(a^2*c)*x
- sqrt(a^2*c*x^2 - c))^2*c^(5/2)*abs(a)*sgn(x) - 3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*a*c^3*sgn(x) + 5*c^(
7/2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^3)*abs(a)

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maple [B]  time = 0.05, size = 378, normalized size = 2.70 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a \left (-6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{4} a^{3} c +6 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{3}+6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{3} a -6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{3} a +6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{2} c -3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{2} c -3 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x \,a^{2}-3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{3} c^{2}+a \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{3 x^{2} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x^2*a*(-6*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^4*a^3*c+6*(-c/a^2)^(1/2
)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^3+6*(-c/a^2)^(1/2)*c^(3/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^3*a-6*(
-c/a^2)^(1/2)*c^(3/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^3*a+6*(-c/a^2)^(1/2)*((a*x-1)*
(a*x+1)*c/a^2)^(1/2)*x^3*a^2*c-3*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^3*a^2*c-3*(-c/a^2)^(1/2)*(c*(a^2*x
^2-1)/a^2)^(3/2)*x*a^2-3*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^3*c^2+a*(c*(a^2*x^2-1)
/a^2)^(3/2)*(-c/a^2)^(1/2))/(-c/a^2)^(1/2)/(c*(a^2*x^2-1)/a^2)^(1/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{2} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^3,x, algorithm="maxima")

[Out]

-integrate((a^2*x^2 - 1)*sqrt(c - c/(a^2*x^2))/((a*x + 1)^2*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{x^3\,{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{4} + x^{3}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{4} + x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)

[Out]

-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**4 + x**3), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**4 + x**3),
x)

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