3.8.52 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx\) [752]

Optimal. Leaf size=111 \[ -\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}{2 x}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}} \]

[Out]

-3/2*a*(c-c/a^2/x^2)^(1/2)-1/2*(a*x+1)*(c-c/a^2/x^2)^(1/2)/x-3/2*a^2*x*arctanh((-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 = 0.27, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6294, 6264, 96, 94, 214} \begin {gather*} -\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}-\frac {(a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{2 x}-\frac {3 a^2 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 \sqrt {1-a x} \sqrt {a x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 94

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 214

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

Rule 6264

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 6294

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/x^(2*p))*(1 - a*x)^p*(1 + a*x)^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] &&  !GtQ[c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 79, normalized size = 0.71 \begin {gather*} \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\left ((1+4 a x) \sqrt {-1+a^2 x^2}\right )+3 a^2 x^2 \text {ArcTan}\left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{2 x \sqrt {-1+a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - c/(a^2*x^2)]*(-((1 + 4*a*x)*Sqrt[-1 + a^2*x^2]) + 3*a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(2*x*Sqrt
[-1 + a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(91)=182\).
time = 0.05, size = 347, normalized size = 3.13

method	result	size
risch	\(-\frac {\left (4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \left (a^{2} x^{2}-1\right )}+\frac {3 a^{2} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\)	\(142\)
default	\(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c \,x^{3}+4 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{3} x +3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c \,x^{2}+4 c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, a \,x^{2}-4 c^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c \,x^{2}+a^{2} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{2}\right )}{2 x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, c}\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.42, size = 177, normalized size = 1.59 \begin {gather*} \left [\frac {3 \, a \sqrt {-c} x \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 (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, \frac {3 \, a \sqrt {c} x \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 (4 \, a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{3} - x^{2}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{3} - x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (91) = 182\).
time = 0.49, size = 195, normalized size = 1.76 \begin {gather*} -{\left (3 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a c \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{2} \mathrm {sgn}\left (x\right ) - 4 \, c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a}\right )} {\left | a \right |} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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