3.10.11 \(\int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx\) [911]

Optimal. Leaf size=124 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \text {ArcSin}(a x)}{a^2 \sqrt {1-a x} \sqrt {1+a x}} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6294, 6264, 81, 52, 41, 222} \begin {gather*} \frac {x \text {ArcSin}(a x) \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {a x+1} \sqrt {1-a x}}+\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - c/(a^2*x^2)]*x)/a^2 + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*
x)^2)/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(a^2*Sqrt[1 - a*x]*Sqrt[1 + a*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 52

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 81

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

Rule 222

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 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]

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 84, normalized size = 0.68 \begin {gather*} \frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (5-3 a x+a^2 x^2\right )-3 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{3 a^2 \sqrt {-1+a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 173, normalized size = 1.40

method	result	size
risch	\(\frac {\left (a^{2} x^{2}-3 a x +5\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}{3 a^{2}}-\frac {\ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{a \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\)	\(125\)
default	\(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3}-3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x +3 c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-6 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {\frac {c \left (a x +1\right ) \left (a x -1\right )}{a^{2}}}\, \sqrt {c}+c x}{\sqrt {c}}\right )+6 \sqrt {\frac {c \left (a x +1\right ) \left (a x -1\right )}{a^{2}}}\, a c \right )}{3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \,a^{3}}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 0.38, size = 204, normalized size = 1.65 \begin {gather*} \left [\frac {2 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \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 )}{6 \, a^{3}}, \frac {{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \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 )}{3 \, a^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.40, size = 117, normalized size = 0.94 \begin {gather*} \frac {1}{6} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (x {\left (\frac {x \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {3 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {5 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} + \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{3} {\left | a \right |}} - \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 10 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{4} {\left | a \right |}}\right )} {\left | a \right |} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(a^2*c*x^2 - c)*(x*(x*sgn(x)/a^2 - 3*sgn(x)/a^3) + 5*sgn(x)/a^4) + 6*sqrt(c)*log(abs(-sqrt(a^2*c)*x
+ sqrt(a^2*c*x^2 - c)))*sgn(x)/(a^3*abs(a)) - (3*a*sqrt(c)*log(abs(c)) + 10*sqrt(-c)*abs(a))*sgn(x)/(a^4*abs(
a)))*abs(a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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