3.720 \(\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx\)

Optimal. Leaf size=130 \[ -\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]

[Out]

Sqrt[c - a^2*c*x^2]/(4*x^4) - (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) + (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) - (4*a^3
*Sqrt[c - a^2*c*x^2])/(3*x) + (7*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/8

________________________________________________________________________________________

Rubi [A]  time = 0.402421, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6152, 1807, 835, 807, 266, 63, 208} \[ -\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^5),x]

[Out]

Sqrt[c - a^2*c*x^2]/(4*x^4) - (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) + (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) - (4*a^3
*Sqrt[c - a^2*c*x^2])/(3*x) + (7*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/8

Rule 6167

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]

Rule 6152

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

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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]

Rubi steps

\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{x^5 \sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{1}{4} \int \frac{8 a c-7 a^2 c x}{x^4 \sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{\int \frac{21 a^2 c^2-16 a^3 c^2 x}{x^3 \sqrt{c-a^2 c x^2}} \, dx}{12 c}\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{\int \frac{32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{24 c^2}\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{8} \left (7 a^4 c\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{16} \left (7 a^4 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{1}{8} \left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.138829, size = 95, normalized size = 0.73 \[ \frac{\left (-32 a^3 x^3+21 a^2 x^2-16 a x+6\right ) \sqrt{c-a^2 c x^2}}{24 x^4}+\frac{7}{8} a^4 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-\frac{7}{8} a^4 \sqrt{c} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^5),x]

[Out]

(Sqrt[c - a^2*c*x^2]*(6 - 16*a*x + 21*a^2*x^2 - 32*a^3*x^3))/(24*x^4) - (7*a^4*Sqrt[c]*Log[x])/8 + (7*a^4*Sqrt
[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/8

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 279, normalized size = 2.2 \begin{align*}{\frac{1}{4\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{2}}{8\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{4}}{8}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }-{\frac{7\,{a}^{4}}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}-2\,{\frac{{a}^{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}-2\,{a}^{5}x\sqrt{-{a}^{2}c{x}^{2}+c}-2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }+2\,{a}^{4}\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}+2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \right ) }-{\frac{2\,a}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)/(a*x+1)*(a*x-1)/x^5,x)

[Out]

1/4/c/x^4*(-a^2*c*x^2+c)^(3/2)+9/8*a^2/c/x^2*(-a^2*c*x^2+c)^(3/2)+7/8*a^4*c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^
2+c)^(1/2))/x)-7/8*a^4*(-a^2*c*x^2+c)^(1/2)-2*a^3/c/x*(-a^2*c*x^2+c)^(3/2)-2*a^5*x*(-a^2*c*x^2+c)^(1/2)-2*a^5*
c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2*a^4*(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2)+2*a^
5*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*(x+1/a)^2+2*(x+1/a)*a*c)^(1/2))-2/3*a/c/x^3*(-a^2*c*x^2+c)^(3
/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x - 1\right )}}{{\left (a x + 1\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*(a*x - 1)/((a*x + 1)*x^5), x)

________________________________________________________________________________________

Fricas [A]  time = 1.66766, size = 416, normalized size = 3.2 \begin{align*} \left [\frac{21 \, a^{4} \sqrt{c} x^{4} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 2 \,{\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac{21 \, a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) -{\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="fricas")

[Out]

[1/48*(21*a^4*sqrt(c)*x^4*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(32*a^3*x^3 - 21*a^
2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4, 1/24*(21*a^4*sqrt(-c)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/
(a^2*c*x^2 - c)) - (32*a^3*x^3 - 21*a^2*x^2 + 16*a*x - 6)*sqrt(-a^2*c*x^2 + c))/x^4]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(1/2)*(a*x-1)/(a*x+1)/x**5,x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x - 1)/(x**5*(a*x + 1)), x)

________________________________________________________________________________________

Giac [B]  time = 1.17523, size = 437, normalized size = 3.36 \begin{align*} -\frac{7 \, a^{4} c \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} + \frac{21 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} - 96 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt{-c} c^{2}{\left | a \right |} - 45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} + 128 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt{-c} c^{3}{\left | a \right |} + 21 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{4} c^{4} - 32 \, a^{3} \sqrt{-c} c^{4}{\left | a \right |}}{12 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1)/x^5,x, algorithm="giac")

[Out]

-7/4*a^4*c*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) + 1/12*(21*(sqrt(-a^2*c)*x - sqr
t(-a^2*c*x^2 + c))^7*a^4*c - 45*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^4*c^2 - 96*(sqrt(-a^2*c)*x - sqrt(
-a^2*c*x^2 + c))^4*a^3*sqrt(-c)*c^2*abs(a) - 45*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^4*c^3 + 128*(sqrt(
-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^3*sqrt(-c)*c^3*abs(a) + 21*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^4*c
^4 - 32*a^3*sqrt(-c)*c^4*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^4