3.11.84 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx\) [1084]

Optimal. Leaf size=101 \[ -\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]

[Out]

-a^3*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))*c^(1/2)-1/3*(-a^2*c*x^2+c)^(1/2)/x^3-a*(-a^2*c*x^2+c)^(1/2)/x^2-5/3
*a^2*(-a^2*c*x^2+c)^(1/2)/x

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6286, 1821, 849, 821, 272, 65, 214} \begin {gather*} -\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \left (-\sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]

[Out]

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

Rule 65

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 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 272

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 821

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 849

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 1821

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 6286

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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]) && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx &=c \int \frac {(1+a x)^2}{x^4 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {1}{3} \int \frac {-6 a c-5 a^2 c x}{x^3 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}+\frac {\int \frac {10 a^2 c^2+6 a^3 c^2 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{6 c}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\left (a^3 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{2} \left (a^3 c\right ) \text {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}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-a \text {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}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 82, normalized size = 0.81 \begin {gather*} -\frac {\left (1+3 a x+5 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \sqrt {c} \log (x)-a^3 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(316\) vs. \(2(85)=170\).
time = 0.07, size = 317, normalized size = 3.14

method	result	size
risch	\(\frac {\left (5 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-3 a x -1\right ) c}{3 x^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-a^{3} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\)	\(87\)
default	\(2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}+2 a^{3} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )-2 a^{3} \left (\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}-\frac {a c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {c \,a^{2}}}\right )+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}\right )\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 140, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {a x + 1} \sqrt {-a x + 1} a^{2} \sqrt {c}}{x} + \frac {a^{4} c^{\frac {3}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) - \frac {2 \, \sqrt {-a^{2} c x^{2} + c} a^{2} c}{x^{2}}}{2 \, a c} - \frac {{\left (2 \, a^{2} \sqrt {c} x^{2} + \sqrt {c}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 164, normalized size = 1.62 \begin {gather*} \left [\frac {3 \, a^{3} \sqrt {c} x^{3} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )}}{6 \, x^{3}}, -\frac {3 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )}}{3 \, x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x**4,x)

[Out]

-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**5 - x**4), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**5 - x**4), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (85) = 170\).
time = 0.43, size = 250, normalized size = 2.48 \begin {gather*} \frac {2 \, a^{3} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, {\left (3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{3} c - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{2} \sqrt {-c} c {\left | a \right |} + 12 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{2} \sqrt {-c} c^{2} {\left | a \right |} - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{3} c^{3} - 5 \, a^{2} \sqrt {-c} c^{3} {\left | a \right |}\right )}}{3 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)

[Out]

-int(((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)), x)

________________________________________________________________________________________