3.1163 \(\int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx\)

Optimal. Leaf size=188 \[ -\frac {4 a^2 \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-a^2 c x^2}}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^3 \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ -\frac {4 a^2 \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-a^2 c x^2}}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^3 \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6150

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

Rule 6153

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 73, normalized size = 0.39 \[ \frac {\sqrt {c-a^2 c x^2} \left (4 a^3 \log (x)-4 a^3 \log (1-a x)-\frac {4 a^2}{x}-\frac {3 a}{2 x^2}-\frac {1}{3 x^3}\right )}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]

[Out]

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

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fricas [A]  time = 1.03, size = 478, normalized size = 2.54 \[ \left [\frac {12 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (24 \, a^{2} x^{2} - {\left (24 \, a^{2} + 9 \, a + 2\right )} x^{3} + 9 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} x^{5} - x^{3}\right )}}, -\frac {24 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (24 \, a^{2} x^{2} - {\left (24 \, a^{2} + 9 \, a + 2\right )} x^{3} + 9 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} x^{5} - x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/6*(12*(a^5*x^5 - a^3*x^3)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4
+ 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*
x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) +
sqrt(-a^2*c*x^2 + c)*(24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3), -
1/6*(24*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*((2*a^2 - 2*a + 1)*x^2 -
2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - sqrt(-a^2*c*x
^2 + c)*(24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*x^5 - x^3)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 81, normalized size = 0.43 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (24 a^{3} \ln \relax (x ) x^{3}-24 \ln \left (a x -1\right ) x^{3} a^{3}-24 a^{2} x^{2}-9 a x -2\right )}{6 \left (a^{2} x^{2}-1\right ) x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(24*a^3*ln(x)*x^3-24*ln(a*x-1)*x^3*a^3-24*a^2*x^2-9*a*x-2)/(a^2
*x^2-1)/x^3

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maxima [A]  time = 0.38, size = 216, normalized size = 1.15 \[ -\frac {1}{2} \, \left (-1\right )^{-2 \, a^{2} c x^{2} + 2 \, c} a^{3} \sqrt {c} \log \left (-2 \, a^{2} c + \frac {2 \, c}{x^{2}}\right ) + \frac {1}{2} \, a^{3} \sqrt {c} \log \left (a x + 1\right ) - \frac {1}{2} \, a^{3} \sqrt {c} \log \left (a x - 1\right ) + \frac {3}{2} \, {\left (a \sqrt {c} \log \left (a x + 1\right ) - a \sqrt {c} \log \left (a x - 1\right ) - \frac {2 \, \sqrt {c}}{x}\right )} a^{2} - \frac {3}{2} \, {\left (\left (-1\right )^{-2 \, a^{2} c x^{2} + 2 \, c} a^{2} \sqrt {c} \log \left (-2 \, a^{2} c + \frac {2 \, c}{x^{2}}\right ) - \frac {a^{2} c}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {c}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} x^{2}}\right )} a - \frac {3 \, a^{2} \sqrt {c} x^{2} + \sqrt {c}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{x^4\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x**4,x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)

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