3.469 \(\int \frac {\tanh ^{-1}(a x)}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ -\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

-1/9/a/c/(-a^2*c*x^2+c)^(3/2)+1/3*x*arctanh(a*x)/c/(-a^2*c*x^2+c)^(3/2)-2/3/a/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*x*a
rctanh(a*x)/c^2/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5960, 5958} \[ -\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2),x]

[Out]

-1/(9*a*c*(c - a^2*c*x^2)^(3/2)) - 2/(3*a*c^2*Sqrt[c - a^2*c*x^2]) + (x*ArcTanh[a*x])/(3*c*(c - a^2*c*x^2)^(3/
2)) + (2*x*ArcTanh[a*x])/(3*c^2*Sqrt[c - a^2*c*x^2])

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}-\frac {2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \tanh ^{-1}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 64, normalized size = 0.61 \[ -\frac {\sqrt {c-a^2 c x^2} \left (\left (6 a^3 x^3-9 a x\right ) \tanh ^{-1}(a x)-6 a^2 x^2+7\right )}{9 a c^3 \left (a^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2),x]

[Out]

-1/9*(Sqrt[c - a^2*c*x^2]*(7 - 6*a^2*x^2 + (-9*a*x + 6*a^3*x^3)*ArcTanh[a*x]))/(a*c^3*(-1 + a^2*x^2)^2)

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 84, normalized size = 0.80 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left (12 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 14\right )}}{18 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*sqrt(-a^2*c*x^2 + c)*(12*a^2*x^2 - 3*(2*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1)) - 14)/(a^5*c^3*x^4 - 2
*a^3*c^3*x^2 + a*c^3)

________________________________________________________________________________________

giac [A]  time = 0.31, size = 111, normalized size = 1.06 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (\frac {2 \, a^{2} x^{2}}{c} - \frac {3}{c}\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, {\left (a^{2} c x^{2} - c\right )}^{2}} - \frac {6 \, a^{2} c x^{2} - 7 \, c}{9 \, {\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} c x^{2} + c} a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(-a^2*c*x^2 + c)*(2*a^2*x^2/c - 3/c)*x*log(-(a*x + 1)/(a*x - 1))/(a^2*c*x^2 - c)^2 - 1/9*(6*a^2*c*x^2
- 7*c)/((a^2*c*x^2 - c)*sqrt(-a^2*c*x^2 + c)*a*c^2)

________________________________________________________________________________________

maple [A]  time = 0.48, size = 160, normalized size = 1.52 \[ \frac {\left (a x +1\right ) \left (-1+3 \arctanh \left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{72 a \left (a x -1\right )^{2} c^{3}}-\frac {3 \left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{8 a \left (a x -1\right ) c^{3}}-\frac {3 \left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{8 a \left (a x +1\right ) c^{3}}+\frac {\left (a x -1\right ) \left (1+3 \arctanh \left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{72 a \left (a x +1\right )^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(a*x+1)*(-1+3*arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x-1)^2/c^3-3/8*(arctanh(a*x)-1)*(-(a*x-1)*(a*
x+1)*c)^(1/2)/a/(a*x-1)/c^3-3/8*(arctanh(a*x)+1)*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x+1)/c^3+1/72*(a*x-1)*(1+3*ar
ctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x+1)^2/c^3

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 90, normalized size = 0.86 \[ -\frac {1}{9} \, a {\left (\frac {6}{\sqrt {-a^{2} c x^{2} + c} a^{2} c^{2}} + \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \operatorname {artanh}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a*(6/(sqrt(-a^2*c*x^2 + c)*a^2*c^2) + 1/((-a^2*c*x^2 + c)^(3/2)*a^2*c)) + 1/3*(2*x/(sqrt(-a^2*c*x^2 + c)*
c^2) + x/((-a^2*c*x^2 + c)^(3/2)*c))*arctanh(a*x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________