3.12.19 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [1119]

Optimal. Leaf size=51 \[ \frac {2 (1+a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c-a^2 c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6276, 667, 197} \begin {gather*} \frac {x}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 (a x+1)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 667

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)*((a + c*x^2)^(p + 1)/(c*(p
+ 1))), x] - Dist[e^2*((p + 2)/(c*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 6276

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 63, normalized size = 1.24 \begin {gather*} -\frac {(-2+a x) \sqrt {1+a x} \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/2} \sqrt {c-a^2 c x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(43)=86\).
time = 0.07, size = 128, normalized size = 2.51

method	result	size
gosper	\(-\frac {\left (a x -2\right ) \left (a x +1\right )^{2}}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a}\)	\(31\)
trager	\(-\frac {\left (a x -2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{3 c^{2} \left (a x -1\right )^{2} a}\)	\(34\)
default	\(-\frac {x}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {1}{3 a c \left (x -\frac {1}{a}\right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{3 a \,c^{2} \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{a}\)	\(128\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (43) = 86\).
time = 0.29, size = 196, normalized size = 3.84 \begin {gather*} \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x + \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x - \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x + \sqrt {-a^{2} c x^{2} + c} a^{2} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x - \sqrt {-a^{2} c x^{2} + c} a^{2} c} + \frac {x}{\sqrt {-a^{2} c x^{2} + c} a c}\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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 47, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\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)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, 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)**(3/2),x)

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (43) = 86\).
time = 0.45, size = 148, normalized size = 2.90 \begin {gather*} -\frac {{\left (a c - 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\left (x\right )}{3 \, {\left (a^{2} c^{\frac {5}{2}} - \sqrt {-a^{2} c} a c^{2}\right )}} + \frac {2 \, {\left (2 \, a^{2} c + 3 \, a \sqrt {c} {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )} + 3 \, {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{2}\right )}}{3 \, {\left (a \sqrt {c} + \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{3} c \mathrm {sgn}\left (x\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)^(3/2),x, algorithm="giac")

[Out]

-1/3*(a*c - 3*sqrt(-a^2*c)*sqrt(c))*sgn(x)/(a^2*c^(5/2) - sqrt(-a^2*c)*a*c^2) + 2/3*(2*a^2*c + 3*a*sqrt(c)*(sq
rt(-a^2*c + c/x^2) - sqrt(c)/x) + 3*(sqrt(-a^2*c + c/x^2) - sqrt(c)/x)^2)/((a*sqrt(c) + sqrt(-a^2*c + c/x^2) -
sqrt(c)/x)^3*c*sgn(x))

________________________________________________________________________________________

Mupad [B]
time = 0.95, size = 33, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-2\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________