3.12.23 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{5/2}} \, dx\) [1123]

Optimal. Leaf size=74 \[ \frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6276, 667, 198, 197} \begin {gather*} \frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 (a x+1)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + a*x))/(5*a*(c - a^2*c*x^2)^(5/2)) + x/(5*c*(c - a^2*c*x^2)^(3/2)) + (2*x)/(5*c^2*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 198

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

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 )^{5/2}} \, dx &=c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.72 \begin {gather*} \frac {2+a x-4 a^2 x^2+2 a^3 x^3}{5 a c^2 (-1+a x)^2 \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + a*x - 4*a^2*x^2 + 2*a^3*x^3)/(5*a*c^2*(-1 + a*x)^2*Sqrt[c - a^2*c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(205\) vs. \(2(62)=124\).
time = 0.06, size = 206, normalized size = 2.78

method	result	size
gosper	\(\frac {\left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right ) \left (a x +1\right )^{2}}{5 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} a}\)	\(47\)
trager	\(-\frac {\left (2 a^{3} x^{3}-4 a^{2} x^{2}+a x +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{5 c^{3} \left (a x -1\right )^{3} a \left (a x +1\right )}\)	\(57\)
default	\(-\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {1}{5 a c \left (x -\frac {1}{a}\right ) \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{6 a^{2} c^{2} \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{3 a^{2} c^{3} \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{a}\)	\(206\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*x/c/(-a^2*c*x^2+c)^(3/2)-2/3*x/c^2/(-a^2*c*x^2+c)^(1/2)-2/a*(1/5/a/c/(x-1/a)/(-c*a^2*(x-1/a)^2-2*c*a*(x-1
/a))^(3/2)-4/5*a*(-1/6*(-2*a^2*c*(x-1/a)-2*a*c)/a^2/c^2/(-c*a^2*(x-1/a)^2-2*c*a*(x-1/a))^(3/2)-1/3/a^2/c^3*(-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)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (62) = 124\).
time = 0.30, size = 218, normalized size = 2.95 \begin {gather*} \frac {1}{5} \, a {\left (\frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} + \frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} a c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} 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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 75, normalized size = 1.01 \begin {gather*} -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \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)**(5/2),x)

[Out]

-Integral(a*x/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) - a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**
3*sqrt(-a**2*c*x**2 + c) + 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) - c**2*sq
rt(-a**2*c*x**2 + c)), x) - Integral(1/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) - a**4*c**2*x**4*sqrt(-a**2*c*x*
*2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) + 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a
**2*c*x**2 + c) - c**2*sqrt(-a**2*c*x**2 + c)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.06, size = 56, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3-4\,a^2\,x^2+a\,x+2\right )}{5\,a\,c^3\,{\left (a\,x-1\right )}^3\,\left (a\,x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c - a^2*c*x^2)^(1/2)*(a*x - 4*a^2*x^2 + 2*a^3*x^3 + 2))/(5*a*c^3*(a*x - 1)^3*(a*x + 1))

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