3.12.54 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^3} \, dx\) [1154]

Optimal. Leaf size=119 \[ \frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}} \]

[Out]

8/35*x/c^3/(-a^2*x^2+1)^(1/2)+1/7/a/c^3/(-a*x+1)^3/(-a^2*x^2+1)^(1/2)+4/35/a/c^3/(-a*x+1)^2/(-a^2*x^2+1)^(1/2)
+4/35/a/c^3/(-a*x+1)/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6273, 669, 673, 197} \begin {gather*} \frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*x)/(35*c^3*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)*Sqrt[1 - a^2*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 669

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

Rule 673

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

Rule 6273

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a^2*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] && IGtQ[(n + 1)/2, 0] &&
!IntegerQ[p - n/2]

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=\frac {\int \frac {1}{(1-a x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4 \int \frac {1}{(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {12 \int \frac {1}{(1-a x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 61, normalized size = 0.51 \begin {gather*} \frac {-13+4 a x+20 a^2 x^2-24 a^3 x^3+8 a^4 x^4}{35 a c^3 (-1+a x)^3 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4)/(35*a*c^3*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [A]
time = 0.08, size = 189, normalized size = 1.59

method	result	size
gosper	\(-\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{2} c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\)	\(63\)
trager	\(-\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{3} \left (a x -1\right )^{4} a \left (a x +1\right )}\)	\(65\)
default	\(-\frac {\frac {1}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}-\frac {4 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{5}\right )}{7}}{c^{3} a^{3}}\)	\(189\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

-1/c^3/a^3*(1/7/a/(x-1/a)^3/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)-4/7*a*(1/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*(x-1/a
)*a)^(1/2)-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-
2*(x-1/a)*a)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.33, size = 144, normalized size = 1.21 \begin {gather*} \frac {13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {-a^{2} x^{2} + 1} + 13}{35 \, {\left (a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, 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)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*
x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*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*} \frac {\int \frac {3 a x}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}

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)**3,x)

[Out]

(Integral(3*a*x/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x*
*2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(a**8*x**8*sqrt(
-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2
*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt
(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1))
, x) + Integral(1/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*
x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3

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

[Out]

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

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Mupad [B]
time = 1.20, size = 125, normalized size = 1.05 \begin {gather*} \frac {29\,\sqrt {1-a^2\,x^2}}{280\,a\,c^3\,{\left (a\,x-1\right )}^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{140\,a\,c^3\,{\left (a\,x-1\right )}^3}+\frac {\sqrt {1-a^2\,x^2}}{14\,a\,c^3\,{\left (a\,x-1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^3}+\frac {29}{280\,a\,c^3}\right )}{\left (a\,x-1\right )\,\left (a\,x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(29*(1 - a^2*x^2)^(1/2))/(280*a*c^3*(a*x - 1)^2) - (13*(1 - a^2*x^2)^(1/2))/(140*a*c^3*(a*x - 1)^3) + (1 - a^2
*x^2)^(1/2)/(14*a*c^3*(a*x - 1)^4) - ((1 - a^2*x^2)^(1/2)*((8*x)/(35*c^3) + 29/(280*a*c^3)))/((a*x - 1)*(a*x +
1))

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