3.6.78 \(\int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\) [578]

Optimal. Leaf size=55 \[ -\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}+\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )} \]

[Out]

-2/15/((a*x-1)/(a*x+1))^(3/2)/a/c^2+1/5/((a*x-1)/(a*x+1))^(3/2)*(-2*a*x+3)/a/c^2/(-a^2*x^2+1)

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \begin {gather*} \frac {(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*E^(3*ArcCoth[a*x]))/(15*a*c^2) + (E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(5*a*c^2*(1 - a^2*x^2))

Rule 6318

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rule 6320

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2
))), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

Rubi steps

\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c}\\ &=-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}+\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 43, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (7-6 a x+2 a^2 x^2\right )}{15 c^2 (-1+a x)^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/15*(Sqrt[1 - 1/(a^2*x^2)]*x*(7 - 6*a*x + 2*a^2*x^2))/(c^2*(-1 + a*x)^3)

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Maple [A]
time = 0.19, size = 52, normalized size = 0.95

method	result	size
gosper	\(-\frac {2 a^{2} x^{2}-6 a x +7}{15 \left (a^{2} x^{2}-1\right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\)	\(49\)
default	\(-\frac {2 a^{2} x^{2}-6 a x +7}{15 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) c^{2} \left (a x -1\right ) a}\)	\(52\)
trager	\(-\frac {\left (a x +1\right ) \left (2 a^{2} x^{2}-6 a x +7\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{15 a \,c^{2} \left (a x -1\right )^{3}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2*a^2*x^2-6*a*x+7)/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/c^2/(a*x-1)/a

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Maxima [A]
time = 0.27, size = 55, normalized size = 1.00 \begin {gather*} \frac {\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*(a*x - 1)/(a*x + 1) - 15*(a*x - 1)^2/(a*x + 1)^2 - 3)/(a*c^2*((a*x - 1)/(a*x + 1))^(5/2))

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Fricas [A]
time = 0.38, size = 77, normalized size = 1.40 \begin {gather*} -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2*a^3*x^3 - 4*a^2*x^2 + a*x + 7)*sqrt((a*x - 1)/(a*x + 1))/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x -
a*c^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 65, normalized size = 1.18 \begin {gather*} -\frac {4 \, {\left (10 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} - 5 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{15 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{5} a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(10*(a + sqrt(a^2 - 1/x^2))^2*x^2 - 5*(a + sqrt(a^2 - 1/x^2))*x + 1)/(((a + sqrt(a^2 - 1/x^2))*x - 1)^5*
a*c^2)

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Mupad [B]
time = 0.04, size = 55, normalized size = 1.00 \begin {gather*} -\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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