3.14.44 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [1344]

Optimal. Leaf size=46 \[ -\frac {e^{n \tanh ^{-1}(a x)} (n-a x)}{a c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

[Out]

-exp(n*arctanh(a*x))*(-a*x+n)/a/c/(-n^2+1)/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6270} \begin {gather*} -\frac {(n-a x) e^{n \tanh ^{-1}(a x)}}{a c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((E^(n*ArcTanh[a*x])*(n - a*x))/(a*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]))

Rule 6270

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n - a*x)*(E^(n*ArcTanh[a*x])/
(a*c*(n^2 - 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 81, normalized size = 1.76 \begin {gather*} \frac {(1-a x)^{\frac {1}{2} (-1-n)} (n-a x) (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a c (-1+n) (1+n) \sqrt {c-a^2 c x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 49, normalized size = 1.07

method	result	size
gosper	\(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a x -n \right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{\left (n^{2}-1\right ) a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 81, normalized size = 1.76 \begin {gather*} -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - n\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n^{2} - a c^{2} - {\left (a^{3} c^{2} n^{2} - a^{3} c^{2}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), 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(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.10, size = 65, normalized size = 1.41 \begin {gather*} -\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {x}{c\,\left (n^2-1\right )}-\frac {n}{a\,c\,\left (n^2-1\right )}\right )}{\sqrt {c-a^2\,c\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atanh(a*x))/(c - a^2*c*x^2)^(3/2),x)

[Out]

-(exp((n*log(a*x + 1))/2 - (n*log(1 - a*x))/2)*(x/(c*(n^2 - 1)) - n/(a*c*(n^2 - 1))))/(c - a^2*c*x^2)^(1/2)

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