3.5.44 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [444]

Optimal. Leaf size=39 \[ \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^2 (2+n)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 37} \begin {gather*} \frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n+2}{2}}}{a c^2 (n+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 6264

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}}}{a c^2 (2+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.38, size = 33, normalized size = 0.85

method	result	size
gosper	\(-\frac {{\mathrm e}^{n \arctanh \left (a x \right )} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2} \left (2+n \right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*c*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 59, normalized size = 1.51 \begin {gather*} \frac {{\left (a x + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n + 2 \, a c^{2} - {\left (a^{2} c^{2} n + 2 \, a^{2} c^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (29) = 58\).
time = 11.30, size = 178, normalized size = 4.56 \begin {gather*} \begin {cases} \frac {x}{c^{2}} & \text {for}\: a = 0 \\\tilde {\infty } x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {a x \operatorname {atanh}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {atanh}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {atanh}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} - \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/c**2, Eq(a, 0)), (zoo*x*exp(oo*n), Eq(a, 1/x)), (-a*x*atanh(a*x)/(a**2*c**2*x*exp(2*atanh(a*x)) -
a*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(a**2*c**2*x*exp(2*atanh(a*x)) - a*c**2*exp(2*atanh(a*x))), Eq(n, -2))
, (-a*x*exp(n*atanh(a*x))/(a**2*c**2*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2) - exp(n*atanh(a*x))/(a**2*c**2
*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2), True))

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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*c*x+c)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.10, size = 37, normalized size = 0.95 \begin {gather*} \frac {{\left (a\,x+1\right )}^{\frac {n}{2}+1}}{a\,c^2\,{\left (1-a\,x\right )}^{\frac {n}{2}+1}\,\left (n+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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