3.14.61 \(\int e^{n \tanh ^{-1}(a x)} x (c-a^2 c x^2)^p \, dx\) [1361]

Optimal. Leaf size=177 \[ -\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}-\frac {2^{\frac {n}{2}+p} n (1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {n}{2}-p,1-\frac {n}{2}+p;2-\frac {n}{2}+p;\frac {1}{2} (1-a x)\right )}{a^2 (1+p) (2-n+2 p)} \]

[Out]

-1/2*(-a*x+1)^(1-1/2*n+p)*(a*x+1)^(1+1/2*n+p)*(-a^2*c*x^2+c)^p/a^2/(1+p)/((-a^2*x^2+1)^p)-2^(1/2*n+p)*n*(-a*x+
1)^(1-1/2*n+p)*(-a^2*c*x^2+c)^p*hypergeom([-1/2*n-p, 1-1/2*n+p],[2-1/2*n+p],-1/2*a*x+1/2)/a^2/(1+p)/(2-n+2*p)/
((-a^2*x^2+1)^p)

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Rubi [A]
time = 0.13, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6288, 6285, 81, 71} \begin {gather*} -\frac {n 2^{\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} \, _2F_1\left (-\frac {n}{2}-p,-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )}{a^2 (p+1) (-n+2 p+2)}-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} (a x+1)^{\frac {n}{2}+p+1}}{2 a^2 (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((1 - a*x)^(1 - n/2 + p)*(1 + a*x)^(1 + n/2 + p)*(c - a^2*c*x^2)^p)/(a^2*(1 + p)*(1 - a^2*x^2)^p) - (2^(n
/2 + p)*n*(1 - a*x)^(1 - n/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1/2*n - p, 1 - n/2 + p, 2 - n/2 + p, (1
- a*x)/2])/(a^2*(1 + p)*(2 - n + 2*p)*(1 - a^2*x^2)^p)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 6285

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

Rule 6288

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

Rubi steps

\begin {align*} \int e^{n \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{n \tanh ^{-1}(a x)} x \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x (1-a x)^{-\frac {n}{2}+p} (1+a x)^{\frac {n}{2}+p} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}+\frac {\left (n \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{-\frac {n}{2}+p} (1+a x)^{\frac {n}{2}+p} \, dx}{2 a (1+p)}\\ &=-\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}-\frac {2^{\frac {n}{2}+p} n (1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {n}{2}-p,1-\frac {n}{2}+p;2-\frac {n}{2}+p;\frac {1}{2} (1-a x)\right )}{a^2 (1+p) (2-n+2 p)}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 136, normalized size = 0.77 \begin {gather*} -\frac {(1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\left ((n-2 (1+p)) (1+a x)^{1+\frac {n}{2}+p}\right )+2^{1+\frac {n}{2}+p} n \, _2F_1\left (-\frac {n}{2}-p,1-\frac {n}{2}+p;2-\frac {n}{2}+p;\frac {1}{2} (1-a x)\right )\right )}{2 a^2 (1+p) (2-n+2 p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((1 - a*x)^(1 - n/2 + p)*(c - a^2*c*x^2)^p*(-((n - 2*(1 + p))*(1 + a*x)^(1 + n/2 + p)) + 2^(1 + n/2 + p)*
n*Hypergeometric2F1[-1/2*n - p, 1 - n/2 + p, 2 - n/2 + p, (1 - a*x)/2]))/(a^2*(1 + p)*(2 - n + 2*p)*(1 - a^2*x
^2)^p)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctanh \left (a x \right )} x \left (-a^{2} c \,x^{2}+c \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x)

[Out]

int(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x)

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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))*x*(-a^2*c*x^2+c)^p,x, algorithm="maxima")

[Out]

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

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Fricas [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))*x*(-a^2*c*x^2+c)^p,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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