3.1361 \(\int e^{n \tanh ^{-1}(a x)} x (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=177 \[ -\frac{n 2^{\frac{n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac{n}{2}+p+1} \text{Hypergeometric2F1}\left (-\frac{n}{2}-p,-\frac{n}{2}+p+1,-\frac{n}{2}+p+2,\frac{1}{2} (1-a x)\right )}{a^2 (p+1) (-n+2 p+2)}-\frac{\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac{n}{2}+p+1} (a x+1)^{\frac{n}{2}+p+1}}{2 a^2 (p+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.183648, antiderivative size = 177, normalized size of antiderivative = 1., 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 = {6153, 6150, 80, 69} \[ -\frac{n 2^{\frac{n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac{n}{2}+p+1} \, _2F_1\left (-\frac{n}{2}-p,-\frac{n}{2}+p+1;-\frac{n}{2}+p+2;\frac{1}{2} (1-a x)\right )}{a^2 (p+1) (-n+2 p+2)}-\frac{\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac{n}{2}+p+1} (a x+1)^{\frac{n}{2}+p+1}}{2 a^2 (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - a*x)^(1 - n/2 + p)*(1 + a*x)^(1 + n/2 + p)*(c - a^2*c*x^2)^p)/(2*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[-n/2 - 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 6153

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]

Rule 6150

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 80

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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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]))

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*}

Mathematica [A]  time = 0.0984656, size = 136, normalized size = 0.77 \[ -\frac{\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac{n}{2}+p+1} \left (n 2^{\frac{n}{2}+p+1} \text{Hypergeometric2F1}\left (-\frac{n}{2}-p,-\frac{n}{2}+p+1,-\frac{n}{2}+p+2,\frac{1}{2} (1-a x)\right )-(n-2 (p+1)) (a x+1)^{\frac{n}{2}+p+1}\right )}{2 a^2 (p+1) (-n+2 p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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*Hy
pergeometric2F1[-n/2 - p, 1 - n/2 + p, 2 - n/2 + p, (1 - a*x)/2]))/(2*a^2*(1 + p)*(2 - n + 2*p)*(1 - a^2*x^2)^
p)

________________________________________________________________________________________

Maple [F]  time = 0.27, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

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)