3.801 \(\int e^{4 \tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^p \, dx\)

Optimal. Leaf size=339 \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (1-2 p),2-p,\frac{1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac{6 a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (3-2 p),2-p,\frac{1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}+\frac{a^4 x^5 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (5-2 p),2-p,\frac{1}{2} (7-2 p),a^2 x^2\right )}{5-2 p}+\frac{2 a^3 x^4 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}+\frac{2 a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-p) (1-a x) (a x+1)} \]

[Out]

(2*a*(c - c/(a^2*x^2))^p*x^2)/((1 - p)*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1 - 2*
p)/2, 2 - p, (3 - 2*p)/2, a^2*x^2])/((1 - 2*p)*(1 - a*x)^p*(1 + a*x)^p) + (6*a^2*(c - c/(a^2*x^2))^p*x^3*Hyper
geometric2F1[(3 - 2*p)/2, 2 - p, (5 - 2*p)/2, a^2*x^2])/((3 - 2*p)*(1 - a*x)^p*(1 + a*x)^p) + (a^4*(c - c/(a^2
*x^2))^p*x^5*Hypergeometric2F1[(5 - 2*p)/2, 2 - p, (7 - 2*p)/2, a^2*x^2])/((5 - 2*p)*(1 - a*x)^p*(1 + a*x)^p)
+ (2*a^3*(c - c/(a^2*x^2))^p*x^4*Hypergeometric2F1[2 - p, 2 - p, 3 - p, a^2*x^2])/((2 - p)*(1 - a*x)^p*(1 + a*
x)^p)

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Rubi [A]  time = 0.342747, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6159, 6129, 127, 95, 125, 364} \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (1-2 p),2-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{6 a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (3-2 p),2-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a^4 x^5 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (5-2 p),2-p;\frac{1}{2} (7-2 p);a^2 x^2\right )}{5-2 p}+\frac{2 a^3 x^4 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (2-p,2-p;3-p;a^2 x^2\right )}{2-p}+\frac{2 a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-p) (1-a x) (a x+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^p,x]

[Out]

(2*a*(c - c/(a^2*x^2))^p*x^2)/((1 - p)*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1 - 2*
p)/2, 2 - p, (3 - 2*p)/2, a^2*x^2])/((1 - 2*p)*(1 - a*x)^p*(1 + a*x)^p) + (6*a^2*(c - c/(a^2*x^2))^p*x^3*Hyper
geometric2F1[(3 - 2*p)/2, 2 - p, (5 - 2*p)/2, a^2*x^2])/((3 - 2*p)*(1 - a*x)^p*(1 + a*x)^p) + (a^4*(c - c/(a^2
*x^2))^p*x^5*Hypergeometric2F1[(5 - 2*p)/2, 2 - p, (7 - 2*p)/2, a^2*x^2])/((5 - 2*p)*(1 - a*x)^p*(1 + a*x)^p)
+ (2*a^3*(c - c/(a^2*x^2))^p*x^4*Hypergeometric2F1[2 - p, 2 - p, 3 - p, a^2*x^2])/((2 - p)*(1 - a*x)^p*(1 + a*
x)^p)

Rule 6159

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

Rule 6129

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])

Rule 127

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

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rule 125

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^p \, dx &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int e^{4 \tanh ^{-1}(a x)} x^{-2 p} (1-a x)^p (1+a x)^p \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{-2+p} (1+a x)^{2+p} \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int \left (4 a x^{1-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+6 a^2 x^{2-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+4 a^3 x^{3-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+a^4 x^{4-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}+x^{-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p}\right ) \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx+\left (4 a \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{1-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx+\left (6 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx+\left (4 a^3 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{3-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx+\left (a^4 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{4-2 p} (1-a x)^{-2+p} (1+a x)^{-2+p} \, dx\\ &=\frac{2 a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{(1-p) (1-a x) (1+a x)}+\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx+\left (6 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{2-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx+\left (4 a^3 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{3-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx+\left (a^4 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} (1-a x)^{-p} (1+a x)^{-p}\right ) \int x^{4-2 p} \left (1-a^2 x^2\right )^{-2+p} \, dx\\ &=\frac{2 a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{(1-p) (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^p x (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac{1}{2} (1-2 p),2-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{6 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^3 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),2-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a^4 \left (c-\frac{c}{a^2 x^2}\right )^p x^5 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (\frac{1}{2} (5-2 p),2-p;\frac{1}{2} (7-2 p);a^2 x^2\right )}{5-2 p}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^p x^4 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (2-p,2-p;3-p;a^2 x^2\right )}{2-p}\\ \end{align*}

Mathematica [C]  time = 0.199298, size = 217, normalized size = 0.64 \[ -\frac{x (1-a x)^{-p} \left (-\left (a^2 x^2-1\right )^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left (4 (a x-1)^p (a x+1)^{2 p} \left (1-a^2 x^2\right )^p \text{Hypergeometric2F1}\left (1-2 p,2-p,2-2 p,\frac{2 a x}{a x+1}\right )+(a x+1) (1-a x)^p \left (a^2 x^2-1\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,\frac{3}{2}-p,a^2 x^2\right )-4 (a x+1) (a x-1)^p \left (1-a^2 x^2\right )^p F_1(1-2 p;1-p,-p;2-2 p;a x,-a x)\right )}{(2 p-1) (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^p,x]

[Out]

-(((c - c/(a^2*x^2))^p*x*(-4*(-1 + a*x)^p*(1 + a*x)*(1 - a^2*x^2)^p*AppellF1[1 - 2*p, 1 - p, -p, 2 - 2*p, a*x,
-(a*x)] + 4*(-1 + a*x)^p*(1 + a*x)^(2*p)*(1 - a^2*x^2)^p*Hypergeometric2F1[1 - 2*p, 2 - p, 2 - 2*p, (2*a*x)/(
1 + a*x)] + (1 - a*x)^p*(1 + a*x)*(-1 + a^2*x^2)^p*Hypergeometric2F1[1/2 - p, -p, 3/2 - p, a^2*x^2]))/((-1 + 2
*p)*(1 - a*x)^p*(1 + a*x)*(-(-1 + a^2*x^2)^2)^p))

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Maple [F]  time = 0.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{4}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x)

[Out]

int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{4}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="maxima")

[Out]

integrate((a*x + 1)^4*(c - c/(a^2*x^2))^p/(a^2*x^2 - 1)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**p,x)

[Out]

Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x + 1)**2/(a*x - 1)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{4}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="giac")

[Out]

integrate((a*x + 1)^4*(c - c/(a^2*x^2))^p/(a^2*x^2 - 1)^2, x)