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

Optimal. Leaf size=217 \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (1-2 p),1-p,\frac{1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac{a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (3-2 p),1-p,\frac{1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}+\frac{a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (1-p,1-p,2-p,a^2 x^2\right )}{1-p} \]

[Out]

((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1 - 2*p)/2, 1 - p, (3 - 2*p)/2, a^2*x^2])/((1 - 2*p)*(1 - a*x)^p*(1
+ a*x)^p) + (a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(3 - 2*p)/2, 1 - p, (5 - 2*p)/2, a^2*x^2])/((3 - 2*
p)*(1 - a*x)^p*(1 + a*x)^p) + (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2])/((1
- p)*(1 - a*x)^p*(1 + a*x)^p)

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Rubi [A]  time = 0.260579, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6159, 6129, 127, 125, 364} \[ \frac{x (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (1-2 p),1-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{a^2 x^3 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (3-2 p),1-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a x^2 (1-a x)^{-p} (a x+1)^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1 - 2*p)/2, 1 - p, (3 - 2*p)/2, a^2*x^2])/((1 - 2*p)*(1 - a*x)^p*(1
+ a*x)^p) + (a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(3 - 2*p)/2, 1 - p, (5 - 2*p)/2, a^2*x^2])/((3 - 2*
p)*(1 - a*x)^p*(1 + a*x)^p) + (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2])/((1
- 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 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^{2 \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^{2 \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)^{-1+p} (1+a x)^{1+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 (2 a x^{1-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+a^2 x^{2-2 p} (1-a x)^{-1+p} (1+a x)^{-1+p}+x^{-2 p} (1-a x)^{-1+p} (1+a x)^{-1+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)^{-1+p} (1+a x)^{-1+p} \, dx+\left (2 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)^{-1+p} (1+a x)^{-1+p} \, dx+\left (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)^{-1+p} (1+a x)^{-1+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} \left (1-a^2 x^2\right )^{-1+p} \, dx+\left (2 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} \left (1-a^2 x^2\right )^{-1+p} \, dx+\left (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 )^{-1+p} \, dx\\ &=\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),1-p;\frac{1}{2} (3-2 p);a^2 x^2\right )}{1-2 p}+\frac{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),1-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2 (1-a x)^{-p} (1+a x)^{-p} \, _2F_1\left (1-p,1-p;2-p;a^2 x^2\right )}{1-p}\\ \end{align*}

Mathematica [C]  time = 0.0983864, size = 142, normalized size = 0.65 \[ \frac{x (1-a x)^{-p} \left (-\left (a^2 x^2-1\right )^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left ((1-a x)^p \left (a^2 x^2-1\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2}-p,-p,\frac{3}{2}-p,a^2 x^2\right )-2 (a x-1)^p \left (1-a^2 x^2\right )^p F_1(1-2 p;1-p,-p;2-2 p;a x,-a x)\right )}{2 p-1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.317, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1} \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)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^p,x)

[Out]

int((a*x+1)^2/(-a^2*x^2+1)*(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 )}^{2}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 11.4331, size = 697, normalized size = 3.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*Piecewise((0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth
(1/(a*x))/a**2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p),
(5/2 - p,), a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) - a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)
*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1),
(0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**
2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,),
a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) - a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)
*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), True)) - Piecewise((0**p*log(a**2*x
**2 - 1)/(2*a) - 0**p*acoth(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper
((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) + a**(-2*p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)
*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*gamma(p + 1/2)*gamma(p + 1)), Abs(a
**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) - 0**p*atanh(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*
pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) + a**(-2*
p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*ga
mma(p + 1/2)*gamma(p + 1)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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