3.807 \(\int e^{-3 \tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^p \, dx\)

Optimal. Leaf size=216 \[ \frac{3 a^2 x^3 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p,\frac{1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}-\frac{a (5-2 p) x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{\sqrt{1-a^2 x^2}}+\frac{x \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.321789, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6160, 6149, 1809, 1808, 364, 807} \[ \frac{3 a^2 x^3 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}-\frac{a (5-2 p) x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,\frac{3}{2}-p;2-p;a^2 x^2\right )}{2 (1-p)}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{\sqrt{1-a^2 x^2}}+\frac{x \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^p/E^(3*ArcTanh[a*x]),x]

[Out]

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

Rule 6160

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/
(1 + (c*x^2)/d)^p, Int[(u*(1 + (c*x^2)/d)^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]

Rule 6149

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 1808

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = Expon[Pq, x]}, Dist[Coeff[Pq,
x, q]/c^q, Int[(c*x)^(m + q)*(a + b*x^2)^p, x], x] + Dist[1/c^q, Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq
- Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] &
& PolyQ[Pq, x] &&  !(IGtQ[m, 0] && ILtQ[p + 1/2, 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])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps

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

Mathematica [A]  time = 0.138352, size = 173, normalized size = 0.8 \[ \frac{1}{2} x \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left (\frac{3 a x \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,a^2 x^2\right )}{p-1}+\frac{6 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{2}-p,\frac{3}{2}-p,\frac{5}{2}-p,a^2 x^2\right )}{3-2 p}+\frac{a^3 x^3 \text{Hypergeometric2F1}\left (\frac{3}{2}-p,2-p,3-p,a^2 x^2\right )}{p-2}+\frac{2 \left (1-a^2 x^2\right )^{p-\frac{1}{2}}}{1-2 p}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))^p/E^(3*ArcTanh[a*x]),x]

[Out]

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

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Maple [F]  time = 0.336, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ax+1 \right ) ^{3}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{p} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

int((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^p/(a*x + 1)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1}{\left (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((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**p/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^p/(a*x + 1)^3, x)