3.1120 \(\int \frac{e^{2 \tanh ^{-1}(a x)}}{x (c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{4 a x+3}{3 c \sqrt{c-a^2 c x^2}} \]

[Out]

(2*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (3 + 4*a*x)/(3*c*Sqrt[c - a^2*c*x^2]) - ArcTanh[Sqrt[c - a^2*c*x^2]/
Sqrt[c]]/c^(3/2)

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Rubi [A]  time = 0.268081, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6151, 1805, 823, 12, 266, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{4 a x+3}{3 c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (3 + 4*a*x)/(3*c*Sqrt[c - a^2*c*x^2]) - ArcTanh[Sqrt[c - a^2*c*x^2]/
Sqrt[c]]/c^(3/2)

Rule 6151

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^(n/2), Int[x^m*(c
+ d*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] ||
GtQ[c, 0]) && IGtQ[n/2, 0]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.111037, size = 75, normalized size = 0.91 \[ \frac{(5-4 a x) \sqrt{c-a^2 c x^2}}{3 c^2 (a x-1)^2}-\frac{\log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )}{c^{3/2}}+\frac{\log (x)}{c^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((5 - 4*a*x)*Sqrt[c - a^2*c*x^2])/(3*c^2*(-1 + a*x)^2) + Log[x]/c^(3/2) - Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]
/c^(3/2)

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Maple [B]  time = 0.041, size = 152, normalized size = 1.9 \begin{align*}{\frac{1}{c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}-{\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{2}{3\,ac} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}}-{\frac{2}{3\,a{c}^{2}} \left ( -2\,{a}^{2}c \left ( x-{a}^{-1} \right ) -2\,ac \right ){\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c/(-a^2*c*x^2+c)^(1/2)-1/c^(3/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-2/3/a/c/(x-1/a)/(-c*a^2*(x-1/a)^
2-2*a*c*(x-1/a))^(1/2)-2/3/a/c^2*(-2*a^2*c*(x-1/a)-2*a*c)/(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)

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

[Out]

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

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Fricas [A]  time = 2.95345, size = 452, normalized size = 5.51 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c}{\left (4 \, a x - 5\right )}}{6 \,{\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}, -\frac{3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} c x^{2} + c}{\left (4 \, a x - 5\right )}}{3 \,{\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a
^2*c*x^2 + c)*(4*a*x - 5))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2), -1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt
(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(4*a*x - 5))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)
]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a x}{- a^{3} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a c x^{2} \sqrt{- a^{2} c x^{2} + c} - c x \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a c x^{2} \sqrt{- a^{2} c x^{2} + c} - c x \sqrt{- a^{2} c x^{2} + c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(a*x/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*sqrt(-a**2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*
c*x**2 + c) - c*x*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*
sqrt(-a**2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*c*x**2 + c) - c*x*sqrt(-a**2*c*x**2 + c)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError