3.1091 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac{1}{2} a c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+\frac{1}{2} a c (4-a x) \sqrt{c-a^2 c x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.285476, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6151, 1807, 815, 844, 217, 203, 266, 63, 208} \[ -\frac{1}{2} a c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+\frac{1}{2} a c (4-a x) \sqrt{c-a^2 c x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1807

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

Rule 815

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

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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

Mathematica [A]  time = 0.15094, size = 124, normalized size = 1.11 \[ -2 a c^{3/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+\frac{1}{2} a c^{3/2} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )+\frac{c \left (a^2 x^2+4 a x-2\right ) \sqrt{c-a^2 c x^2}}{2 x}+2 a c^{3/2} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.042, size = 286, normalized size = 2.6 \begin{align*} -{\frac{1}{cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{a}^{2}x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}-{\frac{3\,cx{a}^{2}}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{3\,{a}^{2}{c}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2\,a}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,a{c}^{3/2}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c}}{x}} \right ) +2\,a\sqrt{-{a}^{2}c{x}^{2}+c}c-{\frac{2\,a}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{a}^{2}c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{a}^{2}{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c/x*(-a^2*c*x^2+c)^(5/2)-a^2*x*(-a^2*c*x^2+c)^(3/2)-3/2*a^2*c*x*(-a^2*c*x^2+c)^(1/2)-3/2*a^2*c^2/(a^2*c)^(1
/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/3*a*(-a^2*c*x^2+c)^(3/2)-2*a*c^(3/2)*ln((2*c+2*c^(1/2)*(-a^
2*c*x^2+c)^(1/2))/x)+2*a*(-a^2*c*x^2+c)^(1/2)*c-2/3*a*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(3/2)+a^2*c*(-c*a^2*(x-
1/a)^2-2*a*c*(x-1/a))^(1/2)*x+a^2*c^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-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^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{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)*(-a^2*c*x^2+c)^(3/2)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.76932, size = 567, normalized size = 5.06 \begin{align*} \left [\frac{a c^{\frac{3}{2}} x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac{3}{2}} x \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) +{\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{2 \, x}, -\frac{8 \, a \sqrt{-c} c x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - a \sqrt{-c} c x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{4 \, 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)*(-a^2*c*x^2+c)^(3/2)/x^2,x, algorithm="fricas")

[Out]

[1/2*(a*c^(3/2)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 2*a*c^(3/2)*x*log(-(a^2*c*x^2 + 2
*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + (a^2*c*x^2 + 4*a*c*x - 2*c)*sqrt(-a^2*c*x^2 + c))/x, -1/4*(8*a*sqr
t(-c)*c*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) - a*sqrt(-c)*c*x*log(2*a^2*c*x^2 - 2*sqrt(-a^2
*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(a^2*c*x^2 + 4*a*c*x - 2*c)*sqrt(-a^2*c*x^2 + c))/x]

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Sympy [C]  time = 8.59299, size = 350, normalized size = 3.12 \begin{align*} a^{2} c \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left (a x \right )} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} + i \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2} - \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c
)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) +
2*a*c*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin
(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a
**2*x**2 + 1) + 1), True)) + c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*s
qrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x
) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True))

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Giac [A]  time = 1.1971, size = 223, normalized size = 1.99 \begin{align*} \frac{4 \, a c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{a^{2} \sqrt{-c} c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{2 \,{\left | a \right |}} + \frac{2 \, a^{2} \sqrt{-c} c^{2}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}{\left | a \right |}} + \frac{1}{2} \, \sqrt{-a^{2} c x^{2} + c}{\left (a^{2} c x + 4 \, a c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) - 1/2*a^2*sqrt(-c)*c*log(abs(-sqrt(
-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/abs(a) + 2*a^2*sqrt(-c)*c^2/(((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c
)*abs(a)) + 1/2*sqrt(-a^2*c*x^2 + c)*(a^2*c*x + 4*a*c)