3.1090 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{3/2}}{x} \, dx\)

Optimal. Leaf size=101 \[ c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+c (a x+1) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.286373, antiderivative size = 101, 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, 1809, 815, 844, 217, 203, 266, 63, 208} \[ c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )+c (a x+1) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(1 + a*x)*Sqrt[c - a^2*c*x^2] - (c - a^2*c*x^2)^(3/2)/3 + c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]
- 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 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 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} \, dx &=c \int \frac{(1+a x)^2 \sqrt{c-a^2 c x^2}}{x} \, dx\\ &=-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}-\frac{\int \frac{\left (-3 a^2 c-6 a^3 c x\right ) \sqrt{c-a^2 c x^2}}{x} \, dx}{3 a^2}\\ &=c (1+a x) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}+\frac{\int \frac{6 a^4 c^3+6 a^5 c^3 x}{x \sqrt{c-a^2 c x^2}} \, dx}{6 a^4 c}\\ &=c (1+a x) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}+c^2 \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx+\left (a c^2\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=c (1+a x) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}+\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )+\left (a 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 )\\ &=c (1+a x) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}+c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-\frac{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^2}\\ &=c (1+a x) \sqrt{c-a^2 c x^2}-\frac{1}{3} \left (c-a^2 c x^2\right )^{3/2}+c^{3/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )-c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.108017, size = 115, normalized size = 1.14 \[ -c^{3/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+c^{3/2} \left (-\tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )\right )+\frac{1}{3} c \left (a^2 x^2+3 a x+2\right ) \sqrt{c-a^2 c x^2}+c^{3/2} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.041, size = 179, normalized size = 1.8 \begin{align*}{\frac{1}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) +\sqrt{-{a}^{2}c{x}^{2}+c}c-{\frac{2}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+ac\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{a{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,x)

[Out]

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

[Out]

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

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Fricas [A]  time = 2.77963, size = 539, normalized size = 5.34 \begin{align*} \left [-c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + \frac{1}{2} \, c^{\frac{3}{2}} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + \frac{1}{3} \,{\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt{-a^{2} c x^{2} + c}, -\sqrt{-c} c \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \frac{1}{2} \, \sqrt{-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + \frac{1}{3} \,{\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt{-a^{2} c x^{2} + 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,x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 8.74049, size = 267, normalized size = 2.64 \begin{align*} a^{2} c \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) + 2 a 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 ) + 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 ) \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,x)

[Out]

a**2*c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True))
+ 2*a*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))
+ 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))

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Giac [A]  time = 1.17446, size = 157, normalized size = 1.55 \begin{align*} \frac{2 \, c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{a \sqrt{-c} c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \frac{1}{3} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (a^{2} c x + 3 \, a c\right )} x + 2 \, 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,x, algorithm="giac")

[Out]

2*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) + a*sqrt(-c)*c*log(abs(-sqrt(-a^2*c)*
x + sqrt(-a^2*c*x^2 + c)))/abs(a) + 1/3*sqrt(-a^2*c*x^2 + c)*((a^2*c*x + 3*a*c)*x + 2*c)