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

Optimal. Leaf size=75 \[ -\frac{1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+c^3 (1-a x) \sqrt{1-a^2 x^2}-c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-c^3 \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.173301, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 1809, 815, 844, 216, 266, 63, 208} \[ -\frac{1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}+c^3 (1-a x) \sqrt{1-a^2 x^2}-c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-c^3 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6128

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

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 216

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

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

Mathematica [A]  time = 0.103919, size = 135, normalized size = 1.8 \[ \frac{c^3 \left (-2 a^4 x^4+6 a^3 x^3-2 a^2 x^2+3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+18 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-6 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-6 a x+4\right )}{6 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(4 - 6*a*x - 2*a^2*x^2 + 6*a^3*x^3 - 2*a^4*x^4 + 3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 18*Sqrt[1 - a^2*x^2]*A
rcSin[Sqrt[1 - a*x]/Sqrt[2]] - 6*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(6*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.036, size = 110, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}{c}^{3}{x}^{2}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{2\,{c}^{3}}{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{c}^{3}ax\sqrt{-{a}^{2}{x}^{2}+1}-{{c}^{3}a\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{c}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c^3*a^2*x^2*(-a^2*x^2+1)^(1/2)+2/3*c^3*(-a^2*x^2+1)^(1/2)-c^3*a*x*(-a^2*x^2+1)^(1/2)-c^3*a/(a^2)^(1/2)*arc
tan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-c^3*arctanh(1/(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.43526, size = 153, normalized size = 2.04 \begin{align*} \frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x^{2} - \sqrt{-a^{2} x^{2} + 1} a c^{3} x - \frac{a c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - c^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{2}{3} \, \sqrt{-a^{2} x^{2} + 1} c^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62816, size = 193, normalized size = 2.57 \begin{align*} 2 \, c^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + \frac{1}{3} \,{\left (a^{2} c^{3} x^{2} - 3 \, a c^{3} x + 2 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 13.2081, size = 226, normalized size = 3.01 \begin{align*} - a^{4} c^{3} \left (\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 a^{3} c^{3} \left (\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a^{3}} & \text{otherwise} \end{cases}\right ) - 2 a c^{3} \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) + c^{3} \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*c**3*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/
4, True)) + 2*a**3*c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) >
1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) - 2*a*c**3*
Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**3
*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))

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Giac [A]  time = 1.25793, size = 128, normalized size = 1.71 \begin{align*} -\frac{a c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a c^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{1}{3} \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, c^{3} +{\left (a^{2} c^{3} x - 3 \, a c^{3}\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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