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

Optimal. Leaf size=83 \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac{1}{2} a c^3 (a x+4) \sqrt{1-a^2 x^2}+2 a c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} a c^3 \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.174132, antiderivative size = 83, 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, 1807, 815, 844, 216, 266, 63, 208} \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac{1}{2} a c^3 (a x+4) \sqrt{1-a^2 x^2}+2 a c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{1}{2} a c^3 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.102182, size = 143, normalized size = 1.72 \[ -\frac{c^3 \left (a^4 x^4-4 a^3 x^3-3 a^2 x^2+2 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+2 a x \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-4 a x \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+4 a x+2\right )}{2 x \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 113, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}{c}^{3}x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{3}{a}^{2}}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-2\,{c}^{3}a\sqrt{-{a}^{2}{x}^{2}+1}-{\frac{{c}^{3}}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+2\,{c}^{3}a{\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^2,x)

[Out]

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

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Maxima [A]  time = 1.44037, size = 157, normalized size = 1.89 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3} x - \frac{a^{2} c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} + 2 \, a c^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1} a c^{3} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.54244, size = 228, normalized size = 2.75 \begin{align*} \frac{2 \, a c^{3} x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 4 \, a c^{3} x \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, a c^{3} x +{\left (a^{2} c^{3} x^{2} - 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \, x} \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^2,x, algorithm="fricas")

[Out]

1/2*(2*a*c^3*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 4*a*c^3*x*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 4*a*c^3*x
+ (a^2*c^3*x^2 - 4*a*c^3*x - 2*c^3)*sqrt(-a^2*x^2 + 1))/x

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Sympy [C]  time = 7.11276, size = 199, normalized size = 2.4 \begin{align*} - a^{4} 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^{3} c^{3} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) - 2 a 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 ) + c^{3} \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \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**2,x)

[Out]

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

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Giac [B]  time = 1.27673, size = 205, normalized size = 2.47 \begin{align*} \frac{a^{4} c^{3} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{a^{2} c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} + \frac{2 \, a^{2} 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{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{3}}{2 \, x{\left | a \right |}} + \frac{1}{2} \,{\left (a^{2} c^{3} x - 4 \, a 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^2,x, algorithm="giac")

[Out]

1/2*a^4*c^3*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - 1/2*a^2*c^3*arcsin(a*x)*sgn(a)/abs(a) + 2*a^2*c^3*log
(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(x
*abs(a)) + 1/2*(a^2*c^3*x - 4*a*c^3)*sqrt(-a^2*x^2 + 1)