3.1143 \(\int \frac{e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2)}{x^3} \, dx\)

Optimal. Leaf size=76 \[ -\frac{3 a c \sqrt{1-a^2 x^2}}{x}-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{7}{2} a^2 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 c \sin ^{-1}(a x) \]

[Out]

-(c*Sqrt[1 - a^2*x^2])/(2*x^2) - (3*a*c*Sqrt[1 - a^2*x^2])/x + a^2*c*ArcSin[a*x] - (7*a^2*c*ArcTanh[Sqrt[1 - a
^2*x^2]])/2

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Rubi [A]  time = 0.194184, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6148, 1807, 844, 216, 266, 63, 208} \[ -\frac{3 a c \sqrt{1-a^2 x^2}}{x}-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{7}{2} a^2 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^2 c \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*Sqrt[1 - a^2*x^2])/(2*x^2) - (3*a*c*Sqrt[1 - a^2*x^2])/x + a^2*c*ArcSin[a*x] - (7*a^2*c*ArcTanh[Sqrt[1 - a
^2*x^2]])/2

Rule 6148

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*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] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

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 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^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^3} \, dx &=c \int \frac{(1+a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} c \int \frac{-6 a-7 a^2 x-2 a^3 x^2}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a c \sqrt{1-a^2 x^2}}{x}+\frac{1}{2} c \int \frac{7 a^2+2 a^3 x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a c \sqrt{1-a^2 x^2}}{x}+\frac{1}{2} \left (7 a^2 c\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (a^3 c\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a c \sqrt{1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)+\frac{1}{4} \left (7 a^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a c \sqrt{1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)-\frac{1}{2} (7 c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{c \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3 a c \sqrt{1-a^2 x^2}}{x}+a^2 c \sin ^{-1}(a x)-\frac{7}{2} a^2 c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0741761, size = 60, normalized size = 0.79 \[ \frac{1}{2} c \left (-\frac{(6 a x+1) \sqrt{1-a^2 x^2}}{x^2}-7 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+2 a^2 \sin ^{-1}(a x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-(((1 + 6*a*x)*Sqrt[1 - a^2*x^2])/x^2) + 2*a^2*ArcSin[a*x] - 7*a^2*ArcTanh[Sqrt[1 - a^2*x^2]]))/2

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Maple [A]  time = 0.045, size = 125, normalized size = 1.6 \begin{align*} 3\,{\frac{{a}^{3}cx}{\sqrt{-{a}^{2}{x}^{2}+1}}}+{{a}^{3}c\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{a}^{2}c}{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-3\,{\frac{ac}{x\sqrt{-{a}^{2}{x}^{2}+1}}}-{\frac{7\,{a}^{2}c}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{c}{2\,{x}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*c*a^3*x/(-a^2*x^2+1)^(1/2)+c*a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/2*c*a^2/(-a^2*x^2+1)
^(1/2)-3*c*a/x/(-a^2*x^2+1)^(1/2)-7/2*c*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))-1/2*c/x^2/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.46518, size = 173, normalized size = 2.28 \begin{align*} \frac{3 \, a^{3} c x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{a^{3} c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{7}{2} \, a^{2} c \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{a^{2} c}{2 \, \sqrt{-a^{2} x^{2} + 1}} - \frac{3 \, a c}{\sqrt{-a^{2} x^{2} + 1} x} - \frac{c}{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^3*c*x/sqrt(-a^2*x^2 + 1) + a^3*c*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) - 7/2*a^2*c*log(2*sqrt(-a^2*x^2 + 1)/ab
s(x) + 2/abs(x)) + 1/2*a^2*c/sqrt(-a^2*x^2 + 1) - 3*a*c/(sqrt(-a^2*x^2 + 1)*x) - 1/2*c/(sqrt(-a^2*x^2 + 1)*x^2
)

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Fricas [A]  time = 2.68276, size = 194, normalized size = 2.55 \begin{align*} -\frac{4 \, a^{2} c x^{2} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 7 \, a^{2} c x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (6 \, a c x + c\right )}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*a^2*c*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 7*a^2*c*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(
-a^2*x^2 + 1)*(6*a*c*x + c))/x^2

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Sympy [C]  time = 7.5597, size = 223, normalized size = 2.93 \begin{align*} a^{3} c \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 ) + 3 a^{2} c \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 ) + 3 a c \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 ) + c \left (\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.18294, size = 242, normalized size = 3.18 \begin{align*} \frac{a^{3} c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{{\left (a^{3} c + \frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c}{x}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}} - \frac{7 \, a^{3} c \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*c*arcsin(a*x)*sgn(a)/abs(a) + 1/8*(a^3*c + 12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*c/x)*a^4*x^2/((sqrt(-a^2*x
^2 + 1)*abs(a) + a)^2*abs(a)) - 7/2*a^3*c*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a)
- 1/8*(12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*c*abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c*abs(a)/(a*x^2))/
a^2