3.313 \(\int \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x^4} \, dx\)

Optimal. Leaf size=88 \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-a^3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a^3 c^3 \sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.178863, antiderivative size = 88, 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, 811, 844, 216, 266, 63, 208} \[ -\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-a^3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a^3 c^3 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^4,x]

[Out]

(a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/x^2 - (c^3*(1 - a^2*x^2)^(3/2))/(3*x^3) - a^3*c^3*ArcSin[a*x] - a^3*c^3*Ar
cTanh[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 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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^4} \, dx &=c \int \frac{(c-a c x)^2 \sqrt{1-a^2 x^2}}{x^4} \, dx\\ &=-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac{1}{3} c \int \frac{\left (6 a c^2-3 a^2 c^2 x\right ) \sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac{1}{12} c \int \frac{12 a^3 c^2-12 a^4 c^2 x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\left (a^3 c^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx-\left (a^4 c^3\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)+\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)-\left (a 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 )\\ &=\frac{a c^3 (1-a x) \sqrt{1-a^2 x^2}}{x^2}-\frac{c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)-a^3 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.104318, size = 156, normalized size = 1.77 \[ \frac{c^3 \left (4 a^4 x^4-6 a^3 x^3-2 a^2 x^2+3 a^3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+18 a^3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-6 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+6 a x-2\right )}{6 x^3 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^4,x]

[Out]

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

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Maple [A]  time = 0.042, size = 119, normalized size = 1.4 \begin{align*} -{{a}^{4}{c}^{3}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{c}^{3}{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{{c}^{3}a}{{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{c}^{3}}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{3}{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}} \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^4,x)

[Out]

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

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Maxima [A]  time = 1.44074, size = 165, normalized size = 1.88 \begin{align*} -\frac{a^{4} c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - a^{3} c^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{3}}{3 \, x} + \frac{\sqrt{-a^{2} x^{2} + 1} a c^{3}}{x^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{3 \, x^{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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65232, size = 225, normalized size = 2.56 \begin{align*} \frac{6 \, a^{3} c^{3} x^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{3} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (2 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x + c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{3 \, x^{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^4,x, algorithm="fricas")

[Out]

1/3*(6*a^3*c^3*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2
*a^2*c^3*x^2 - 3*a*c^3*x + c^3)*sqrt(-a^2*x^2 + 1))/x^3

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Sympy [C]  time = 6.57562, size = 279, normalized size = 3.17 \begin{align*} - a^{4} 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 ) + 2 a^{3} 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 ) - 2 a c^{3} \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 ) + c^{3} \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \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**4,x)

[Out]

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

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Giac [B]  time = 1.23791, size = 338, normalized size = 3.84 \begin{align*} -\frac{a^{4} c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a^{4} 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 (a^{4} c^{3} - \frac{6 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} - \frac{\frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{3}}{x} - \frac{6 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{3}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}}{x^{3}}}{24 \, a^{2}{\left | a \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^4,x, algorithm="giac")

[Out]

-a^4*c^3*arcsin(a*x)*sgn(a)/abs(a) - a^4*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs
(a) + 1/24*(a^4*c^3 - 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^
2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - 1/24*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^3/x - 6*
(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^3/x^2 + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/x^3)/(a^2*abs(a))