3.324 \(\int \frac{e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x^5} \, dx\)

Optimal. Leaf size=110 \[ -\frac{11 a^2 c^4 \sqrt{1-a^2 x^2}}{8 x^2}+\frac{a c^4 \sqrt{1-a^2 x^2}}{x^3}-\frac{c^4 \sqrt{1-a^2 x^2}}{4 x^4}+\frac{13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^4 c^4 \sin ^{-1}(a x) \]

[Out]

-(c^4*Sqrt[1 - a^2*x^2])/(4*x^4) + (a*c^4*Sqrt[1 - a^2*x^2])/x^3 - (11*a^2*c^4*Sqrt[1 - a^2*x^2])/(8*x^2) + a^
4*c^4*ArcSin[a*x] + (13*a^4*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

________________________________________________________________________________________

Rubi [A]  time = 0.247726, antiderivative size = 116, normalized size of antiderivative = 1.05, number of steps used = 9, 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} \[ \frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac{a^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}+\frac{13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+a^4 c^4 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^5,x]

[Out]

-(a^2*c^4*(13 - 8*a*x)*Sqrt[1 - a^2*x^2])/(8*x^2) - (c^4*(1 - a^2*x^2)^(3/2))/(4*x^4) + (a*c^4*(1 - a^2*x^2)^(
3/2))/x^3 + a^4*c^4*ArcSin[a*x] + (13*a^4*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8

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)^4}{x^5} \, dx &=c \int \frac{(c-a c x)^3 \sqrt{1-a^2 x^2}}{x^5} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac{1}{4} c \int \frac{\sqrt{1-a^2 x^2} \left (12 a c^3-13 a^2 c^3 x+4 a^3 c^3 x^2\right )}{x^4} \, dx\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+\frac{1}{12} c \int \frac{\left (39 a^2 c^3-12 a^3 c^3 x\right ) \sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{a^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac{1}{48} c \int \frac{78 a^4 c^3-48 a^5 c^3 x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}-\frac{1}{8} \left (13 a^4 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+\left (a^5 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)-\frac{1}{16} \left (13 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac{1}{8} \left (13 a^2 c^4\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^2 c^4 (13-8 a x) \sqrt{1-a^2 x^2}}{8 x^2}-\frac{c^4 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac{a c^4 \left (1-a^2 x^2\right )^{3/2}}{x^3}+a^4 c^4 \sin ^{-1}(a x)+\frac{13}{8} a^4 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.210056, size = 125, normalized size = 1.14 \[ \frac{1}{16} c^4 \left (-\frac{2 \left (-11 a^4 x^4+8 a^3 x^3+9 a^2 x^2+29 a^4 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-8 a x+2\right )}{x^4 \sqrt{1-a^2 x^2}}+26 a^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-13 a^4 \sin ^{-1}(a x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^5,x]

[Out]

(c^4*(-13*a^4*ArcSin[a*x] - (2*(2 - 8*a*x + 9*a^2*x^2 + 8*a^3*x^3 - 11*a^4*x^4 + 29*a^4*x^4*Sqrt[1 - a^2*x^2]*
ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(x^4*Sqrt[1 - a^2*x^2]) + 26*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]))/16

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 118, normalized size = 1.1 \begin{align*}{{c}^{4}{a}^{5}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{{c}^{4}}{4\,{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{11\,{a}^{2}{c}^{4}}{8\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{13\,{c}^{4}{a}^{4}}{8}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{a{c}^{4}}{{x}^{3}}\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)^4/x^5,x)

[Out]

c^4*a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/4*c^4*(-a^2*x^2+1)^(1/2)/x^4-11/8*a^2*c^4*(-a^2
*x^2+1)^(1/2)/x^2+13/8*c^4*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))+a*c^4*(-a^2*x^2+1)^(1/2)/x^3

________________________________________________________________________________________

Maxima [A]  time = 1.42836, size = 163, normalized size = 1.48 \begin{align*} \frac{a^{5} c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{13}{8} \, a^{4} c^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{11 \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4}}{8 \, x^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} a c^{4}}{x^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{4 \, x^{4}} \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)^4/x^5,x, algorithm="maxima")

[Out]

a^5*c^4*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) + 13/8*a^4*c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 11/8*sq
rt(-a^2*x^2 + 1)*a^2*c^4/x^2 + sqrt(-a^2*x^2 + 1)*a*c^4/x^3 - 1/4*sqrt(-a^2*x^2 + 1)*c^4/x^4

________________________________________________________________________________________

Fricas [A]  time = 1.54803, size = 234, normalized size = 2.13 \begin{align*} -\frac{16 \, a^{4} c^{4} x^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 13 \, a^{4} c^{4} x^{4} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) +{\left (11 \, a^{2} c^{4} x^{2} - 8 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{8 \, x^{4}} \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)^4/x^5,x, algorithm="fricas")

[Out]

-1/8*(16*a^4*c^4*x^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 13*a^4*c^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) +
(11*a^2*c^4*x^2 - 8*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/x^4

________________________________________________________________________________________

Sympy [C]  time = 11.2849, size = 505, normalized size = 4.59 \begin{align*} \text{result too large to display} \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)**4/x**5,x)

[Out]

a**5*c**4*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**4*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**3*c**4*Pi
ecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 2*a**2*c**4*Piecewi
se((-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)) - 3*a*c**4*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)) + c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*s
qrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(
a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(
a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))

________________________________________________________________________________________

Giac [B]  time = 1.17904, size = 427, normalized size = 3.88 \begin{align*} \frac{a^{5} c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{13 \, a^{5} c^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} + \frac{{\left (a^{5} c^{4} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{3} c^{4}}{x} + \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a c^{4}}{x^{2}} - \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a x^{3}}\right )} a^{8} x^{4}}{64 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}{\left | a \right |}} + \frac{\frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{5} c^{4}{\left | a \right |}}{x} - \frac{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{3} c^{4}{\left | a \right |}}{x^{2}} + \frac{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} a c^{4}{\left | a \right |}}{x^{3}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}{\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \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)^4/x^5,x, algorithm="giac")

[Out]

a^5*c^4*arcsin(a*x)*sgn(a)/abs(a) + 13/8*a^5*c^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))
/abs(a) + 1/64*(a^5*c^4 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3*c^4/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a
*c^4/x^2 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a))
+ 1/64*(8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*c^4*abs(a)/x - 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^3*c^4*abs
(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c^4*abs(a)/x^3 - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4*abs(a)/
(a*x^4))/a^4