3.609 \(\int e^{-3 \coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

Optimal. Leaf size=145 \[ -\frac {1}{3} a^2 c x^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}+\frac {5}{6} a c x^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{2} c x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {5 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{2 a} \]

[Out]

5/2*c*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a+5/6*a*c*(1-1/a/x)^(3/2)*x^2*(1+1/a/x)^(1/2)-1/3*a^2*c*(1-1/a/
x)^(5/2)*x^3*(1+1/a/x)^(1/2)-5/2*c*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6191, 6195, 94, 92, 208} \[ -\frac {1}{3} a^2 c x^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}+\frac {5}{6} a c x^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {5}{2} c x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {5 c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/2 + (5*a*c*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2)/6 - (a^2*c*
(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3)/3 + (5*c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(2*a)

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 94

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

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]

Rule 6191

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^(2*p)*(1 -
1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &
& IntegerQ[p]

Rule 6195

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 61, normalized size = 0.42 \[ \frac {c \left (a x \sqrt {1-\frac {1}{a^2 x^2}} \left (-2 a^2 x^2+9 a x-22\right )+15 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-22 + 9*a*x - 2*a^2*x^2) + 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)

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fricas [A]  time = 0.56, size = 92, normalized size = 0.63 \[ \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 13 \, a c x + 22 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*c*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c*x^3 - 7*a^2*
c*x^2 + 13*a*c*x + 22*c)*sqrt((a*x - 1)/(a*x + 1)))/a

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giac [A]  time = 0.18, size = 82, normalized size = 0.57 \[ -\frac {5 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, a c x \mathrm {sgn}\left (a x + 1\right ) - 9 \, c \mathrm {sgn}\left (a x + 1\right )\right )} x + \frac {22 \, c \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*c*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) - 1/6*sqrt(a^2*x^2 - 1)*((2*a*c*x*sgn(a*x +
1) - 9*c*sgn(a*x + 1))*x + 22*c*sgn(a*x + 1)/a)

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maple [A]  time = 0.05, size = 183, normalized size = 1.26 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c \left (9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -24 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+24 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{6 \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)^2*c*(9*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x*a-2*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(
1/2)-9*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a-24*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)+24*a*ln(
(a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2)))/(a*x-1)/((a*x-1)*(a*x+1))^(1/2)/a/(a^2)^(1/2)

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maxima [A]  time = 0.31, size = 171, normalized size = 1.18 \[ \frac {1}{6} \, a {\left (\frac {2 \, {\left (33 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 40 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}} + \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*(2*(33*c*((a*x - 1)/(a*x + 1))^(5/2) - 40*c*((a*x - 1)/(a*x + 1))^(3/2) + 15*c*sqrt((a*x - 1)/(a*x + 1))
)/(3*(a*x - 1)*a^2/(a*x + 1) - 3*(a*x - 1)^2*a^2/(a*x + 1)^2 + (a*x - 1)^3*a^2/(a*x + 1)^3 - a^2) + 15*c*log(s
qrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)

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mupad [B]  time = 0.07, size = 133, normalized size = 0.92 \[ \frac {5\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {5\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {40\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+11\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a - (5*c*((a*x - 1)/(a*x + 1))^(1/2) - (40*c*((a*x - 1)/(a*x + 1))^(3
/2))/3 + 11*c*((a*x - 1)/(a*x + 1))^(5/2))/(a - (3*a*(a*x - 1))/(a*x + 1) + (3*a*(a*x - 1)^2)/(a*x + 1)^2 - (a
*(a*x - 1)^3)/(a*x + 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - c \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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