\(\int e^{-3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [430]

Optimal result

Integrand size = 22, antiderivative size = 135 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {33 c^3 \csc ^{-1}(a x)}{2 a}-\frac {6 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

33/2*c^3*arccsc(a*x)/a-6*c^3*arctanh((1-1/a^2/x^2)^(1/2))/a+32*c^3*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2)+6*c^3*(1-1/
a^2/x^2)^(1/2)/a-1/2*c^3*(1-1/a^2/x^2)^(1/2)/a^2/x+c^3*x*(1-1/a^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6312, 1819, 1821, 1823, 858, 222, 272, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {6 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^3 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+\frac {33 c^3 \csc ^{-1}(a x)}{2 a} \]

[In]

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

[Out]

(6*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (32*c^3*(a - x^(-1)))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (c^3*Sqrt[1 - 1/(a^2*x^2
)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x + (33*c^3*ArcCsc[a*x])/(2*a) - (6*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)
]])/a

Rule 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 222

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 272

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 858

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 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1821

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 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6312

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^6}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-c^6+\frac {6 c^6 x}{a}+\frac {16 c^6 x^2}{a^2}-\frac {6 c^6 x^3}{a^3}+\frac {c^6 x^4}{a^4}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\text {Subst}\left (\int \frac {-\frac {6 c^6}{a}-\frac {16 c^6 x}{a^2}+\frac {6 c^6 x^2}{a^3}-\frac {c^6 x^3}{a^4}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^2 \text {Subst}\left (\int \frac {\frac {12 c^6}{a^3}+\frac {33 c^6 x}{a^4}-\frac {12 c^6 x^2}{a^5}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 c^3} \\ & = \frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {a^4 \text {Subst}\left (\int \frac {-\frac {12 c^6}{a^5}-\frac {33 c^6 x}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 c^3} \\ & = \frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\left (33 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {\left (6 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {33 c^3 \csc ^{-1}(a x)}{2 a}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a} \\ & = \frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {33 c^3 \csc ^{-1}(a x)}{2 a}-\left (6 a c^3\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = \frac {6 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {33 c^3 \csc ^{-1}(a x)}{2 a}-\frac {6 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.51 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.91 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (420 a^2 \sqrt {1+\frac {1}{a x}} x^2-3465 a^3 \sqrt {1+\frac {1}{a x}} x^3+16800 a^4 \sqrt {1+\frac {1}{a x}} x^4+17955 a^5 \sqrt {1+\frac {1}{a x}} x^5-32340 a^6 \sqrt {1+\frac {1}{a x}} x^6+630 a^7 \sqrt {1+\frac {1}{a x}} x^7+44730 a^5 \sqrt {1-\frac {1}{a x}} x^5 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+44730 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-2520 a^5 \sqrt {1-\frac {1}{a x}} x^5 \arcsin \left (\frac {1}{a x}\right )-2520 a^6 \sqrt {1-\frac {1}{a x}} x^6 \arcsin \left (\frac {1}{a x}\right )-3780 a^6 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {1+\frac {1}{a x}} x^6 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+126 \sqrt {2} a^2 x^2 (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+90 \sqrt {2} a x (-1+a x)^4 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )-70 \sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+280 \sqrt {2} a x \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )-350 \sqrt {2} a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+350 \sqrt {2} a^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )-280 \sqrt {2} a^5 x^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+70 \sqrt {2} a^6 x^6 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )\right )}{630 a^6 \sqrt {1-\frac {1}{a x}} x^5 (1+a x)} \]

[In]

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

[Out]

(c^3*(420*a^2*Sqrt[1 + 1/(a*x)]*x^2 - 3465*a^3*Sqrt[1 + 1/(a*x)]*x^3 + 16800*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 17955
*a^5*Sqrt[1 + 1/(a*x)]*x^5 - 32340*a^6*Sqrt[1 + 1/(a*x)]*x^6 + 630*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 44730*a^5*Sqrt[
1 - 1/(a*x)]*x^5*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 44730*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/
Sqrt[2]] - 2520*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[1/(a*x)] - 2520*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 3
780*a^6*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 126*Sqrt[2]*a^2*x^2*(-1 +
a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 90*Sqrt[2]*a*x*(-1 + a*x)^4*(1 + a*x)*Hy
pergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] - 70*Sqrt[2]*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2
] + 280*Sqrt[2]*a*x*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 350*Sqrt[2]*a^2*x^2*Hypergeometric2F1
[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 350*Sqrt[2]*a^4*x^4*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 2
80*Sqrt[2]*a^5*x^5*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 70*Sqrt[2]*a^6*x^6*Hypergeometric2F1[3
/2, 9/2, 11/2, (1 - 1/(a*x))/2]))/(630*a^6*Sqrt[1 - 1/(a*x)]*x^5*(1 + a*x))

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.34

method	result	size
risch	\(\frac {\left (a x +1\right ) \left (12 a x -1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2} a^{3}}+\frac {\left (-\frac {6 a^{3} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {33 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}+a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}+\frac {32 a \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{3} \left (a x -1\right )}\)	\(181\)
default	\(-\frac {\left (-12 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+12 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-57 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-33 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{4} x^{4}+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+23 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-78 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-66 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+32 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+10 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -33 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-33 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a^{3} \sqrt {a^{2}}\, x^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\)	\(450\)

[In]

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

[Out]

1/2*(a*x+1)*(12*a*x-1)/x^2*c^3/a^3*((a*x-1)/(a*x+1))^(1/2)+(-6*a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^
2)^(1/2)+33/2*a^2*arctan(1/(a^2*x^2-1)^(1/2))+a^2*((a*x-1)*(a*x+1))^(1/2)+32*a/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1
/a))^(1/2))*c^3/a^3/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {66 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \]

[In]

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

[Out]

-1/2*(66*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + 12*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 1
2*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^3*x^3 + 78*a^2*c^3*x^2 + 11*a*c^3*x - c^3)*sqrt((a
*x - 1)/(a*x + 1)))/(a^3*x^2)

Sympy [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \left (- \frac {4 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\right )\, dx + \int \frac {6 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\, dx + \int \left (- \frac {4 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{4} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{3}} \]

[In]

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

[Out]

c**3*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**4 + x**3), x) + Integral(-4*a*sqrt(a*x/(a*x + 1) - 1/(a
*x + 1))/(a*x**3 + x**2), x) + Integral(6*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-
4*a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**4*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*
x + 1), x))/a**3

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-{\left (\frac {33 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {6 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {6 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {32 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {11 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 6 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 13 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \]

[In]

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

[Out]

-(33*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 6*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 6*c^3*log(sqrt
((a*x - 1)/(a*x + 1)) - 1)/a^2 - 32*c^3*sqrt((a*x - 1)/(a*x + 1))/a^2 + (11*c^3*((a*x - 1)/(a*x + 1))^(5/2) -
6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 13*c^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a
^2/(a*x + 1)^2 - (a*x - 1)^3*a^2/(a*x + 1)^3 + a^2))*a

Giac [F]

\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

undef

Mupad [B] (verification not implemented)

Time = 3.91 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.41 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {13\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+6\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-11\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a+\frac {a\,\left (a\,x-1\right )}{a\,x+1}-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {32\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {33\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,12{}\mathrm {i}}{a} \]

[In]

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

[Out]

(13*c^3*((a*x - 1)/(a*x + 1))^(1/2) + 6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 11*c^3*((a*x - 1)/(a*x + 1))^(5/2))/
(a + (a*(a*x - 1))/(a*x + 1) - (a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) + (32*c^3*((a*x - 1)
/(a*x + 1))^(1/2))/a - (33*c^3*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a + (c^3*atan(((a*x - 1)/(a*x + 1))^(1/2)*1i
)*12i)/a