\(\int e^{-3 \text {arctanh}(a x)} (c-\frac {c}{a x})^3 \, dx\) [534]

Optimal result

Integrand size = 22, antiderivative size = 140 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {32 c^3 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {c^3 \sqrt {1-a^2 x^2}}{a}+\frac {c^3 \sqrt {1-a^2 x^2}}{2 a^3 x^2}-\frac {6 c^3 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {6 c^3 \arcsin (a x)}{a}+\frac {33 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 0.77 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.13 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {c^3 \left (-63 \sqrt {2} a^2 x^2 (-1+a x)^3 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},\frac {1}{2} (1-a x)\right )+45 \sqrt {2} a^2 x^2 (-1+a x)^4 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {1}{2} (1-a x)\right )+35 \left (-2 \sqrt {1+a x}+24 a x \sqrt {1+a x}-670 a^2 x^2 \sqrt {1+a x}+406 a^3 x^3 \sqrt {1+a x}+290 a^4 x^4 \sqrt {1+a x}-54 a^5 x^5 \sqrt {1+a x}+6 a^6 x^6 \sqrt {1+a x}-48 a^2 x^2 \sqrt {1-a x} \arcsin (a x)-48 a^3 x^3 \sqrt {1-a x} \arcsin (a x)+864 a^2 x^2 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+864 a^3 x^3 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-66 a^2 x^2 \sqrt {1+a x} \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\sqrt {2} a^2 x^2 (-1+a x)^5 (1+a x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} (1-a x)\right )\right )\right )}{140 a^3 x^2 \sqrt {1-a x} (1+a x)} \] Input:

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

Output:

-1/140*(c^3*(-63*Sqrt[2]*a^2*x^2*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[ 
3/2, 5/2, 7/2, (1 - a*x)/2] + 45*Sqrt[2]*a^2*x^2*(-1 + a*x)^4*(1 + a*x)*Hy 
pergeometric2F1[3/2, 7/2, 9/2, (1 - a*x)/2] + 35*(-2*Sqrt[1 + a*x] + 24*a* 
x*Sqrt[1 + a*x] - 670*a^2*x^2*Sqrt[1 + a*x] + 406*a^3*x^3*Sqrt[1 + a*x] + 
290*a^4*x^4*Sqrt[1 + a*x] - 54*a^5*x^5*Sqrt[1 + a*x] + 6*a^6*x^6*Sqrt[1 + 
a*x] - 48*a^2*x^2*Sqrt[1 - a*x]*ArcSin[a*x] - 48*a^3*x^3*Sqrt[1 - a*x]*Arc 
Sin[a*x] + 864*a^2*x^2*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] + 864*a 
^3*x^3*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 66*a^2*x^2*Sqrt[1 + a 
*x]*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]] - Sqrt[2]*a^2*x^2*(-1 + a 
*x)^5*(1 + a*x)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - a*x)/2])))/(a^3*x^2 
*Sqrt[1 - a*x]*(1 + a*x))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6681, 6678, 528, 2338, 2338, 2340, 27, 538, 223, 243, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx\)

\(\Big \downarrow \) 6681

\(\displaystyle -\frac {c^3 \int \frac {e^{-3 \text {arctanh}(a x)} (1-a x)^3}{x^3}dx}{a^3}\)

\(\Big \downarrow \) 6678

\(\displaystyle -\frac {c^3 \int \frac {(1-a x)^6}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx}{a^3}\)

\(\Big \downarrow \) 528

\(\displaystyle -\frac {c^3 \left (\int \frac {-a^4 x^4+6 a^3 x^3+16 a^2 x^2-6 a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}\right )}{a^3}\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} \int \frac {2 x^3 a^4-12 x^2 a^3-33 x a^2+12 a}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (\int \frac {-2 x^2 a^4+12 x a^3+33 a^2}{x \sqrt {1-a^2 x^2}}dx+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (-\frac {\int -\frac {3 a^4 (4 a x+11)}{x \sqrt {1-a^2 x^2}}dx}{a^2}+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \int \frac {4 a x+11}{x \sqrt {1-a^2 x^2}}dx+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \left (4 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+11 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx\right )+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \left (11 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+4 \arcsin (a x)\right )+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \left (\frac {11}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+4 \arcsin (a x)\right )+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \left (4 \arcsin (a x)-\frac {11 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} \left (3 a^2 \left (4 \arcsin (a x)-11 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+2 a^2 \sqrt {1-a^2 x^2}+\frac {12 a \sqrt {1-a^2 x^2}}{x}\right )+\frac {32 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )}{a^3}\)

Input:

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

Output:

-((c^3*((32*a^2*(1 - a*x))/Sqrt[1 - a^2*x^2] - Sqrt[1 - a^2*x^2]/(2*x^2) + 
 (2*a^2*Sqrt[1 - a^2*x^2] + (12*a*Sqrt[1 - a^2*x^2])/x + 3*a^2*(4*ArcSin[a 
*x] - 11*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/a^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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 223

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 528

Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy 
mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b 
*x^2])), x] + Simp[c^2/a   Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) 
^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr 
eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

rule 2338

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]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[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] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

rule 2340

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] + Simp[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 6678

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[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 6681

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

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {\left (12 a^{3} x^{3}-a^{2} x^{2}-12 a x +1\right ) c^{3}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3}}-\frac {\left (-\frac {33 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {6 a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {32 a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+a^{2} \sqrt {-a^{2} x^{2}+1}\right ) c^{3}}{a^{3}}\)	\(153\)
default	\(\frac {c^{3} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {33 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}+6 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )-\frac {8 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-4 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )+18 a^{2} \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}\)	\(456\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.26 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {66 \, a^{3} c^{3} x^{3} + 66 \, a^{2} c^{3} x^{2} - 24 \, {\left (a^{3} c^{3} x^{3} + a^{2} c^{3} x^{2}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 33 \, {\left (a^{3} c^{3} x^{3} + a^{2} c^{3} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a^{4} x^{3} + a^{3} x^{2}\right )}} \] Input:

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

Output:

-1/2*(66*a^3*c^3*x^3 + 66*a^2*c^3*x^2 - 24*(a^3*c^3*x^3 + a^2*c^3*x^2)*arc 
tan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 33*(a^3*c^3*x^3 + a^2*c^3*x^2)*log(( 
sqrt(-a^2*x^2 + 1) - 1)/x) + (2*a^3*c^3*x^3 + 78*a^2*c^3*x^2 + 11*a*c^3*x 
- c^3)*sqrt(-a^2*x^2 + 1))/(a^4*x^3 + a^3*x^2)
 

Sympy [F]

\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right )\, dx + \int \frac {3 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right )\, dx + \int \left (- \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right )\, dx + \int \frac {3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\, dx + \int \left (- \frac {a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right )\, dx\right )}{a^{3}} \] Input:

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

Output:

c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + 
 x**3), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 
+ 3*a*x**4 + x**3), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3* 
x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(-2*a**3*x**3*sqrt(-a* 
*2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(3* 
a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3) 
, x) + Integral(-a**5*x**5*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 
 3*a*x**4 + x**3), x))/a**3
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (125) = 250\).

Time = 0.16 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.89 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {6 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {{\left (c^{3} - \frac {23 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac {536 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} + \frac {33 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3} {\left | a \right |}}{a^{2} x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a^{4} x^{2}}}{8 \, a^{2}} \] Input:

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

Output:

-6*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*(c^3 - 23*(sqrt(-a^2*x^2 + 1)*abs(a 
) + a)*c^3/(a^2*x) - 536*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4*x^2))* 
a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a 
)/(a^2*x) + 1)*abs(a)) + 33/2*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) 
 - 2*a)/(a^2*abs(x)))/abs(a) - sqrt(-a^2*x^2 + 1)*c^3/a - 1/8*(24*(sqrt(-a 
^2*x^2 + 1)*abs(a) + a)*c^3*abs(a)/(a^2*x) - (sqrt(-a^2*x^2 + 1)*abs(a) + 
a)^2*c^3*abs(a)/(a^4*x^2))/a^2
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.15 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {32\,c^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {6\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {6\,c^3\,\sqrt {1-a^2\,x^2}}{a^2\,x}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,33{}\mathrm {i}}{2\,a} \] Input:

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

Output:

(32*c^3*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/ 
2)) - (c^3*atan((1 - a^2*x^2)^(1/2)*1i)*33i)/(2*a) - (c^3*(1 - a^2*x^2)^(1 
/2))/a - (6*c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (6*c^3*(1 - a^2*x^2) 
^(1/2))/(a^2*x) + (c^3*(1 - a^2*x^2)^(1/2))/(2*a^3*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.66 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (-48 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+48 \mathit {asin} \left (a x \right ) a^{3} x^{3}+48 \mathit {asin} \left (a x \right ) a^{2} x^{2}-132 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}+8 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+561 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+44 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+132 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}+132 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}+8 a^{4} x^{4}+55 a^{3} x^{3}-517 a^{2} x^{2}-48 a x +4\right )}{8 a^{3} x^{2} \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:

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

Output:

(c**3*( - 48*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 48*asin(a*x)*a** 
3*x**3 + 48*asin(a*x)*a**2*x**2 - 132*sqrt( - a**2*x**2 + 1)*log(tan(asin( 
a*x)/2))*a**2*x**2 + 8*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 561*sqrt( - a**2 
*x**2 + 1)*a**2*x**2 + 44*sqrt( - a**2*x**2 + 1)*a*x - 4*sqrt( - a**2*x**2 
 + 1) + 132*log(tan(asin(a*x)/2))*a**3*x**3 + 132*log(tan(asin(a*x)/2))*a* 
*2*x**2 + 8*a**4*x**4 + 55*a**3*x**3 - 517*a**2*x**2 - 48*a*x + 4))/(8*a** 
3*x**2*(sqrt( - a**2*x**2 + 1) - a*x - 1))