\(\int e^{-3 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [707]

Optimal result

Integrand size = 20, antiderivative size = 74 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \arcsin (a x)}{a}-\frac {3 c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{a} \] Output:

-c*(-a^2*x^2+1)^(1/2)/a+c*(-a^2*x^2+1)^(1/2)/a^2/x-3*c*arcsin(a*x)/a-3*c*a 
rctanh((-a^2*x^2+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c \left ((-1+a x) \sqrt {1-a^2 x^2}+3 a x \arcsin (a x)+3 a x \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{a^2 x} \] Input:

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

Output:

-((c*((-1 + a*x)*Sqrt[1 - a^2*x^2] + 3*a*x*ArcSin[a*x] + 3*a*x*ArcTanh[Sqr 
t[1 - a^2*x^2]]))/(a^2*x))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6707, 6699, 540, 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.

\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx\)

\(\Big \downarrow \) 6707

\(\displaystyle -\frac {c \int \frac {e^{-3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )}{x^2}dx}{a^2}\)

\(\Big \downarrow \) 6699

\(\displaystyle -\frac {c \int \frac {(1-a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx}{a^2}\)

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

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 6699

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

rule 6707

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.34

method	result	size
risch	\(-\frac {\left (a^{2} x^{2}-1\right ) c}{x \sqrt {-a^{2} x^{2}+1}\, a^{2}}-\frac {\left (a \sqrt {-a^{2} x^{2}+1}+3 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c}{a^{2}}\)	\(99\)
default	\(\frac {c \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )+3 a \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}-9 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}\)	\(278\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {6 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c x - \sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}}{a^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

c*(Integral(-sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Inte 
gral(a*x*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Integral 
(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Integr 
al(-a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x))/a**2
 

Maxima [F]

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

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.76 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {a^{2} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {3 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {3 \, c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, a^{2} x {\left | a \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c\,\sqrt {1-a^2\,x^2}}{a}+\frac {c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{a} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (-3 \mathit {asin} \left (a x \right ) a x -\sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x +a x \right )}{a^{2} x} \] Input:

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

Output:

(c*( - 3*asin(a*x)*a*x - sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 
1) + 3*log(tan(asin(a*x)/2))*a*x + a*x))/(a**2*x)