\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{3/2}}{x} \, dx\) [1112]

Optimal result

Integrand size = 27, antiderivative size = 101 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=c (1+a x) \sqrt {c-a^2 c x^2}-\frac {1}{3} \left (c-a^2 c x^2\right )^{3/2}+c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.14 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=\frac {1}{3} c \left (2+3 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}-c^{3/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )+c^{3/2} \log (x)-c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

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

Output:

(c*(2 + 3*a*x + a^2*x^2)*Sqrt[c - a^2*c*x^2])/3 - c^(3/2)*ArcTan[(a*x*Sqrt 
[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] + c^(3/2)*Log[x] - c^(3/2)*Log[ 
c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6701, 541, 27, 535, 27, 538, 224, 216, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

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 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 6701

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^(n/2)   Int[x^m*(c + d*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]) && IGtQ[n/2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(204\) vs. \(2(83)=166\).

Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.03

method	result	size
default	\(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}+2 a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\)	\(205\)

Input:

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

Output:

1/3*(-a^2*c*x^2+c)^(3/2)+c*((-a^2*c*x^2+c)^(1/2)-c^(1/2)*ln((2*c+2*c^(1/2) 
*(-a^2*c*x^2+c)^(1/2))/x))-2/3*(-(x-1/a)^2*a^2*c-2*(x-1/a)*a*c)^(3/2)+2*a* 
c*(-1/4*(-2*(x-1/a)*a^2*c-2*a*c)/a^2/c*(-(x-1/a)^2*a^2*c-2*(x-1/a)*a*c)^(1 
/2)+1/2*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-(x-1/a)^2*a^2*c-2*(x-1/a) 
*a*c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.19 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=\left [-c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, c^{\frac {3}{2}} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + \frac {1}{3} \, {\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}, \sqrt {-c} c \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) + \frac {1}{2} \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + \frac {1}{3} \, {\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}\right ] \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.82 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.71 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=a^{2} c \left (\begin {cases} \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{3} - \frac {\sqrt {- a^{2} c x^{2} + c}}{3 a^{2}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} i \sqrt {c} \sqrt {a^{2} x^{2} - 1} - \sqrt {c} \log {\left (a x \right )} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} + i \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {c} \sqrt {- a^{2} x^{2} + 1} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} - \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

a**2*c*Piecewise((x**2*sqrt(-a**2*c*x**2 + c)/3 - sqrt(-a**2*c*x**2 + c)/( 
3*a**2), Ne(a**2*c, 0)), (sqrt(c)*x**2/2, True)) + 2*a*c*Piecewise((c*Piec 
ewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a** 
2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/2 + x*sqrt(-a**2*c*x 
**2 + c)/2, Ne(a**2*c, 0)), (sqrt(c)*x, True)) + c*Piecewise((I*sqrt(c)*sq 
rt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c 
)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt 
(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True))
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=\frac {2 \, c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {a \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{2} c x + 3 \, a c\right )} x + 2 \, c\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x} \, dx=\frac {\sqrt {c}\, c \left (3 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+3 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )-2\right )}{3} \] Input:

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

Output:

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