\(\int \frac {e^{-3 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [577]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {2 (1-a x)^{3/2}}{a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1+a x}}+\frac {3 (1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {3 (1-a x)^{3/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.34 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {2 x (1-a x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(3/2)),x]
 

Output:

(2*x*(1 - a*x)^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, -(a*x)])/(5*(c - c/( 
a*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6684, 6678, 516, 57, 60, 63, 222}

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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 516

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 63

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

\(\Big \downarrow \) 222

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

Input:

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(3/2)),x]
 

Output:

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

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 222

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

rule 516

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free 
Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || 
(GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[n]))
 

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 6684

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.09

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+3 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a x +6 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}+3 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \sqrt {a}\, c^{2} \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\)	\(143\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c}+\frac {\left (-\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{a^{4} c \left (x +\frac {1}{a}\right )}\right ) a \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c}\)	\(229\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.24 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}}, \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(3*(a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 
 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1) 
) + 4*(a^2*x^2 + 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c 
^2*x^2 - a*c^2), 1/2*(3*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)* 
a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^ 
2 + 3*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - a*c^ 
2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (3 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x +3 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )-\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x -3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a \,c^{2} \left (a x +1\right )} \] Input:

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

Output:

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