\(\int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx\) [270]

Optimal result

Integrand size = 20, antiderivative size = 116 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\frac {16 c^2 \sqrt {c-a c x}}{a}+\frac {8 c (c-a c x)^{3/2}}{3 a}+\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {16 \sqrt {2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Output:

16*c^2*(-a*c*x+c)^(1/2)/a+8/3*c*(-a*c*x+c)^(3/2)/a+4/5*(-a*c*x+c)^(5/2)/a+ 
2/7*(-a*c*x+c)^(7/2)/a/c-16*2^(1/2)*c^(5/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2 
^(1/2)/c^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.69 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=-\frac {2 c^2 \left (\sqrt {c-a c x} \left (-1037+269 a x-87 a^2 x^2+15 a^3 x^3\right )+840 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{105 a} \] Input:

Integrate[(c - a*c*x)^(5/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

(-2*c^2*(Sqrt[c - a*c*x]*(-1037 + 269*a*x - 87*a^2*x^2 + 15*a^3*x^3) + 840 
*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(105*a)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6680, 35, 60, 60, 60, 60, 73, 219}

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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {(1-a x) (c-a c x)^{5/2}}{a x+1}dx\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {\int \frac {(c-a c x)^{7/2}}{a x+1}dx}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \int \frac {(c-a c x)^{5/2}}{a x+1}dx+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \int \frac {(c-a c x)^{3/2}}{a x+1}dx+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \left (2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\)

Input:

Int[(c - a*c*x)^(5/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

((2*(c - a*c*x)^(7/2))/(7*a) + 2*c*((2*(c - a*c*x)^(5/2))/(5*a) + 2*c*((2* 
(c - a*c*x)^(3/2))/(3*a) + 2*c*((2*Sqrt[c - a*c*x])/a - (2*Sqrt[2]*Sqrt[c] 
*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a))))/c
 

Defintions of rubi rules used

rule 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*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 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6680

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(-\frac {2 \left (56 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {\left (15 a^{3} x^{3}-87 a^{2} x^{2}+269 a x -1037\right ) \sqrt {-c \left (a x -1\right )}}{15}\right ) c^{2}}{7 a}\)	\(71\)
risch	\(\frac {2 \left (15 a^{3} x^{3}-87 a^{2} x^{2}+269 a x -1037\right ) \left (a x -1\right ) c^{3}}{105 a \sqrt {-c \left (a x -1\right )}}-\frac {16 \sqrt {2}\, c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a}\)	\(76\)
derivativedivides	\(\frac {\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {4 \left (-a c x +c \right )^{\frac {5}{2}} c}{5}+\frac {8 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+16 \sqrt {-a c x +c}\, c^{3}-16 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\)	\(87\)
default	\(-\frac {2 \left (-\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {2 \left (-a c x +c \right )^{\frac {5}{2}} c}{5}-\frac {4 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-8 \sqrt {-a c x +c}\, c^{3}+8 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\)	\(87\)

Input:

int((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

-2/7*(56*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))+1 
/15*(15*a^3*x^3-87*a^2*x^2+269*a*x-1037)*(-c*(a*x-1))^(1/2))*c^2/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.62 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\left [\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {5}{2}} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}, -\frac {2 \, {\left (840 \, \sqrt {2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}\right ] \] Input:

integrate((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
 

Output:

[2/105*(420*sqrt(2)*c^(5/2)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c 
) - 3*c)/(a*x + 1)) - (15*a^3*c^2*x^3 - 87*a^2*c^2*x^2 + 269*a*c^2*x - 103 
7*c^2)*sqrt(-a*c*x + c))/a, -2/105*(840*sqrt(2)*sqrt(-c)*c^2*arctan(sqrt(2 
)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + (15*a^3*c^2*x^3 - 87*a^2*c^2*x^ 
2 + 269*a*c^2*x - 1037*c^2)*sqrt(-a*c*x + c))/a]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 6.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.11 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\begin {cases} - \frac {2 \left (- \frac {8 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 8 c^{3} \sqrt {- a c x + c} - \frac {4 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} - \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a c} & \text {for}\: a c \neq 0 \\c^{\frac {5}{2}} \left (- x + 2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )\right ) & \text {otherwise} \end {cases} \] Input:

integrate((-a*c*x+c)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

Piecewise((-2*(-8*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c))) 
/sqrt(-c) - 8*c**3*sqrt(-a*c*x + c) - 4*c**2*(-a*c*x + c)**(3/2)/3 - 2*c*( 
-a*c*x + c)**(5/2)/5 - (-a*c*x + c)**(7/2)/7)/(a*c), Ne(a*c, 0)), (c**(5/2 
)*(-x + 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True))), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 42 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 840 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a c} \] Input:

integrate((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
 

Output:

2/105*(420*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt 
(2)*sqrt(c) + sqrt(-a*c*x + c))) + 15*(-a*c*x + c)^(7/2) + 42*(-a*c*x + c) 
^(5/2)*c + 140*(-a*c*x + c)^(3/2)*c^2 + 840*sqrt(-a*c*x + c)*c^3)/(a*c)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\frac {16 \, \sqrt {2} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left (15 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{6} c^{6} - 42 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{6} c^{7} - 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{6} c^{8} - 840 \, \sqrt {-a c x + c} a^{6} c^{9}\right )}}{105 \, a^{7} c^{7}} \] Input:

integrate((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

16*sqrt(2)*c^3*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) 
- 2/105*(15*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^6*c^6 - 42*(a*c*x - c)^2*sqrt 
(-a*c*x + c)*a^6*c^7 - 140*(-a*c*x + c)^(3/2)*a^6*c^8 - 840*sqrt(-a*c*x + 
c)*a^6*c^9)/(a^7*c^7)
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}+\frac {8\,c\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}+\frac {16\,c^2\,\sqrt {c-a\,c\,x}}{a}+\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c}+\frac {\sqrt {2}\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,16{}\mathrm {i}}{a} \] Input:

int(-((a^2*x^2 - 1)*(c - a*c*x)^(5/2))/(a*x + 1)^2,x)
 

Output:

(4*(c - a*c*x)^(5/2))/(5*a) + (8*c*(c - a*c*x)^(3/2))/(3*a) + (16*c^2*(c - 
 a*c*x)^(1/2))/a + (2*(c - a*c*x)^(7/2))/(7*a*c) + (2^(1/2)*c^(5/2)*atan(( 
2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*16i)/a
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 \sqrt {c}\, c^{2} \left (-15 \sqrt {-a x +1}\, a^{3} x^{3}+87 \sqrt {-a x +1}\, a^{2} x^{2}-269 \sqrt {-a x +1}\, a x +1037 \sqrt {-a x +1}+420 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-420 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )\right )}{105 a} \] Input:

int((-a*c*x+c)^(5/2)/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(2*sqrt(c)*c**2*( - 15*sqrt( - a*x + 1)*a**3*x**3 + 87*sqrt( - a*x + 1)*a* 
*2*x**2 - 269*sqrt( - a*x + 1)*a*x + 1037*sqrt( - a*x + 1) + 420*sqrt(2)*l 
og(sqrt( - a*x + 1) - sqrt(2)) - 420*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2 
))))/(105*a)