\(\int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx\) [269]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {32 c^3 \sqrt {c-a c x}}{a}+\frac {16 c^2 (c-a c x)^{3/2}}{3 a}+\frac {8 c (c-a c x)^{5/2}}{5 a}+\frac {4 (c-a c x)^{7/2}}{7 a}+\frac {2 (c-a c x)^{9/2}}{9 a c}-\frac {32 \sqrt {2} c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Output:

32*c^3*(-a*c*x+c)^(1/2)/a+16/3*c^2*(-a*c*x+c)^(3/2)/a+8/5*c*(-a*c*x+c)^(5/ 
2)/a+4/7*(-a*c*x+c)^(7/2)/a+2/9*(-a*c*x+c)^(9/2)/a/c-32*2^(1/2)*c^(7/2)*ar 
ctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 c^3 \left (\sqrt {c-a c x} \left (6257-1754 a x+732 a^2 x^2-230 a^3 x^3+35 a^4 x^4\right )-5040 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{315 a} \] Input:

Integrate[(c - a*c*x)^(7/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

(2*c^3*(Sqrt[c - a*c*x]*(6257 - 1754*a*x + 732*a^2*x^2 - 230*a^3*x^3 + 35* 
a^4*x^4) - 5040*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])] 
))/(315*a)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6680, 35, 60, 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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx\)

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 35

\(\displaystyle \frac {\int \frac {(c-a c x)^{9/2}}{a x+1}dx}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \int \frac {(c-a c x)^{7/2}}{a x+1}dx+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \int \frac {(c-a c x)^{5/2}}{a x+1}dx+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \left (2 c \int \frac {(c-a c x)^{3/2}}{a x+1}dx+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 c \left (2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}\right )+\frac {2 (c-a c x)^{9/2}}{9 a}}{c}\)

Input:

Int[(c - a*c*x)^(7/2)/E^(2*ArcTanh[a*x]),x]
 

Output:

((2*(c - a*c*x)^(9/2))/(9*a) + 2*c*((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 = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57

method	result	size
pseudoelliptic	\(-\frac {32 c^{3} \left (\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\frac {\left (35 a^{4} x^{4}-230 a^{3} x^{3}+732 a^{2} x^{2}-1754 a x +6257\right ) \sqrt {-c \left (a x -1\right )}}{5040}\right )}{a}\)	\(78\)
risch	\(-\frac {2 \left (35 a^{4} x^{4}-230 a^{3} x^{3}+732 a^{2} x^{2}-1754 a x +6257\right ) \left (a x -1\right ) c^{4}}{315 a \sqrt {-c \left (a x -1\right )}}-\frac {32 \sqrt {2}\, c^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a}\)	\(84\)
derivativedivides	\(\frac {\frac {2 \left (-a c x +c \right )^{\frac {9}{2}}}{9}+\frac {4 c \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {8 c^{2} \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {16 c^{3} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+32 \sqrt {-a c x +c}\, c^{4}-32 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\)	\(101\)
default	\(-\frac {2 \left (-\frac {\left (-a c x +c \right )^{\frac {9}{2}}}{9}-\frac {2 c \left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {4 c^{2} \left (-a c x +c \right )^{\frac {5}{2}}}{5}-\frac {8 c^{3} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-16 \sqrt {-a c x +c}\, c^{4}+16 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\)	\(101\)

Input:

int((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

-32*c^3*(c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))-1 
/5040*(35*a^4*x^4-230*a^3*x^3+732*a^2*x^2-1754*a*x+6257)*(-c*(a*x-1))^(1/2 
))/a
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\left [\frac {2 \, {\left (2520 \, \sqrt {2} c^{\frac {7}{2}} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right )} \sqrt {-a c x + c}\right )}}{315 \, a}, -\frac {2 \, {\left (5040 \, \sqrt {2} \sqrt {-c} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) - {\left (35 \, a^{4} c^{3} x^{4} - 230 \, a^{3} c^{3} x^{3} + 732 \, a^{2} c^{3} x^{2} - 1754 \, a c^{3} x + 6257 \, c^{3}\right )} \sqrt {-a c x + c}\right )}}{315 \, a}\right ] \] Input:

integrate((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
 

Output:

[2/315*(2520*sqrt(2)*c^(7/2)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt( 
c) - 3*c)/(a*x + 1)) + (35*a^4*c^3*x^4 - 230*a^3*c^3*x^3 + 732*a^2*c^3*x^2 
 - 1754*a*c^3*x + 6257*c^3)*sqrt(-a*c*x + c))/a, -2/315*(5040*sqrt(2)*sqrt 
(-c)*c^3*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) - (35*a^4*c 
^3*x^4 - 230*a^3*c^3*x^3 + 732*a^2*c^3*x^2 - 1754*a*c^3*x + 6257*c^3)*sqrt 
(-a*c*x + c))/a]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 6.86 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\begin {cases} - \frac {2 \left (- \frac {16 \sqrt {2} c^{5} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 16 c^{4} \sqrt {- a c x + c} - \frac {8 c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {4 c^{2} \left (- a c x + c\right )^{\frac {5}{2}}}{5} - \frac {2 c \left (- a c x + c\right )^{\frac {7}{2}}}{7} - \frac {\left (- a c x + c\right )^{\frac {9}{2}}}{9}\right )}{a c} & \text {for}\: a c \neq 0 \\c^{\frac {7}{2}} \left (- x + 2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )\right ) & \text {otherwise} \end {cases} \] Input:

integrate((-a*c*x+c)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

Piecewise((-2*(-16*sqrt(2)*c**5*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)) 
)/sqrt(-c) - 16*c**4*sqrt(-a*c*x + c) - 8*c**3*(-a*c*x + c)**(3/2)/3 - 4*c 
**2*(-a*c*x + c)**(5/2)/5 - 2*c*(-a*c*x + c)**(7/2)/7 - (-a*c*x + c)**(9/2 
)/9)/(a*c), Ne(a*c, 0)), (c**(7/2)*(-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 = 123, normalized size of antiderivative = 0.90 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \, {\left (2520 \, \sqrt {2} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 35 \, {\left (-a c x + c\right )}^{\frac {9}{2}} + 90 \, {\left (-a c x + c\right )}^{\frac {7}{2}} c + 252 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c^{2} + 840 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 5040 \, \sqrt {-a c x + c} c^{4}\right )}}{315 \, a c} \] Input:

integrate((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
 

Output:

2/315*(2520*sqrt(2)*c^(9/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqr 
t(2)*sqrt(c) + sqrt(-a*c*x + c))) + 35*(-a*c*x + c)^(9/2) + 90*(-a*c*x + c 
)^(7/2)*c + 252*(-a*c*x + c)^(5/2)*c^2 + 840*(-a*c*x + c)^(3/2)*c^3 + 5040 
*sqrt(-a*c*x + c)*c^4)/(a*c)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {32 \, \sqrt {2} c^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left (35 \, {\left (a c x - c\right )}^{4} \sqrt {-a c x + c} a^{8} c^{8} - 90 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{8} c^{9} + 252 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{8} c^{10} + 840 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{8} c^{11} + 5040 \, \sqrt {-a c x + c} a^{8} c^{12}\right )}}{315 \, a^{9} c^{9}} \] Input:

integrate((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

32*sqrt(2)*c^4*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) 
+ 2/315*(35*(a*c*x - c)^4*sqrt(-a*c*x + c)*a^8*c^8 - 90*(a*c*x - c)^3*sqrt 
(-a*c*x + c)*a^8*c^9 + 252*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^8*c^10 + 840*( 
-a*c*x + c)^(3/2)*a^8*c^11 + 5040*sqrt(-a*c*x + c)*a^8*c^12)/(a^9*c^9)
 

Mupad [B] (verification not implemented)

Time = 22.65 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {4\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a}+\frac {8\,c\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}+\frac {32\,c^3\,\sqrt {c-a\,c\,x}}{a}+\frac {16\,c^2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}+\frac {2\,{\left (c-a\,c\,x\right )}^{9/2}}{9\,a\,c}+\frac {\sqrt {2}\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,32{}\mathrm {i}}{a} \] Input:

int(-((a^2*x^2 - 1)*(c - a*c*x)^(7/2))/(a*x + 1)^2,x)
 

Output:

(4*(c - a*c*x)^(7/2))/(7*a) + (8*c*(c - a*c*x)^(5/2))/(5*a) + (32*c^3*(c - 
 a*c*x)^(1/2))/a + (16*c^2*(c - a*c*x)^(3/2))/(3*a) + (2*(c - a*c*x)^(9/2) 
)/(9*a*c) + (2^(1/2)*c^(7/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2 
)))*32i)/a
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \sqrt {c}\, c^{3} \left (35 \sqrt {-a x +1}\, a^{4} x^{4}-230 \sqrt {-a x +1}\, a^{3} x^{3}+732 \sqrt {-a x +1}\, a^{2} x^{2}-1754 \sqrt {-a x +1}\, a x +6257 \sqrt {-a x +1}+2520 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-2520 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )\right )}{315 a} \] Input:

int((-a*c*x+c)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(2*sqrt(c)*c**3*(35*sqrt( - a*x + 1)*a**4*x**4 - 230*sqrt( - a*x + 1)*a**3 
*x**3 + 732*sqrt( - a*x + 1)*a**2*x**2 - 1754*sqrt( - a*x + 1)*a*x + 6257* 
sqrt( - a*x + 1) + 2520*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2)) - 2520*sqr 
t(2)*log(sqrt( - a*x + 1) + sqrt(2))))/(315*a)