\(\int e^{-2 \text {arctanh}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [562]

Optimal result

Integrand size = 24, antiderivative size = 164 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {21 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {11 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Output:

21*c^3*(c-c/a/x)^(1/2)/a+5/3*c^2*(c-c/a/x)^(3/2)/a-3/5*c*(c-c/a/x)^(5/2)/a 
-(c-c/a/x)^(7/2)*x+11*c^(7/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a-32*2^(1/2 
)*c^(7/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.76 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (6-52 a x+376 a^2 x^2-15 a^3 x^3\right )}{15 a^3 x^2}+\frac {11 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {32 \sqrt {2} c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Input:

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

Output:

(c^3*Sqrt[c - c/(a*x)]*(6 - 52*a*x + 376*a^2*x^2 - 15*a^3*x^3))/(15*a^3*x^ 
2) + (11*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (32*Sqrt[2]*c^(7/ 
2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6683, 1035, 281, 899, 109, 27, 171, 27, 171, 27, 171, 27, 174, 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.

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {a \left (-\frac {\int \frac {c^2 \left (11 a+\frac {3}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} x}{2 a \left (a+\frac {1}{x}\right )}d\frac {1}{x}}{a}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (-\frac {c^2 \int \frac {\left (11 a+\frac {3}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} x}{a+\frac {1}{x}}d\frac {1}{x}}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {a \left (-\frac {c^2 \left (\frac {2}{5} \int \frac {5 c \left (11 a-\frac {5}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2} x}{2 \left (a+\frac {1}{x}\right )}d\frac {1}{x}+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \int \frac {\left (11 a-\frac {5}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2} x}{a+\frac {1}{x}}d\frac {1}{x}+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (\frac {2}{3} \int \frac {3 c \left (11 a-\frac {21}{x}\right ) \sqrt {c-\frac {c}{a x}} x}{2 \left (a+\frac {1}{x}\right )}d\frac {1}{x}-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \int \frac {\left (11 a-\frac {21}{x}\right ) \sqrt {c-\frac {c}{a x}} x}{a+\frac {1}{x}}d\frac {1}{x}-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (2 \int \frac {c \left (11 a-\frac {53}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-42 \sqrt {c-\frac {c}{a x}}\right )-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \int \frac {\left (11 a-\frac {53}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-42 \sqrt {c-\frac {c}{a x}}\right )-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (11 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-64 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\right )-42 \sqrt {c-\frac {c}{a x}}\right )-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (\frac {128 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {22 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}\right )-42 \sqrt {c-\frac {c}{a x}}\right )-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {a \left (-\frac {c^2 \left (c \left (c \left (c \left (\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {22 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-42 \sqrt {c-\frac {c}{a x}}\right )-\frac {10}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {6}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )}{2 a^2}-\frac {c x \left (c-\frac {c}{a x}\right )^{7/2}}{a}\right )}{c}\)

Input:

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

Output:

(a*(-((c*(c - c/(a*x))^(7/2)*x)/a) - (c^2*((6*(c - c/(a*x))^(5/2))/5 + c*( 
(-10*(c - c/(a*x))^(3/2))/3 + c*(-42*Sqrt[c - c/(a*x)] + c*((-22*ArcTanh[S 
qrt[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] + (64*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)] 
/(Sqrt[2]*Sqrt[c])])/Sqrt[c])))))/(2*a^2)))/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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 281

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

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

rule 1035

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) 
 + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c 
 + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ 
mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

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, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !G 
tQ[c, 0]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {\left (15 a^{4} x^{4}-391 a^{3} x^{3}+428 a^{2} x^{2}-58 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} a^{3} \left (a x -1\right )}-\frac {\left (-\frac {11 a^{3} \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {16 a^{2} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) c^{3} \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a^{3} \left (a x -1\right )}\)	\(224\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (-1110 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{4}+480 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, x^{4}+660 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+555 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-480 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{4}-720 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-92 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{30 x^{3} a^{\frac {7}{2}} \sqrt {x \left (a x -1\right )}\, \sqrt {\frac {1}{a}}}\)	\(281\)

Input:

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

Output:

-1/15*(15*a^4*x^4-391*a^3*x^3+428*a^2*x^2-58*a*x+6)/x^2*c^3/a^3*(c*(a*x-1) 
/a/x)^(1/2)/(a*x-1)-(-11/2*a^3*ln((-1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c* 
x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)-16*a^2*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a 
*c+2*2^(1/2)*c^(1/2)*((x+1/a)^2*a^2*c-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))* 
c^3/a^3*(c*(a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.05 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {480 \, \sqrt {2} a^{2} c^{\frac {7}{2}} x^{2} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 165 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{30 \, a^{3} x^{2}}, \frac {480 \, \sqrt {2} a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 165 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - {\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{15 \, a^{3} x^{2}}\right ] \] Input:

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

Output:

[1/30*(480*sqrt(2)*a^2*c^(7/2)*x^2*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x 
- c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) + 165*a^2*c^(7/2)*x^2*log(-2*a*c*x - 
 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(15*a^3*c^3*x^3 - 376*a^2* 
c^3*x^2 + 52*a*c^3*x - 6*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2), 1/15*(48 
0*sqrt(2)*a^2*sqrt(-c)*c^3*x^2*arctan(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c 
)/(a*x))/(a*c*x - c)) - 165*a^2*sqrt(-c)*c^3*x^2*arctan(a*sqrt(-c)*x*sqrt( 
(a*c*x - c)/(a*x))/(a*c*x - c)) - (15*a^3*c^3*x^3 - 376*a^2*c^3*x^2 + 52*a 
*c^3*x - 6*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2)]
 

Sympy [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=- \int \left (- \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {6 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left (- \frac {4 c^{3} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right )\, dx - \int \frac {c^{3} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \frac {a c^{3} x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \] Input:

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

Output:

-Integral(-4*c**3*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(6*c**3*sqrt(c 
 - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(-4*c**3*sqrt(c - c/(a*x))/(a* 
*3*x**3 + a**2*x**2), x) - Integral(c**3*sqrt(c - c/(a*x))/(a**4*x**4 + a* 
*3*x**3), x) - Integral(a*c**3*x*sqrt(c - c/(a*x))/(a*x + 1), x)
 

Maxima [F]

\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=-\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.20 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (-60 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}+1504 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}-208 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x +24 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-960 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right ) a^{3} x^{3}-960 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right ) a^{3} x^{3}+960 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right ) a^{3} x^{3}+660 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}-709 a^{3} x^{3}\right )}{60 a^{4} x^{3}} \] Input:

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

Output:

(sqrt(c)*c**3*( - 60*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**3*x**3 + 1504*sqrt(x 
)*sqrt(a)*sqrt(a*x - 1)*a**2*x**2 - 208*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x 
+ 24*sqrt(x)*sqrt(a)*sqrt(a*x - 1) - 960*sqrt(2)*log(sqrt(a*x - 1) + sqrt( 
x)*sqrt(a) - sqrt(2)*i + i)*a**3*x**3 - 960*sqrt(2)*log(sqrt(a*x - 1) + sq 
rt(x)*sqrt(a) + sqrt(2)*i - i)*a**3*x**3 + 960*sqrt(2)*log(2*sqrt(x)*sqrt( 
a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2)*a**3*x**3 + 660*log(sqrt(a*x - 1 
) + sqrt(x)*sqrt(a))*a**3*x**3 - 709*a**3*x**3))/(60*a**4*x**3)