\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a x})^{9/2} \, dx\) [542]

Optimal result

Integrand size = 22, antiderivative size = 263 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=-\frac {18 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x}}{35 (1-a x)^{9/2}}+\frac {227 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 \sqrt {1+a x}}{105 (1-a x)^{9/2}}-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {7 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}} \] Output:

-18/35*a^3*(c-c/a/x)^(9/2)*x^4*(a*x+1)^(1/2)/(-a*x+1)^(9/2)+227/105*a^4*(c 
-c/a/x)^(9/2)*x^5*(a*x+1)^(1/2)/(-a*x+1)^(9/2)-10/21*a^2*(c-c/a/x)^(9/2)*x 
^3*(a*x+1)^(1/2)/(-a*x+1)^(5/2)+2/5*a*(c-c/a/x)^(9/2)*x^2*(a*x+1)^(1/2)/(- 
a*x+1)^(3/2)-2/7*(c-c/a/x)^(9/2)*x*(a*x+1)^(1/2)/(-a*x+1)^(1/2)-7*a^(7/2)* 
(c-c/a/x)^(9/2)*x^(9/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x+1)^(9/2)
 

Mathematica [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.38 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+a x} \left (-30+162 a x-601 a^2 x^2-688 a^3 x^3+105 a^4 x^4\right )+245 a^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-a x\right )\right )}{105 a^4 x^3 \sqrt {1-a x}} \] Input:

Integrate[E^ArcTanh[a*x]*(c - c/(a*x))^(9/2),x]
 

Output:

(c^4*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x]*(-30 + 162*a*x - 601*a^2*x^2 - 688*a 
^3*x^3 + 105*a^4*x^4) + 245*a^2*x^2*Hypergeometric2F1[-3/2, -3/2, -1/2, -( 
a*x)]))/(105*a^4*x^3*Sqrt[1 - a*x])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.64, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6684, 6679, 108, 27, 167, 27, 167, 27, 160, 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.

\(\displaystyle \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx\)

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (\frac {2}{5} \int -\frac {a (1-a x)^2 (59 a x+25)}{2 x^{5/2} \sqrt {a x+1}}dx-\frac {14 (1-a x)^3 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \int \frac {(1-a x)^2 (59 a x+25)}{x^{5/2} \sqrt {a x+1}}dx-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {2}{3} \int \frac {a (27-227 a x) (1-a x)}{2 x^{3/2} \sqrt {a x+1}}dx-\frac {50 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {1}{3} a \int \frac {(27-227 a x) (1-a x)}{x^{3/2} \sqrt {a x+1}}dx-\frac {50 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {1}{3} a \left (-\frac {735}{2} a \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx-\frac {\sqrt {a x+1} (54-227 a x)}{\sqrt {x}}\right )-\frac {50 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {1}{3} a \left (-735 a \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}-\frac {\sqrt {a x+1} (54-227 a x)}{\sqrt {x}}\right )-\frac {50 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {x^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {1}{3} a \left (-735 \sqrt {a} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-\frac {\sqrt {a x+1} (54-227 a x)}{\sqrt {x}}\right )-\frac {50 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {14 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right ) \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}\)

Input:

Int[E^ArcTanh[a*x]*(c - c/(a*x))^(9/2),x]
 

Output:

((c - c/(a*x))^(9/2)*x^(9/2)*((-2*(1 - a*x)^4*Sqrt[1 + a*x])/(7*x^(7/2)) - 
 (a*((-14*(1 - a*x)^3*Sqrt[1 + a*x])/(5*x^(5/2)) - (a*((-50*(1 - a*x)^2*Sq 
rt[1 + a*x])/(3*x^(3/2)) + (a*(-(((54 - 227*a*x)*Sqrt[1 + a*x])/Sqrt[x]) - 
 735*Sqrt[a]*ArcSinh[Sqrt[a]*Sqrt[x]]))/3))/5))/7))/(1 - a*x)^(9/2)
 

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 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 108

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

rule 160

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

rule 167

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

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 6679

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

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 = 0.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \sqrt {-a^{2} x^{2}+1}\, \left (210 a^{\frac {9}{2}} \sqrt {-x \left (a x +1\right )}\, x^{4}+735 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{4} x^{4}+584 a^{\frac {7}{2}} x^{3} \sqrt {-x \left (a x +1\right )}-712 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}+324 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}-60 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{210 x^{3} a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\)	\(172\)
risch	\(\frac {\left (105 a^{5} x^{5}+397 a^{4} x^{4}-64 a^{3} x^{3}-194 a^{2} x^{2}+132 a x -30\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{105 x^{3} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{4}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(208\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-1/210*(c*(a*x-1)/a/x)^(1/2)/x^3*c^4/a^(9/2)*(-a^2*x^2+1)^(1/2)*(210*a^(9/ 
2)*(-x*(a*x+1))^(1/2)*x^4+735*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1 
/2))*a^4*x^4+584*a^(7/2)*x^3*(-x*(a*x+1))^(1/2)-712*a^(5/2)*x^2*(-x*(a*x+1 
))^(1/2)+324*a^(3/2)*x*(-x*(a*x+1))^(1/2)-60*a^(1/2)*(-x*(a*x+1))^(1/2))/( 
a*x-1)/(-x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.47 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{420 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x, algorithm="fricas" 
)
 

Output:

[1/420*(735*(a^4*c^4*x^4 - a^3*c^4*x^3)*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*(105*a^4*c^4*x^4 + 292*a^3*c^4*x^3 - 356*a^2*c^4*x^ 
2 + 162*a*c^4*x - 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5 
*x^4 - a^4*x^3), 1/210*(735*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*arctan(2*s 
qrt(-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*(105*a^4*c^4*x^4 + 292*a^3*c^4*x^3 - 356*a^2*c^4*x^2 + 162*a*c^ 
4*x - 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x 
^3)]
 

Sympy [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(9/2),x)
 

Output:

Integral((-c*(-1 + 1/(a*x)))**(9/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), 
x)
 

Maxima [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x, algorithm="maxima" 
)
 

Output:

integrate((a*x + 1)*(c - c/(a*x))^(9/2)/sqrt(-a^2*x^2 + 1), x)
 

Giac [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x, algorithm="giac")
 

Output:

integrate((a*x + 1)*(c - c/(a*x))^(9/2)/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(((c - c/(a*x))^(9/2)*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(((c - c/(a*x))^(9/2)*(a*x + 1))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.48 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} \left (735 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{4} x^{4}-105 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{4} i \,x^{4}-292 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} i \,x^{3}+356 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}-162 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +30 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{105 a^{5} x^{4}} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a/x)^(9/2),x)
 

Output:

(sqrt(c)*c**4*(735*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a**4* 
x**4 - 105*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**4*i*x**4 - 292*sqrt(x)*sqrt(a) 
*sqrt(a*x + 1)*a**3*i*x**3 + 356*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*i*x**2 
 - 162*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x + 30*sqrt(x)*sqrt(a)*sqrt(a*x + 
 1)*i))/(105*a**5*x**4)