\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [528]

Optimal result

Integrand size = 22, antiderivative size = 219 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=-\frac {6 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3 \sqrt {1+a x}}{5 (1-a x)^{7/2}}-\frac {31 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4 \sqrt {1+a x}}{15 (1-a x)^{7/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}{3 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}{5 \sqrt {1-a x}}+\frac {5 a^{5/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{7/2}} \] Output:

-6/5*a^2*(c-c/a/x)^(7/2)*x^3*(a*x+1)^(1/2)/(-a*x+1)^(7/2)-31/15*a^3*(c-c/a 
/x)^(7/2)*x^4*(a*x+1)^(1/2)/(-a*x+1)^(7/2)+2/3*a*(c-c/a/x)^(7/2)*x^2*(a*x+ 
1)^(1/2)/(-a*x+1)^(3/2)-2/5*(c-c/a/x)^(7/2)*x*(a*x+1)^(1/2)/(-a*x+1)^(1/2) 
+5*a^(5/2)*(c-c/a/x)^(7/2)*x^(7/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x+1)^(7/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.43 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+a x} \left (6-28 a x+56 a^2 x^2+15 a^3 x^3\right )-75 a^{5/2} x^{5/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{15 a^3 x^2 \sqrt {1-a x}} \] Input:

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

Output:

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x]*(6 - 28*a*x + 56*a^2*x^2 + 15*a^3*x^ 
3) - 75*a^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]]))/(15*a^3*x^2*Sqrt[1 - a* 
x])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.63, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6684, 6679, 108, 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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (\frac {2}{3} \int -\frac {a (1-a x) (31 a x+9)}{2 x^{3/2} \sqrt {a x+1}}dx-\frac {10 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \int \frac {(1-a x) (31 a x+9)}{x^{3/2} \sqrt {a x+1}}dx-\frac {10 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (\frac {75}{2} a \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx-\frac {\sqrt {a x+1} (31 a x+18)}{\sqrt {x}}\right )-\frac {10 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (75 a \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}-\frac {\sqrt {a x+1} (31 a x+18)}{\sqrt {x}}\right )-\frac {10 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {x^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (75 \sqrt {a} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-\frac {\sqrt {a x+1} (31 a x+18)}{\sqrt {x}}\right )-\frac {10 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}{(1-a x)^{7/2}}\)

Input:

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

Output:

((c - c/(a*x))^(7/2)*x^(7/2)*((-2*(1 - a*x)^3*Sqrt[1 + a*x])/(5*x^(5/2)) - 
 (a*((-10*(1 - a*x)^2*Sqrt[1 + a*x])/(3*x^(3/2)) - (a*(-((Sqrt[1 + a*x]*(1 
8 + 31*a*x))/Sqrt[x]) + 75*Sqrt[a]*ArcSinh[Sqrt[a]*Sqrt[x]]))/3))/5))/(1 - 
 a*x)^(7/2)
 

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 = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \sqrt {-a^{2} x^{2}+1}\, \left (30 a^{\frac {7}{2}} x^{3} \sqrt {-x \left (a x +1\right )}+75 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{3} x^{3}+112 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}-56 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+12 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{30 x^{2} a^{\frac {7}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\)	\(154\)
risch	\(\frac {\left (15 a^{4} x^{4}+71 a^{3} x^{3}+28 a^{2} x^{2}-22 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{15 x^{2} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{3}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(200\)

Input:

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

Output:

-1/30*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(-a^2*x^2+1)^(1/2)*(30*a^(7/2) 
*x^3*(-x*(a*x+1))^(1/2)+75*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2) 
)*a^3*x^3+112*a^(5/2)*x^2*(-x*(a*x+1))^(1/2)-56*a^(3/2)*x*(-x*(a*x+1))^(1/ 
2)+12*a^(1/2)*(-x*(a*x+1))^(1/2))/(a*x-1)/(-x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.66 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (15 \, a^{3} c^{3} x^{3} + 56 \, a^{2} c^{3} x^{2} - 28 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/60*(75*(a^3*c^3*x^3 - a^2*c^3*x^2)*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*(15*a^3*c^3*x^3 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c^3 
)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1/30*(7 
5*(a^3*c^3*x^3 - a^2*c^3*x^2)*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*(15*a^3*c^3*x^3 
 + 56*a^2*c^3*x^2 - 28*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c 
)/(a*x)))/(a^4*x^3 - a^3*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.49 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (75 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{3} x^{3}-15 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} i \,x^{3}-56 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}+28 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x -6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{15 a^{4} x^{3}} \] Input:

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

Output:

(sqrt(c)*c**3*(75*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a**3*x 
**3 - 15*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*i*x**3 - 56*sqrt(x)*sqrt(a)*sq 
rt(a*x + 1)*a**2*i*x**2 + 28*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x - 6*sqrt( 
x)*sqrt(a)*sqrt(a*x + 1)*i))/(15*a**4*x**3)