\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) [234]

Optimal result

Integrand size = 18, antiderivative size = 250 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{32 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}+\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{32 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]

[Out]

-1/6*a^4*(1-1/a/x)^(7/2)*(1+1/a/x)^(3/2)*x^2/(a-1/x)^3/(-a*c*x+c)^(7/2)+1/16*a^4*(1-1/a/x)^(7/2)*(1+1/a/x)^(3/
2)*x^3/(a-1/x)^2/(-a*c*x+c)^(7/2)-1/64*a^(5/2)*(1-1/a/x)^(7/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(
1/2))/(1/x)^(7/2)/(-a*c*x+c)^(7/2)*2^(1/2)-1/32*a^3*(1-1/a/x)^(7/2)*x^3*(1+1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(7/
2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{32 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}+\frac {a^4 x^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {a^4 x^2 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {a^3 x^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}{32 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}} \]

[In]

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

[Out]

-1/6*(a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^2)/((a - x^(-1))^3*(c - a*c*x)^(7/2)) - (a^3*(1 - 1/(a*x))
^(7/2)*Sqrt[1 + 1/(a*x)]*x^3)/(32*(a - x^(-1))*(c - a*c*x)^(7/2)) + (a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/
2)*x^3)/(16*(a - x^(-1))^2*(c - a*c*x)^(7/2)) - (a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(S
qrt[a]*Sqrt[1 + 1/(a*x)])])/(32*Sqrt[2]*(x^(-1))^(7/2)*(c - a*c*x)^(7/2))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6311

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

Rule 6316

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}} \\ & = -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {x^{3/2} \sqrt {1+\frac {x}{a}}}{\left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {x} \sqrt {1+\frac {x}{a}}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{4 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}+\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{32 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{32 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}+\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{64 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{32 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}+\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {\left (a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{32 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ & = -\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{32 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}+\frac {a^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2} x^3}{16 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{32 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.56 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (\frac {2 \sqrt {a} \sqrt {1+\frac {1}{a x}} \left (25+10 a x-3 a^2 x^2\right )}{\sqrt {\frac {1}{x}}}+3 \sqrt {2} (-1+a x)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{192 \sqrt {a} c^3 \sqrt {\frac {1}{x}} (-1+a x)^3 \sqrt {c-a c x}} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*((2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(25 + 10*a*x - 3*a^2*x^2))/Sqrt[x^(-1)] + 3*Sqrt[2]*(-1 + a*x
)^3*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(192*Sqrt[a]*c^3*Sqrt[x^(-1)]*(-1 + a*x)^3*S
qrt[c - a*c*x])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}+6 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -20 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -50 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{192 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right )^{3} c^{\frac {9}{2}} \sqrt {-c \left (a x +1\right )}\, a}\)	\(219\)

[In]

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

[Out]

1/192*(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^3*c*x^3+9*2^(1/2)*arctan
(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^2*c*x^2+6*a^2*x^2*(-c*(a*x+1))^(1/2)*c^(1/2)-9*2^(1/2)*arctan(1/2*(
-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a*c*x-20*a*x*c^(1/2)*(-c*(a*x+1))^(1/2)+3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(
1/2)*2^(1/2)/c^(1/2))*c-50*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)^3/c^(9/2)/(-c*(a*x+1))^
(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 35 \, a x - 25\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 35 \, a x - 25\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

[In]

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

[Out]

[-1/384*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2
)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*(3*a^3*x^3 -
7*a^2*x^2 - 35*a*x - 25)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4
*x^2 - 4*a^2*c^4*x + a*c^4), -1/192*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(sq
rt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*(3*a^3*x^3 - 7*a^2*x^2 - 35*a*x - 25
)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c
^4)]

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.42 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left (3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} + 16 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c - 12 \, \sqrt {-a c x - c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c^{2}}}{192 \, a {\left | c \right |}} \]

[In]

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

[Out]

1/192*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(5/2) - 2*(3*(a*c*x + c)^2*sqrt(-a*c*x - c) +
16*(-a*c*x - c)^(3/2)*c - 12*sqrt(-a*c*x - c)*c^2)/((a*c*x - c)^3*c^2))/(a*abs(c))

Mupad [F(-1)]

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

[In]

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

[Out]

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