\(\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx\) [273]

Optimal result

Integrand size = 20, antiderivative size = 195 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}} \]

[Out]

-8/5*(-a*c*x+c)^(3/2)/a/(1-1/a/x)^(3/2)/(1+1/a/x)^(1/2)+184/5*(-a*c*x+c)^(3/2)/a^3/(1-1/a/x)^(3/2)/x^2/(1+1/a/
x)^(1/2)+16*(-a*c*x+c)^(3/2)/a^2/(1-1/a/x)^(3/2)/x/(1+1/a/x)^(1/2)+2/5*(a-1/x)^3*x*(-a*c*x+c)^(3/2)/a^3/(1-1/a
/x)^(3/2)/(1+1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}} \]

[In]

Int[(c - a*c*x)^(3/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(-8*(c - a*c*x)^(3/2))/(5*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + (184*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*
x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2) + (16*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(
a - x^(-1))^3*x*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

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

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 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 {(c-a c x)^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (12 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (8 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (92 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2}} \\ & = -\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.29 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 c \sqrt {c-a c x} \left (91+43 a x-7 a^2 x^2+a^3 x^3\right )}{5 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[In]

Integrate[(c - a*c*x)^(3/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(-2*c*Sqrt[c - a*c*x]*(91 + 43*a*x - 7*a^2*x^2 + a^3*x^3))/(5*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.32

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (a^{3} x^{3}-7 a^{2} x^{2}+43 a x +91\right ) \left (-a c x +c \right )^{\frac {3}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{5 a \left (a x -1\right )^{3}}\)	\(63\)
default	\(-\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c \left (a^{3} x^{3}-7 a^{2} x^{2}+43 a x +91\right )}{5 \left (a x -1\right )^{2} a}\)	\(65\)
risch	\(\frac {2 \left (a^{2} x^{2}-8 a x +51\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{5 a \sqrt {-c \left (a x -1\right )}}+\frac {16 c^{2} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\)	\(86\)

[In]

int((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/5*(a*x+1)*(a^3*x^3-7*a^2*x^2+43*a*x+91)*(-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.32 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{2} x - a\right )}} \]

[In]

integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

-2/5*(a^3*c*x^3 - 7*a^2*c*x^2 + 43*a*c*x + 91*c)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((-a*c*x+c)**(3/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.48 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (a^{4} \sqrt {-c} c x^{4} - 6 \, a^{3} \sqrt {-c} c x^{3} + 36 \, a^{2} \sqrt {-c} c x^{2} + 134 \, a \sqrt {-c} c x + 91 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}^{2}}{5 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]

[In]

integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

-2/5*(a^4*sqrt(-c)*c*x^4 - 6*a^3*sqrt(-c)*c*x^3 + 36*a^2*sqrt(-c)*c*x^2 + 134*a*sqrt(-c)*c*x + 91*sqrt(-c)*c)*
(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))

Giac [F(-2)]

Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.42 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.42 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^2\,x^2-6\,a\,x+37\right )}{5\,a}-\frac {256\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{5\,a\,\left (a\,x-1\right )} \]

[In]

int((c - a*c*x)^(3/2)*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

- (2*c*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(a^2*x^2 - 6*a*x + 37))/(5*a) - (256*c*(c - a*c*x)^(1/2)*
((a*x - 1)/(a*x + 1))^(1/2))/(5*a*(a*x - 1))