3.232 \(\int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\)

Optimal. Leaf size=128 \[ -\frac {a x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac {a x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))*(c - a*c*x)^(3/2))) - (Sqrt[a]*(1 - 1/(a*x))^(3/2)
*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[2]*(x^(-1))^(3/2)*(c - a*c*x)^(3/2))

Rule 93

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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*} \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \operatorname {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 )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 116, normalized size = 0.91 \[ \frac {x \sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {\frac {1}{a x}+1}+\sqrt {2} \sqrt {\frac {1}{x}} (a x-1) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )\right )}{2 \sqrt {a} c (a x-1) \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-
1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(2*Sqrt[a]*c*(-1 + a*x)*Sqrt[c - a*c*x])

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fricas [A]  time = 0.47, size = 281, normalized size = 2.20 \[ \left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(sqrt(2)*(a^2*x^2 - 2*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*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1
)/(a*x + 1)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), -1/2*(sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*arctan(sqrt(2)*
sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) + 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/
(a*x + 1)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]

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giac [A]  time = 0.18, size = 71, normalized size = 0.55 \[ -\frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {-a c x - c}}{a c x - c}}{2 \, a c \mathrm {sgn}\left (-a c x - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/sqrt(c) + 2*sqrt(-a*c*x - c)/(a*c*x - c))/(a*c*sgn(
-a*c*x - c))

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maple [A]  time = 0.06, size = 118, normalized size = 0.92 \[ -\frac {\sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c -\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, c^{\frac {5}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(-c*(a*x-1))^(1/2)*(2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2)
)*x*a*c-2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c+2*(-c*(a*x+1))^(1/2)*c^(1/2))/(-c*(a*x+1))^(1
/2)/c^(5/2)/a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a c x + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c-a\,c\,x\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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