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

Optimal. Leaf size=76 \[ -\frac {\sqrt {2} \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

[Out]

-(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)

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Rubi [A]  time = 0.18, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6176, 6181, 93, 206} \[ -\frac {\sqrt {2} \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]

[Out]

-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((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 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 {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {\left (2 \left (1-\frac {1}{a x}\right )^{3/2}\right ) \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 {\sqrt {2} \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 )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 76, normalized size = 1.00 \[ -\frac {\sqrt {2} \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]

[Out]

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

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fricas [A]  time = 0.59, size = 141, normalized size = 1.86 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, a c}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{a c^{\frac {3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 64, normalized size = 0.84 \[ \frac {{\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{a \sqrt {c}} - \frac {\sqrt {2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right )}{a \sqrt {c}}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 78, normalized size = 1.03 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, c^{\frac {3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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