3.578 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\)
Optimal. Leaf size=42 \[ \frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt {c-\frac {c}{a x}} \]
[Out]
2/3*a*(c-c/a/x)^(3/2)/c-4*a*(c-c/a/x)^(1/2)
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Rubi [A] time = 0.22, antiderivative size = 42, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used =
{6133, 25, 514, 444, 43} \[ \frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt {c-\frac {c}{a x}} \]
Antiderivative was successfully verified.
[In]
Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^2,x]
[Out]
-4*a*Sqrt[c - c/(a*x)] + (2*a*(c - c/(a*x))^(3/2))/(3*c)
Rule 25
Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] && !(IntegerQ[m] && NegQ[n])
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 444
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]
Rule 514
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
| !IntegerQ[p])
Rule 6133
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[(u*(c + d/x)^p*(1 + a*x)^(n/
2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] &
& !GtQ[c, 0]
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^2 (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^3} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^2} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {a+x}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \left (\frac {2 a}{\sqrt {c-\frac {c x}{a}}}-\frac {a \sqrt {c-\frac {c x}{a}}}{c}\right ) \, dx,x,\frac {1}{x}\right )}{a}\\ &=-4 a \sqrt {c-\frac {c}{a x}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.67 \[ -\frac {2 (5 a x+1) \sqrt {c-\frac {c}{a x}}}{3 x} \]
Antiderivative was successfully verified.
[In]
Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^2,x]
[Out]
(-2*Sqrt[c - c/(a*x)]*(1 + 5*a*x))/(3*x)
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fricas [A] time = 0.41, size = 28, normalized size = 0.67 \[ -\frac {2 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^2,x, algorithm="fricas")
[Out]
-2/3*(5*a*x + 1)*sqrt((a*c*x - c)/(a*x))/x
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Unable to divide, perhaps due to rounding error%%%{%%%{6,[2,1,5]%%%}+%%%{-6,[1,1,4]%%%}+%%%{-6,[0,1,3]%%%
},[4]%%%}+%%%{%%{[%%%{-9,[2,0,4]%%%}+%%%{9,[1,0,3]%%%}+%%%{9,[0,0,2]%%%},0,%%%{9,[4,1,6]%%%}+%%%{-18,[3,1,5]%%
%}+%%%{9,[2,1,4]%%%}+%%%{-9,[0,1,2]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,
[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[3]%%%}+%%%{%%%{18,[
4,1,5]%%%}+%%%{-36,[3,1,4]%%%}+%%%{18,[1,1,2]%%%},[2]%%%}+%%%{%%{[%%%{-3,[4,0,4]%%%}+%%%{6,[3,0,3]%%%}+%%%{-3,
[1,0,1]%%%},0,%%%{3,[6,1,6]%%%}+%%%{-9,[5,1,5]%%%}+%%%{9,[4,1,4]%%%}+%%%{-3,[3,1,3]%%%}+%%%{-3,[2,1,2]%%%}+%%%
{3,[1,1,1]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3
]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[1]%%%} / %%%{%%%{256,[8,8,8]%%%}+%%%{-1024,[
7,8,7]%%%}+%%%{1536,[6,8,6]%%%}+%%%{-1024,[5,8,5]%%%}+%%%{256,[4,8,4]%%%},[4]%%%}+%%%{%%{[%%%{-384,[8,7,7]%%%}
+%%%{1536,[7,7,6]%%%}+%%%{-2304,[6,7,5]%%%}+%%%{1536,[5,7,4]%%%}+%%%{-384,[4,7,3]%%%},0,%%%{384,[10,8,9]%%%}+%
%%{-1920,[9,8,8]%%%}+%%%{4224,[8,8,7]%%%}+%%%{-5376,[7,8,6]%%%}+%%%{4224,[6,8,5]%%%}+%%%{-1920,[5,8,4]%%%}+%%%
{384,[4,8,3]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2
,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[3]%%%}+%%%{%%%{768,[10,8,8]%%%}+%%%{-3840,
[9,8,7]%%%}+%%%{7680,[8,8,6]%%%}+%%%{-7680,[7,8,5]%%%}+%%%{3840,[6,8,4]%%%}+%%%{-768,[5,8,3]%%%},[2]%%%}+%%%{%
%{[%%%{-128,[10,7,7]%%%}+%%%{640,[9,7,6]%%%}+%%%{-1280,[8,7,5]%%%}+%%%{1280,[7,7,4]%%%}+%%%{-640,[6,7,3]%%%}+%
%%{128,[5,7,2]%%%},0,%%%{128,[12,8,9]%%%}+%%%{-768,[11,8,8]%%%}+%%%{2048,[10,8,7]%%%}+%%%{-3200,[9,8,6]%%%}+%%
%{3200,[8,8,5]%%%}+%%%{-2048,[7,8,4]%%%}+%%%{768,[6,8,3]%%%}+%%%{-128,[5,8,2]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%
{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+
%%%{1,[0,2,0]%%%}]%%},[1]%%%} Error: Bad Argument Value
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maple [A] time = 0.03, size = 27, normalized size = 0.64 \[ -\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (5 a x +1\right )}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^2,x)
[Out]
-2/3*(c*(a*x-1)/a/x)^(1/2)*(5*a*x+1)/x
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^2,x, algorithm="maxima")
[Out]
-integrate((a*x + 1)^2*sqrt(c - c/(a*x))/((a^2*x^2 - 1)*x^2), x)
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mupad [B] time = 0.90, size = 24, normalized size = 0.57 \[ -\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\left (5\,a\,x+1\right )}{3\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(x^2*(a^2*x^2 - 1)),x)
[Out]
-(2*(c - c/(a*x))^(1/2)*(5*a*x + 1))/(3*x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {c - \frac {c}{a x}}}{a x^{3} - x^{2}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{3} - x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**2,x)
[Out]
-Integral(sqrt(c - c/(a*x))/(a*x**3 - x**2), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**3 - x**2), x)
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