3.577 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\)

Optimal. Leaf size=47 \[ -2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6133, 25, 514, 446, 80, 63, 208} \[ -2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x,x]

[Out]

-2*Sqrt[c - c/(a*x)] - 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

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

Rule 208

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

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} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^2} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-2 \sqrt {c-\frac {c}{a x}}+c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-2 \sqrt {c-\frac {c}{a x}}-(2 a) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 1.00 \[ -2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x,x]

[Out]

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

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fricas [A]  time = 0.51, size = 111, normalized size = 2.36 \[ \left [\sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, \sqrt {\frac {a c x - c}{a x}}, 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, \sqrt {\frac {a c x - c}{a x}}\right ] \]

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,x, algorithm="fricas")

[Out]

[sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*sqrt((a*c*x - c)/(a*x)), 2*sqrt(-c)*arc
tan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*sqrt((a*c*x - c)/(a*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,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)]Warning, choosing root of [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]%%%}] at parameters values [86,-97,-82]
Warning, choosing root of [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]%%%}] at parameters values [7,-27,26]Warning
, choosing root of [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]%%%}] at parameters values [-89,63,-49]Warning, cho
osing root of [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]%%%}] at parameters values [-86,-64,-30]Warning, choosin
g root of [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]%%%}] at parameters values [70,22,42]Warning, choosing root
of [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]%%%}] at parameters values [56,-9,-13]Sign error (%%%{sqrt(c)*a,0%%
%}+%%%{2*sqrt(-a*c)*abs(a),1/2%%%}+%%%{-2*sqrt(c)*a^2,1%%%}+%%%{-a*sqrt(-a*c)*abs(a),3/2%%%}+%%%{-a^2*sqrt(-a*
c)*abs(a)/4,5/2%%%}+%%%{undef,7/2%%%})Limit: Max order reached or unable to make series expansion Error: Bad A
rgument Value

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maple [B]  time = 0.04, size = 98, normalized size = 2.09 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-2 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}+2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}-\ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a \right )}{x \sqrt {\left (a x -1\right ) x}\, \sqrt {a}} \]

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,x)

[Out]

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

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

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,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2}{x\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 9.65, size = 39, normalized size = 0.83 \[ \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - 2 \sqrt {c - \frac {c}{a x}} \]

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,x)

[Out]

2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sqrt(-c) - 2*sqrt(c - c/(a*x))

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