3.586 \(\int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\)

Optimal. Leaf size=154 \[ -\frac {2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {a} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}} \]

[Out]

-2*arcsinh(a^(1/2)*x^(1/2))*a^(1/2)*(c-c/a/x)^(1/2)*x^(1/2)/(-a*x+1)^(1/2)+4*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(
a*x+1)^(1/2))*2^(1/2)*a^(1/2)*(c-c/a/x)^(1/2)*x^(1/2)/(-a*x+1)^(1/2)-2*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a*x+1)^
(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6134, 6129, 98, 157, 54, 215, 93, 206} \[ -\frac {2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}-\frac {2 \sqrt {a} \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {a} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x,x]

[Out]

(-2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x] - (2*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt
[x]])/Sqrt[1 - a*x] + (4*Sqrt[2]*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 +
a*x]])/Sqrt[1 - a*x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 98

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6134

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(x^p*(c + d/x)^p)/(1 + (c*
x)/d)^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*
d^2, 0] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1+a x)^{3/2}}{x^{3/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-\frac {3 a}{2}-\frac {a^2 x}{2}}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {\left (a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x}}+\frac {\left (4 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-a x}}+\frac {\left (8 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 103, normalized size = 0.67 \[ -\frac {2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {a x+1}+\sqrt {a} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-2 \sqrt {2} \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )\right )}{\sqrt {1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x,x]

[Out]

(-2*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x] + Sqrt[a]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]] - 2*Sqrt[2]*Sqrt[a]*Sqrt[x]*Ar
cTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/Sqrt[1 - a*x]

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fricas [A]  time = 0.91, size = 429, normalized size = 2.79 \[ \left [\frac {2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*s
qrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + (a*x - 1)*sqrt(-c
)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(
a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1), -(2*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sq
rt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (a*x - 1)*sqrt(c)*
arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 +
1)*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]

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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)^3/(-a^2*x^2+1)^(3/2)*(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)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.06, size = 166, normalized size = 1.08 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (-\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x a +2 \sqrt {-\left (a x +1\right ) x}\, \sqrt {a}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}+4 \sqrt {a}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x \right ) \sqrt {2}}{2 \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {a}\, \sqrt {-\frac {1}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(a*x-1)/a/x)^(1/2)*(-a^2*x^2+1)^(1/2)*(-arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*2^(1/2)*(-1/a)
^(1/2)*x*a+2*(-(a*x+1)*x)^(1/2)*a^(1/2)*2^(1/2)*(-1/a)^(1/2)+4*a^(1/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)
^(1/2)*a-3*a*x-1)/(a*x-1))*x)*2^(1/2)/(a*x-1)/(-(a*x+1)*x)^(1/2)/a^(1/2)/(-1/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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