3.6.87 \(\int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) [587]

Optimal. Leaf size=147 \[ -\frac {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 147, 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 = {6269, 6264, 96, 95, 212} \begin {gather*} \frac {4 \sqrt {2} a^{3/2} \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}}-\frac {4 a \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x] - (2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(3*x*Sqrt[1 - a*x
]) + (4*Sqrt[2]*a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a
*x]

Rule 95

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 212

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

Rule 6264

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 6269

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^2} \, 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^{5/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1+a x)^{3/2}}{x^{5/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {\left (2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{3/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {\left (4 a^2 \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 {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {\left (8 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {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 {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{3/2} \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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 93, normalized size = 0.63 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x} (1+7 a x)+6 \sqrt {2} a^{3/2} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{3 x \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c - c/(a*x)]*(-(Sqrt[1 + a*x]*(1 + 7*a*x)) + 6*Sqrt[2]*a^(3/2)*x^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x
])/Sqrt[1 + a*x]]))/(3*x*Sqrt[1 - a*x])

________________________________________________________________________________________

Maple [A]
time = 0.88, size = 150, normalized size = 1.02

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 a x \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}+6 a \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}+\sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\right ) \sqrt {2}}{3 x \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\)	\(150\)
risch	\(-\frac {2 \left (7 a^{2} x^{2}+8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {4 a \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-2 c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(206\)

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^2,x,method=_RETURNVERBOSE)

[Out]

1/3*(c*(a*x-1)/a/x)^(1/2)/x*(-a^2*x^2+1)^(1/2)*(7*a*x*(-(a*x+1)*x)^(1/2)*2^(1/2)*(-1/a)^(1/2)+6*a*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+(-(a*x+1)*x)^(1/2)*2^(1/2)*(-1/a)^(1/2))*2^(1/2)/(a*
x-1)/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 309, normalized size = 2.10 \begin {gather*} \left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\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 ) + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, -\frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\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 ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \end {gather*}

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^2,x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(2)*(a^2*x^2 - a*x)*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)*sqrt(-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)) + 2*sqrt(-a^2
*x^2 + 1)*(7*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x), -2/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*arctan(2*s
qrt(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)) - sqrt(-a^2*x^2 + 1
)*(7*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x)]

________________________________________________________________________________________

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

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**2,x)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________