3.6.10 \(\int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [510]

Optimal. Leaf size=152 \[ \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}+\frac {5 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {1-\frac {1}{a x}}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6317, 6314, 100, 162, 65, 214, 212} \begin {gather*} \frac {x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {5 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \sqrt {1-\frac {1}{a x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/Sqrt[1 - 1/(a*x)] + (5*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(
a*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*Sqrt[1 - 1/(a*x)])

Rule 65

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 100

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 162

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

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 214

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

Rule 6314

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

Rule 6317

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.61 \begin {gather*} \frac {\sqrt {c-\frac {c}{a x}} \left (a \sqrt {1+\frac {1}{a x}} x+5 \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )-4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{a \sqrt {1-\frac {1}{a x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - c/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x + 5*ArcTanh[Sqrt[1 + 1/(a*x)]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]
/Sqrt[2]]))/(a*Sqrt[1 - 1/(a*x)])

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Maple [A]
time = 0.13, size = 160, normalized size = 1.05

method	result	size
default	\(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+5 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-4 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}}\)	\(160\)
risch	\(\frac {x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(2*(x*(a*x+1))^(1/2)*a^(3/2)*(1/a)^(1/2)+5
*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2)-4*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*(x*(a
*x+1))^(1/2)*a+3*a*x+1)/(a*x-1))*a^(1/2))/(x*(a*x+1))^(1/2)/a^(3/2)/(1/a)^(1/2)

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

[Out]

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

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Fricas [A]
time = 0.44, size = 512, normalized size = 3.37 \begin {gather*} \left [\frac {4 \, \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^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 5 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {4 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 5 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*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^3*x^3 + 4*a^2*x
^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) +
5*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x
+ 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(
a*x)))/(a^2*x - a), 1/2*(4*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)
/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 5*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*
x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(a^2*x^2 + a*x)*s
qrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]

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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 (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

[Out]

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

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