### 3.313 $$\int e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=163 $-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}}$

[Out]

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

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Rubi [A]  time = 0.187324, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6176, 6181, 94, 93, 206} $-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6176

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

Rule 6181

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

Rule 94

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[p, 1] &&  !SumSimplerQ[m, 1])

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 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])

Rubi steps

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

Mathematica [A]  time = 0.0599873, size = 105, normalized size = 0.64 $\frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+7)-6 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{3 a^{3/2} \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.194, size = 107, normalized size = 0.7 \begin{align*} -{\frac{2\,ax-2}{ \left ( 3\,ax+3 \right ) a}\sqrt{-c \left ( ax-1 \right ) } \left ( 6\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -xa\sqrt{-c \left ( ax+1 \right ) }-7\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59122, size = 598, normalized size = 3.67 \begin{align*} \left [\frac{2 \,{\left (3 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{3 \,{\left (a^{2} x - a\right )}}, -\frac{2 \,{\left (6 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{3 \,{\left (a^{2} x - a\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*s
qrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (a^2*x^2 + 8*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)
/(a*x + 1)))/(a^2*x - a), -2/3*(6*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x
- 1)/(a*x + 1))/(a*c*x - c)) - (a^2*x^2 + 8*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*x - a)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.17755, size = 144, normalized size = 0.88 \begin{align*} -\frac{12 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 16 \, \sqrt{2} \sqrt{-c}}{3 \, a \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (6 \, \sqrt{2} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) +{\left (-a c x - c\right )}^{\frac{3}{2}} - 6 \, \sqrt{-a c x - c} c\right )}}{3 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*I*sqrt(2)*sqrt(-c)*arctan(-I) - 16*sqrt(2)*sqrt(-c))/(a*sgn(c)) - 2/3*(6*sqrt(2)*c^(3/2)*arctan(1/2*s
qrt(2)*sqrt(-a*c*x - c)/sqrt(c)) + (-a*c*x - c)^(3/2) - 6*sqrt(-a*c*x - c)*c)/(a*c*sgn(-a*c*x - c))