### 3.440 $$\int e^{\coth ^{-1}(a x)} (c-\frac{c}{a x})^{5/2} \, dx$$

Optimal. Leaf size=157 $\frac{c^4 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}-\frac{3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}$

[Out]

(-2*c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a*(c - c/(a*x))^(3/2)) + (3*c^3*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a*x)
]) + (c^4*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) - (3*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/
Sqrt[c - c/(a*x)]])/a

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Rubi [A]  time = 0.132817, antiderivative size = 189, normalized size of antiderivative = 1.2, number of steps used = 7, number of rules used = 7, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.318, Rules used = {6182, 6179, 89, 80, 50, 63, 208} $-\frac{2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{3 a \left (1-\frac{1}{a x}\right )^{5/2}}+\frac{x \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{5/2}}{\left (1-\frac{1}{a x}\right )^{5/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{5/2}}{a \left (1-\frac{1}{a x}\right )^{5/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{5/2}}$

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^(5/2),x]

[Out]

(3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2))/(a*(1 - 1/(a*x))^(5/2)) - (2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2)
)/(3*a*(1 - 1/(a*x))^(5/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2)*x)/(1 - 1/(a*x))^(5/2) - (3*(c - c/(a*x
))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(5/2))

Rule 6182

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])

Rule 6179

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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0736338, size = 89, normalized size = 0.57 $\frac{c^2 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2+10 a x-2\right )-9 a x \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{3 a^2 x \sqrt{1-\frac{1}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(5/2),x]

[Out]

(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-2 + 10*a*x + 3*a^2*x^2) - 9*a*x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^
2*Sqrt[1 - 1/(a*x)]*x)

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Maple [A]  time = 0.18, size = 132, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}}{6\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 6\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+20\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(5/2),x)

[Out]

1/6/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*c^2*(6*a^(5/2)*x^2*((a*x+1)*x)^(1/2)+20*a^(3/2)*x*((a*x+1)*x
)^(1/2)-9*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*x^2*a^2-4*((a*x+1)*x)^(1/2)*a^(1/2))/x/a^(5/2)
/((a*x+1)*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(5/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a*x))^(5/2)/sqrt((a*x - 1)/(a*x + 1)), x)

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Fricas [A]  time = 2.18703, size = 790, normalized size = 5.03 \begin{align*} \left [\frac{9 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{12 \,{\left (a^{3} x^{2} - a^{2} x\right )}}, \frac{9 \,{\left (a^{2} c^{2} x^{2} - a c^{2} x\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 (3 \, a^{3} c^{2} x^{3} + 13 \, a^{2} c^{2} x^{2} + 8 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(9*(a^2*c^2*x^2 - a*c^2*x)*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*(3*a^3*c^2*x^3 + 13*a^2*c^2*x^2 + 8*a*c
^2*x - 2*c^2)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(9*(a^2*c^2*x^2 - a*c^
2*x)*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*(3*a^3*c^2*x^3 + 13*a^2*c^2*x^2 + 8*a*c^2*x - 2*c^2)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x
- c)/(a*x)))/(a^3*x^2 - a^2*x)]

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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))**(1/2)*(c-c/a/x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{5}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(5/2),x, algorithm="giac")

[Out]

integrate((c - c/(a*x))^(5/2)/sqrt((a*x - 1)/(a*x + 1)), x)