### 3.514 $$\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx$$

Optimal. Leaf size=209 $-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right )$

[Out]

(4*a^3*c^3*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(5/2)) + (2*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a
*x))^(3/2)) - (2*a^3*c^2*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(3/2)) + (4*a^3*c*Sqrt[1 - 1/(a^2*x^2)])/Sq
rt[c - c/(a*x)] - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])]

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Rubi [A]  time = 0.476001, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6178, 1639, 795, 665, 661, 208} $-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]

[Out]

(4*a^3*c^3*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(5/2)) + (2*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a
*x))^(3/2)) - (2*a^3*c^2*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(3/2)) + (4*a^3*c*Sqrt[1 - 1/(a^2*x^2)])/Sq
rt[c - c/(a*x)] - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])]

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 661

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

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 \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{1}{7} \left (2 a^4 c\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{3 c^2}{2 a^2}-\frac{5 c^2 x}{a^3}\right ) \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}-\left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}-\left (2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-\left (4 a^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{c x}{a}} \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}+\left (8 a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )\\ &=\frac{4 a^3 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right )\\ \end{align*}

Mathematica [A]  time = 0.283286, size = 170, normalized size = 0.81 $\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (52 a^3 x^3+16 a^2 x^2+9 a x+3\right ) \sqrt{c-\frac{c}{a x}}}{21 x^2 (a x-1)}-2 \sqrt{2} a^3 \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+2 \sqrt{2} a^3 \sqrt{c} \log \left ((a x-1)^2\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 + 9*a*x + 16*a^2*x^2 + 52*a^3*x^3))/(21*x^2*(-1 + a*x)) + 2*Sq
rt[2]*a^3*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^3*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sq
rt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]

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Maple [A]  time = 0.187, size = 187, normalized size = 0.9 \begin{align*} -{\frac{2\,ax-2}{ \left ( 21\,ax+21 \right ){x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 21\,{a}^{3}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ){x}^{4}-52\,{a}^{3}\sqrt{{a}^{-1}}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-16\,{a}^{2}\sqrt{{a}^{-1}}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-9\,a\sqrt{{a}^{-1}}x\sqrt{ \left ( ax+1 \right ) x}-3\,\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

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^4,x)

[Out]

-2/21/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)/x^3*(21*a^3*2^(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^4-52*a^3*(1/a)^(1/2)*x^3*((a*x+1)*x)^(1/2)-16*a^2*(1/a)^(1/2)*x^2*
((a*x+1)*x)^(1/2)-9*a*(1/a)^(1/2)*x*((a*x+1)*x)^(1/2)-3*(1/a)^(1/2)*((a*x+1)*x)^(1/2))/((a*x+1)*x)^(1/2)/(1/a)
^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x^{4} \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)*(c-c/a/x)^(1/2)/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(c - c/(a*x))/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)

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Fricas [A]  time = 2.01236, size = 867, normalized size = 4.15 \begin{align*} \left [\frac{21 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\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 ) + 2 \,{\left (52 \, a^{4} x^{4} + 68 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{21 \,{\left (a x^{4} - x^{3}\right )}}, \frac{2 \,{\left (21 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\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 ) +{\left (52 \, a^{4} x^{4} + 68 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}\right )}}{21 \,{\left (a x^{4} - x^{3}\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)*(c-c/a/x)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*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)) + 2*(52*a^4*x^4 + 68*a^3*x^3 + 25*a^2*x^2 + 12*a*x + 3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(
a*x)))/(a*x^4 - x^3), 2/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*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)) + (52*a^4*x^4 + 68*a^3*x^3 + 25
*a^2*x^2 + 12*a*x + 3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3)]

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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)*(c-c/a/x)**(1/2)/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x^{4} \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)*(c-c/a/x)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(c - c/(a*x))/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)