### 3.505 $$\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx$$

Optimal. Leaf size=96 $-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{8 a^3 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{10 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+4 a^3 \sqrt{c-\frac{c}{a x}}$

[Out]

4*a^3*Sqrt[c - c/(a*x)] - (10*a^3*(c - c/(a*x))^(3/2))/(3*c) + (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^3*(c
- c/(a*x))^(7/2))/(7*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.351501, antiderivative size = 96, 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 = {6167, 6133, 25, 514, 446, 77} $-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\frac{8 a^3 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{10 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+4 a^3 \sqrt{c-\frac{c}{a x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]

[Out]

4*a^3*Sqrt[c - c/(a*x)] - (10*a^3*(c - c/(a*x))^(3/2))/(3*c) + (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^3*(c
- c/(a*x))^(7/2))/(7*c^3)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6133

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

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1+a x)}{x^4 (1-a x)} \, dx\\ &=\frac{c \int \frac{1+a x}{\sqrt{c-\frac{c}{a x}} x^5} \, dx}{a}\\ &=\frac{c \int \frac{a+\frac{1}{x}}{\sqrt{c-\frac{c}{a x}} x^4} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{x^2 (a+x)}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \left (\frac{2 a^3}{\sqrt{c-\frac{c x}{a}}}-\frac{5 a^3 \sqrt{c-\frac{c x}{a}}}{c}+\frac{4 a^3 \left (c-\frac{c x}{a}\right )^{3/2}}{c^2}-\frac{a^3 \left (c-\frac{c x}{a}\right )^{5/2}}{c^3}\right ) \, dx,x,\frac{1}{x}\right )}{a}\\ &=4 a^3 \sqrt{c-\frac{c}{a x}}-\frac{10 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\frac{8 a^3 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0510289, size = 44, normalized size = 0.46 $\frac{2 \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]

[Out]

(2*Sqrt[c - c/(a*x)]*(15 + 39*a*x + 52*a^2*x^2 + 104*a^3*x^3))/(105*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.116, size = 43, normalized size = 0.5 \begin{align*}{\frac{208\,{x}^{3}{a}^{3}+104\,{a}^{2}{x}^{2}+78\,ax+30}{105\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(104*a^3*x^3+52*a^2*x^2+39*a*x+15)*(c*(a*x-1)/a/x)^(1/2)/x^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a x}}}{{\left (a x - 1\right )} x^{4}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.55228, size = 103, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt{\frac{a c x - c}{a x}}}{105 \, x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(1/2)/x^4,x, algorithm="fricas")

[Out]

2/105*(104*a^3*x^3 + 52*a^2*x^2 + 39*a*x + 15)*sqrt((a*c*x - c)/(a*x))/x^3

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{x^{4} \left (a x - 1\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x**4*(a*x - 1)), x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(1/2)/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError