### 3.437 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^5} \, dx$$

Optimal. Leaf size=138 $-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5}$

[Out]

(-2*(a + x^(-1)))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(5/2)) - (10*a + 13/x)/(15*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2)) - (
30*a + 41/x)/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^5 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x
^2)]])/(a*c^5)

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Rubi [A]  time = 0.389577, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} $-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]

[Out]

(-2*(a + x^(-1)))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(5/2)) - (10*a + 13/x)/(15*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2)) - (
30*a + 41/x)/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^5 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x
^2)]])/(a*c^5)

Rule 6177

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

Rule 852

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

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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

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 \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^5} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (c-\frac{c x}{a}\right )^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^2}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-5 c^2-\frac{10 c^2 x}{a}-\frac{8 c^2 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 c^2+\frac{30 c^2 x}{a}+\frac{26 c^2 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-15 c^2-\frac{30 c^2 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^5}\\ &=-\frac{2 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^5}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^5}\\ \end{align*}

Mathematica [A]  time = 0.0808256, size = 104, normalized size = 0.75 $\frac{15 a^4 x^4-76 a^3 x^3+32 a^2 x^2+30 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+82 a x-56}{15 a^2 c^5 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]

[Out]

(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[
1 - 1/(a^2*x^2)]])/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)

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Maple [B]  time = 0.144, size = 615, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x)

[Out]

1/30*(60*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^6*a^7+75*(a^2)^(1/2)*((a*x-1)*(a*x+1))^
(1/2)*x^6*a^6-120*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6-45*(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(3/2)*x^4*a^4-150*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-60*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+
1))^(1/2))/(a^2)^(1/2))*x^4*a^5-2*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3-75*(a^2)^(1/2)*((a*x-1)*(a*x+1))
^(1/2)*x^4*a^4+240*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+64*(a^2)^(1/2)*((a*x-1)
*(a*x+1))^(3/2)*x^2*a^2+300*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-60*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+14*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-75*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
1/2)*x^2*a^2-120*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-37*((a*x-1)*(a*x+1))^(3/2)*
(a^2)^(1/2)-150*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+60*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a
^2)^(1/2))+75*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/a*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/c^5/((a*x-1)*(a*x+1))
^(1/2)/(a*x-1)^5

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Maxima [A]  time = 1.05843, size = 238, normalized size = 1.72 \begin{align*} \frac{1}{120} \, a{\left (\frac{\frac{32 \,{\left (a x - 1\right )}}{a x + 1} + \frac{310 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{585 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac{15 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="maxima")

[Out]

1/120*a*((32*(a*x - 1)/(a*x + 1) + 310*(a*x - 1)^2/(a*x + 1)^2 - 585*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^5*((a
*x - 1)/(a*x + 1))^(7/2) - a^2*c^5*((a*x - 1)/(a*x + 1))^(5/2)) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*
c^5) - 240*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^5) + 15*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^5))

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Fricas [A]  time = 1.84735, size = 387, normalized size = 2.8 \begin{align*} \frac{30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="fricas")

[Out]

1/15*(30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt((a*x - 1
)/(a*x + 1)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)

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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(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a x}\right )}^{5}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^5, x)