3.225 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx\)

Optimal. Leaf size=125 \[ -\frac {46 \left (a+\frac {1}{x}\right )}{35 a^2 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {35 a+\frac {13}{x}}{35 a^2 c^6 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]

[Out]

-46/35*(a+1/x)/a^2/c^6/(1-1/a^2/x^2)^(3/2)+24/35*(a+1/x)^2/a^3/c^6/(1-1/a^2/x^2)^(5/2)-1/7*(a+1/x)^3/a^4/c^6/(
1-1/a^2/x^2)^(7/2)+1/35*(35*a+13/x)/a^2/c^6/(1-1/a^2/x^2)^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6175, 6178, 852, 1635, 637} \[ -\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {46 \left (a+\frac {1}{x}\right )}{35 a^2 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {35 a+\frac {13}{x}}{35 a^2 c^6 \sqrt {1-\frac {1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6),x]

[Out]

(-46*(a + x^(-1)))/(35*a^2*c^6*(1 - 1/(a^2*x^2))^(3/2)) + (24*(a + x^(-1))^2)/(35*a^3*c^6*(1 - 1/(a^2*x^2))^(5
/2)) - (a + x^(-1))^3/(7*a^4*c^6*(1 - 1/(a^2*x^2))^(7/2)) + (35*a + 13/x)/(35*a^2*c^6*Sqrt[1 - 1/(a^2*x^2)])

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

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 1635

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

Rule 6175

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

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]

Rubi steps

\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx &=\frac {\int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^6 x^6} \, dx}{a^6 c^6}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1-\frac {x}{a}\right )^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^6 c^6}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (1+\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{a^6 c^6}\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2 \left (3 a^4+7 a^3 x+7 a^2 x^2+7 a x^3\right )}{\left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 a^6 c^6}\\ &=\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right ) \left (33 a^4+70 a^3 x+35 a^2 x^2\right )}{\left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 a^6 c^6}\\ &=-\frac {46 \left (a+\frac {1}{x}\right )}{35 a^2 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {39 a^4+105 a^3 x}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 a^6 c^6}\\ &=-\frac {46 \left (a+\frac {1}{x}\right )}{35 a^2 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {24 \left (a+\frac {1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\left (a+\frac {1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {35 a+\frac {13}{x}}{35 a^2 c^6 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 66, normalized size = 0.53 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right )}{35 c^6 (a x-1)^4 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4))/(35*c^6*(-1 + a*x)^4*(1 + a*x))

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fricas [A]  time = 0.62, size = 96, normalized size = 0.77 \[ \frac {{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{6} x^{4} - 4 \, a^{4} c^{6} x^{3} + 6 \, a^{3} c^{6} x^{2} - 4 \, a^{2} c^{6} x + a c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="fricas")

[Out]

1/35*(8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*x - 13)*sqrt((a*x - 1)/(a*x + 1))/(a^5*c^6*x^4 - 4*a^4*c^6*x^3
+ 6*a^3*c^6*x^2 - 4*a^2*c^6*x + a*c^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a c x - c\right )}^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(a*c*x - c)^6, x)

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maple [A]  time = 0.04, size = 66, normalized size = 0.53 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 x^{4} a^{4}-24 x^{3} a^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{5} c^{6} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x)

[Out]

1/35*((a*x-1)/(a*x+1))^(3/2)*(8*a^4*x^4-24*a^3*x^3+20*a^2*x^2+4*a*x-13)*(a*x+1)/(a*x-1)^5/c^6/a

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maxima [A]  time = 0.31, size = 97, normalized size = 0.78 \[ \frac {1}{560} \, a {\left (\frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{6}} + \frac {\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="maxima")

[Out]

1/560*a*(35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^6) + (28*(a*x - 1)/(a*x + 1) - 70*(a*x - 1)^2/(a*x + 1)^2 + 140*(
a*x - 1)^3/(a*x + 1)^3 - 5)/(a^2*c^6*((a*x - 1)/(a*x + 1))^(7/2)))

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mupad [B]  time = 0.07, size = 60, normalized size = 0.48 \[ \frac {8\,a^4\,x^4-24\,a^3\,x^3+20\,a^2\,x^2+4\,a\,x-13}{35\,a\,c^6\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a*c*x)^6,x)

[Out]

(4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4 - 13)/(35*a*c^6*(a*x + 1)^4*((a*x - 1)/(a*x + 1))^(7/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\, dx}{c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c)**6,x)

[Out]

(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9*a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**
3 + 9*a**2*x**2 - 5*a*x + 1), x) + Integral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9
*a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + 9*a**2*x**2 - 5*a*x + 1), x))/c**6

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