3.2.66 \(\int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx\) [166]

Optimal. Leaf size=133 \[ \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3} \]

[Out]

1/9*a^5*(1-1/a^2/x^2)^(3/2)/c^5/(a-1/x)^6-8/21*a^4*(1-1/a^2/x^2)^(3/2)/c^5/(a-1/x)^5+47/105*a^3*(1-1/a^2/x^2)^
(3/2)/c^5/(a-1/x)^4-58/315*a^2*(1-1/a^2/x^2)^(3/2)/c^5/(a-1/x)^3

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Rubi [A]
time = 0.23, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6310, 6313, 1653, 807, 673, 665} \begin {gather*} -\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - a*c*x)^5,x]

[Out]

(a^5*(1 - 1/(a^2*x^2))^(3/2))/(9*c^5*(a - x^(-1))^6) - (8*a^4*(1 - 1/(a^2*x^2))^(3/2))/(21*c^5*(a - x^(-1))^5)
+ (47*a^3*(1 - 1/(a^2*x^2))^(3/2))/(105*c^5*(a - x^(-1))^4) - (58*a^2*(1 - 1/(a^2*x^2))^(3/2))/(315*c^5*(a -
x^(-1))^3)

Rule 665

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

Rule 673

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

Rule 807

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

Rule 1653

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 6310

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 6313

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^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\frac {\int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^5 x^5} \, dx}{a^5 c^5}\\ &=\frac {\text {Subst}\left (\int \frac {x^3 \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{a^5 c^5}\\ &=\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {\text {Subst}\left (\int \frac {\left (\frac {4}{a^2}-\frac {7 x}{a^3}+\frac {2 x^2}{a^4}\right ) \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{c^5}\\ &=\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {a^4 \text {Subst}\left (\int \frac {\left (\frac {18}{a^6}-\frac {20 x}{a^7}\right ) \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{2 c^5}\\ &=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}+\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {29 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^5} \, dx,x,\frac {1}{x}\right )}{3 a^2 c^5}\\ &=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{21 a^2 c^5}\\ &=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{105 a^2 c^5}\\ &=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 59, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-58-25 a x+21 a^2 x^2-10 a^3 x^3+2 a^4 x^4\right )}{315 c^5 (-1+a x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*(Sqrt[1 - 1/(a^2*x^2)]*x*(-58 - 25*a*x + 21*a^2*x^2 - 10*a^3*x^3 + 2*a^4*x^4))/(c^5*(-1 + a*x)^5)

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Maple [A]
time = 0.13, size = 58, normalized size = 0.44

method	result	size
gosper	\(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(58\)
default	\(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(58\)
trager	\(-\frac {\left (a x +1\right ) \left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}-25 a x -58\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{315 a \,c^{5} \left (a x -1\right )^{5}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(2*a^3*x^3-12*a^2*x^2+33*a*x-58)*(a*x+1)/(a*x-1)^4/c^5/((a*x-1)/(a*x+1))^(1/2)/a

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Maxima [A]
time = 0.27, size = 71, normalized size = 0.53 \begin {gather*} -\frac {\frac {135 \, {\left (a x - 1\right )}}{a x + 1} - \frac {189 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {105 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35}{2520 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(135*(a*x - 1)/(a*x + 1) - 189*(a*x - 1)^2/(a*x + 1)^2 + 105*(a*x - 1)^3/(a*x + 1)^3 - 35)/(a*c^5*((a*
x - 1)/(a*x + 1))^(9/2))

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Fricas [A]
time = 0.33, size = 116, normalized size = 0.87 \begin {gather*} -\frac {{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 83 \, a x - 58\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(2*a^5*x^5 - 8*a^4*x^4 + 11*a^3*x^3 - 4*a^2*x^2 - 83*a*x - 58)*sqrt((a*x - 1)/(a*x + 1))/(a^6*c^5*x^5 -
5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*a^2*c^5*x - a*c^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {1}{a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 5*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 10*
a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 10*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 5*a*x*sqrt(a*x/
(a*x + 1) - 1/(a*x + 1)) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c**5

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Giac [A]
time = 0.47, size = 125, normalized size = 0.94 \begin {gather*} \frac {4 \, {\left (315 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{5} x^{5} + 189 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{4} x^{4} + 84 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 36 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 9 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{315 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{9} a c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/315*(315*(a + sqrt(a^2 - 1/x^2))^5*x^5 + 189*(a + sqrt(a^2 - 1/x^2))^4*x^4 + 84*(a + sqrt(a^2 - 1/x^2))^3*x^
3 - 36*(a + sqrt(a^2 - 1/x^2))^2*x^2 + 9*(a + sqrt(a^2 - 1/x^2))*x - 1)/(((a + sqrt(a^2 - 1/x^2))*x - 1)^9*a*c
^5)

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Mupad [B]
time = 1.17, size = 72, normalized size = 0.54 \begin {gather*} \frac {\frac {3\,{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {3\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{9}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*(a*x - 1)^2)/(5*(a*x + 1)^2) - (a*x - 1)^3/(3*(a*x + 1)^3) - (3*(a*x - 1))/(7*(a*x + 1)) + 1/9)/(8*a*c^5*(
(a*x - 1)/(a*x + 1))^(9/2))

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