3.269 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx\)

Optimal. Leaf size=104 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}+\frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {1}{5 a c^2 (c-a c x)^{5/2}} \]

[Out]

1/5/a/c^2/(-a*c*x+c)^(5/2)+1/6/a/c^3/(-a*c*x+c)^(3/2)-1/8*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))/a/c^(9
/2)*2^(1/2)+1/4/a/c^4/(-a*c*x+c)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6130, 21, 51, 63, 206} \[ \frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {1}{5 a c^2 (c-a c x)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(9/2)),x]

[Out]

1/(5*a*c^2*(c - a*c*x)^(5/2)) + 1/(6*a*c^3*(c - a*c*x)^(3/2)) + 1/(4*a*c^4*Sqrt[c - a*c*x]) - ArcTanh[Sqrt[c -
a*c*x]/(Sqrt[2]*Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))

Rule 21

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6130

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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0]
)

Rubi steps

\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx &=\int \frac {1-a x}{(1+a x) (c-a c x)^{9/2}} \, dx\\ &=\frac {\int \frac {1}{(1+a x) (c-a c x)^{7/2}} \, dx}{c}\\ &=\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {\int \frac {1}{(1+a x) (c-a c x)^{5/2}} \, dx}{2 c^2}\\ &=\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {\int \frac {1}{(1+a x) (c-a c x)^{3/2}} \, dx}{4 c^3}\\ &=\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {1}{4 a c^4 \sqrt {c-a c x}}+\frac {\int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{8 c^4}\\ &=\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {\operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{4 a c^5}\\ &=\frac {1}{5 a c^2 (c-a c x)^{5/2}}+\frac {1}{6 a c^3 (c-a c x)^{3/2}}+\frac {1}{4 a c^4 \sqrt {c-a c x}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 39, normalized size = 0.38 \[ \frac {\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (1-a x)\right )}{5 a c^2 (c-a c x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(9/2)),x]

[Out]

Hypergeometric2F1[-5/2, 1, -3/2, (1 - a*x)/2]/(5*a*c^2*(c - a*c*x)^(5/2))

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fricas [A]  time = 0.59, size = 252, normalized size = 2.42 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - 4 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, \frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 2 \, {\left (15 \, a^{2} x^{2} - 40 \, a x + 37\right )} \sqrt {-a c x + c}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(9/2),x, algorithm="fricas")

[Out]

[1/240*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) -
3*c)/(a*x + 1)) - 4*(15*a^2*x^2 - 40*a*x + 37)*sqrt(-a*c*x + c))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x -
a*c^5), 1/120*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt
(-c)/c) - 2*(15*a^2*x^2 - 40*a*x + 37)*sqrt(-a*c*x + c))/(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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giac [A]  time = 0.17, size = 93, normalized size = 0.89 \[ \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{8 \, a \sqrt {-c} c^{4}} + \frac {15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}}{60 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(9/2),x, algorithm="giac")

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)*c^4) + 1/60*(15*(a*c*x - c)^2 - 10*(a*c*
x - c)*c + 12*c^2)/((a*c*x - c)^2*sqrt(-a*c*x + c)*a*c^4)

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maple [A]  time = 0.04, size = 78, normalized size = 0.75 \[ \frac {\frac {1}{4 c^{3} \sqrt {-a c x +c}}+\frac {1}{6 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}+\frac {1}{5 c \left (-a c x +c \right )^{\frac {5}{2}}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {7}{2}}}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(9/2),x)

[Out]

2/c/a*(1/8/c^3/(-a*c*x+c)^(1/2)+1/12/c^2/(-a*c*x+c)^(3/2)+1/10/c/(-a*c*x+c)^(5/2)-1/16/c^(7/2)*2^(1/2)*arctanh
(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2)))

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maxima [A]  time = 0.42, size = 101, normalized size = 0.97 \[ \frac {\frac {15 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} + \frac {4 \, {\left (15 \, {\left (a c x - c\right )}^{2} - 10 \, {\left (a c x - c\right )} c + 12 \, c^{2}\right )}}{{\left (-a c x + c\right )}^{\frac {5}{2}} c^{3}}}{240 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/240*(15*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(7/2) + 4*
(15*(a*c*x - c)^2 - 10*(a*c*x - c)*c + 12*c^2)/((-a*c*x + c)^(5/2)*c^3))/(a*c)

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mupad [B]  time = 0.09, size = 78, normalized size = 0.75 \[ \frac {\frac {c-a\,c\,x}{6\,c^2}+\frac {1}{5\,c}+\frac {{\left (c-a\,c\,x\right )}^2}{4\,c^3}}{a\,c\,{\left (c-a\,c\,x\right )}^{5/2}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{8\,a\,c^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a^2*x^2 - 1)/((c - a*c*x)^(9/2)*(a*x + 1)^2),x)

[Out]

((c - a*c*x)/(6*c^2) + 1/(5*c) + (c - a*c*x)^2/(4*c^3))/(a*c*(c - a*c*x)^(5/2)) - (2^(1/2)*atanh((2^(1/2)*(c -
a*c*x)^(1/2))/(2*c^(1/2))))/(8*a*c^(9/2))

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sympy [A]  time = 40.71, size = 99, normalized size = 0.95 \[ \frac {1}{5 a c^{2} \left (- a c x + c\right )^{\frac {5}{2}}} + \frac {1}{6 a c^{3} \left (- a c x + c\right )^{\frac {3}{2}}} + \frac {1}{4 a c^{4} \sqrt {- a c x + c}} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{8 a c^{4} \sqrt {- c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(9/2),x)

[Out]

1/(5*a*c**2*(-a*c*x + c)**(5/2)) + 1/(6*a*c**3*(-a*c*x + c)**(3/2)) + 1/(4*a*c**4*sqrt(-a*c*x + c)) + sqrt(2)*
atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(8*a*c**4*sqrt(-c))

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