3.260 \(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\)

Optimal. Leaf size=193 \[ -\frac {3 a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}+\frac {3 a^3 x^3 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {a^3 x^2 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}} \]

[Out]

-3/32*a^(5/2)*(1-1/a/x)^(7/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(1/2))/(1/x)^(7/2)/(-a*c*x+c)^(7/2
)*2^(1/2)-1/4*a^3*(1-1/a/x)^(7/2)*x^2*(1+1/a/x)^(1/2)/(a-1/x)^2/(-a*c*x+c)^(7/2)+3/16*a^3*(1-1/a/x)^(7/2)*x^3*
(1+1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(7/2)

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Rubi [A]  time = 0.20, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6176, 6181, 94, 93, 206} \[ \frac {3 a^3 x^3 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {a^3 x^2 \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{7/2}}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}-\frac {3 a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)),x]

[Out]

-(a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^2)/(4*(a - x^(-1))^2*(c - a*c*x)^(7/2)) + (3*a^3*(1 - 1/(a*x))^(
7/2)*Sqrt[1 + 1/(a*x)]*x^3)/(16*(a - x^(-1))*(c - a*c*x)^(7/2)) - (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt
[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(16*Sqrt[2]*(x^(-1))^(7/2)*(c - a*c*x)^(7/2))

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

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

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 6176

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

Rule 6181

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

Rubi steps

\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \, dx}{(c-a c x)^{7/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{7/2} \operatorname {Subst}\left (\int \frac {x^{3/2}}{\left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {\left (3 a \left (1-\frac {1}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {3 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {\left (3 a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{32 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {3 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {\left (3 a^2 \left (1-\frac {1}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{16 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ &=-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {3 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {3 a^{5/2} \left (1-\frac {1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 125, normalized size = 0.65 \[ \frac {x \sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {\frac {1}{a x}+1} (7-3 a x)+3 \sqrt {2} \sqrt {\frac {1}{x}} (a x-1)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )\right )}{32 \sqrt {a} c^3 (a x-1)^2 \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 - 3*a*x) + 3*Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)^2*ArcTanh[(S
qrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(32*Sqrt[a]*c^3*(-1 + a*x)^2*Sqrt[c - a*c*x])

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fricas [A]  time = 0.53, size = 341, normalized size = 1.77 \[ \left [-\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, {\left (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{32 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="fricas")

[Out]

[-1/64*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a*c*x
+ c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*(3*a^2*x^2 - 4*a*x - 7)*s
qrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), -1/32*(3*sqrt(
2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))
/(a*c*x - c)) - 2*(3*a^2*x^2 - 4*a*x - 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*c^4*x^3 - 3*a^3*c^4
*x^2 + 3*a^2*c^4*x - a*c^4)]

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giac [A]  time = 0.23, size = 88, normalized size = 0.46 \[ \frac {{\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (-a c x - c\right )}^{\frac {3}{2}} + 10 \, \sqrt {-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2} c^{2}}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{32 \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="giac")

[Out]

1/32*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(5/2) + 2*(3*(-a*c*x - c)^(3/2) + 10*sqrt(-a*c*
x - c)*c)/((a*c*x - c)^2*c^2))*abs(c)*sgn(a*x + 1)/(a*c^2)

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maple [A]  time = 0.06, size = 172, normalized size = 0.89 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x^{2} a^{2} c +6 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c +6 x a \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -14 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{32 c^{\frac {9}{2}} \left (a x -1\right )^{3} \sqrt {-c \left (a x +1\right )}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x)

[Out]

1/32*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1
/2))*x^2*a^2*c+6*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c+6*x*a*(-c*(a*x+1))^(1/2)*c^(1/2)
-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c-14*(-c*(a*x+1))^(1/2)*c^(1/2))/c^(9/2)/(a*x-1)^3/(
-c*(a*x+1))^(1/2)/a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (-a c x + c\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a\,c\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(7/2),x)

[Out]

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(7/2),x)

[Out]

Timed out

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