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

Optimal. Leaf size=359 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {11 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{24 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {51 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

x*(1-1/a^2/x^2)^(1/2)/c^3/(c-c/a^2/x^2)^(1/2)+1/24*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)^3/(c-c/a^2/x^2)^(1/2)-11
/32*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)^2/(c-c/a^2/x^2)^(1/2)+3/2*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)/(c-c/a^2/x
^2)^(1/2)+1/32*(1-1/a^2/x^2)^(1/2)/a/c^3/(a*x+1)^2/(c-c/a^2/x^2)^(1/2)-5/16*(1-1/a^2/x^2)^(1/2)/a/c^3/(a*x+1)/
(c-c/a^2/x^2)^(1/2)+51/32*ln(-a*x+1)*(1-1/a^2/x^2)^(1/2)/a/c^3/(c-c/a^2/x^2)^(1/2)-19/32*ln(a*x+1)*(1-1/a^2/x^
2)^(1/2)/a/c^3/(c-c/a^2/x^2)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6197, 6193, 88} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {11 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{24 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {51 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(24*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)^3) - (11*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (3*Sqrt[1 - 1/(a^2*x^
2)])/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*
x)^2) - (5*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1
- a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*
x^2)])

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6193

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

Rule 6197

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

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^7}{(-1+a x)^4 (1+a x)^3} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^7}+\frac {1}{8 a^7 (-1+a x)^4}+\frac {11}{16 a^7 (-1+a x)^3}+\frac {3}{2 a^7 (-1+a x)^2}+\frac {51}{32 a^7 (-1+a x)}-\frac {1}{16 a^7 (1+a x)^3}+\frac {5}{16 a^7 (1+a x)^2}-\frac {19}{32 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{24 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {11 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {5 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {51 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 104, normalized size = 0.29 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (96 a x+\frac {144}{1-a x}-\frac {30}{a x+1}-\frac {33}{(a x-1)^2}+\frac {3}{(a x+1)^2}-\frac {4}{(a x-1)^3}+153 \log (1-a x)-57 \log (a x+1)\right )}{96 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 1/(a^2*x^2))^(7/2)*(96*a*x + 144/(1 - a*x) - 4/(-1 + a*x)^3 - 33/(-1 + a*x)^2 + 3/(1 + a*x)^2 - 30/(1 +
a*x) + 153*Log[1 - a*x] - 57*Log[1 + a*x]))/(96*a*(c - c/(a^2*x^2))^(7/2))

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fricas [A]  time = 0.60, size = 205, normalized size = 0.57 \[ \frac {{\left (96 \, a^{6} x^{6} - 96 \, a^{5} x^{5} - 366 \, a^{4} x^{4} + 222 \, a^{3} x^{3} + 338 \, a^{2} x^{2} - 122 \, a x - 57 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right ) + 153 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x - 1\right ) - 88\right )} \sqrt {a^{2} c}}{96 \, {\left (a^{7} c^{4} x^{5} - a^{6} c^{4} x^{4} - 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} + a^{3} c^{4} x - a^{2} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(96*a^6*x^6 - 96*a^5*x^5 - 366*a^4*x^4 + 222*a^3*x^3 + 338*a^2*x^2 - 122*a*x - 57*(a^5*x^5 - a^4*x^4 - 2*
a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*x + 1) + 153*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*
x - 1) - 88)*sqrt(a^2*c)/(a^7*c^4*x^5 - a^6*c^4*x^4 - 2*a^5*c^4*x^3 + 2*a^4*c^4*x^2 + a^3*c^4*x - a^2*c^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 247, normalized size = 0.69 \[ \frac {\left (a x -1\right ) \left (a x +1\right ) \left (96 x^{6} a^{6}+153 \ln \left (a x -1\right ) x^{5} a^{5}-57 \ln \left (a x +1\right ) x^{5} a^{5}-96 x^{5} a^{5}-153 \ln \left (a x -1\right ) x^{4} a^{4}+57 \ln \left (a x +1\right ) x^{4} a^{4}-366 x^{4} a^{4}-306 \ln \left (a x -1\right ) x^{3} a^{3}+114 a^{3} x^{3} \ln \left (a x +1\right )+222 x^{3} a^{3}+306 \ln \left (a x -1\right ) x^{2} a^{2}-114 \ln \left (a x +1\right ) x^{2} a^{2}+338 a^{2} x^{2}+153 \ln \left (a x -1\right ) x a -57 a x \ln \left (a x +1\right )-122 a x -153 \ln \left (a x -1\right )+57 \ln \left (a x +1\right )-88\right )}{96 \sqrt {\frac {a x -1}{a x +1}}\, a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96/((a*x-1)/(a*x+1))^(1/2)*(a*x-1)*(a*x+1)*(96*x^6*a^6+153*ln(a*x-1)*x^5*a^5-57*ln(a*x+1)*x^5*a^5-96*x^5*a^5
-153*ln(a*x-1)*x^4*a^4+57*ln(a*x+1)*x^4*a^4-366*x^4*a^4-306*ln(a*x-1)*x^3*a^3+114*a^3*x^3*ln(a*x+1)+222*x^3*a^
3+306*ln(a*x-1)*x^2*a^2-114*ln(a*x+1)*x^2*a^2+338*a^2*x^2+153*ln(a*x-1)*x*a-57*a*x*ln(a*x+1)-122*a*x-153*ln(a*
x-1)+57*ln(a*x+1)-88)/a^8/x^7/(c*(a^2*x^2-1)/a^2/x^2)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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