3.854 \(\int \frac {e^{3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\)

Optimal. Leaf size=360 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

x*(1-1/a^2/x^2)^(1/2)/c^3/(c-c/a^2/x^2)^(1/2)-1/16*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)^4/(c-c/a^2/x^2)^(1/2)+1/
2*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)^3/(c-c/a^2/x^2)^(1/2)-59/32*(1-1/a^2/x^2)^(1/2)/a/c^3/(-a*x+1)^2/(c-c/a^2
/x^2)^(1/2)+75/16*(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
)/(c-c/a^2/x^2)^(1/2)+201/64*ln(-a*x+1)*(1-1/a^2/x^2)^(1/2)/a/c^3/(c-c/a^2/x^2)^(1/2)-9/64*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 = 0.21, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6197, 6193, 88} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*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)]/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*
(1 - a*x)^4) + Sqrt[1 - 1/(a^2*x^2)]/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^3) - (59*Sqrt[1 - 1/(a^2*x^2)])/
(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (75*Sqrt[1 - 1/(a^2*x^2)])/(16*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)) + (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 -
 a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (9*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(64*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^{3 \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^{3 \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)^5 (1+a x)^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 \left (\frac {1}{a^7}+\frac {1}{4 a^7 (-1+a x)^5}+\frac {3}{2 a^7 (-1+a x)^4}+\frac {59}{16 a^7 (-1+a x)^3}+\frac {75}{16 a^7 (-1+a x)^2}+\frac {201}{64 a^7 (-1+a x)}+\frac {1}{32 a^7 (1+a x)^2}-\frac {9}{64 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}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^4}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {59 \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 {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 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)}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 140, normalized size = 0.39 \[ \frac {a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (\frac {75}{16 a^8 (1-a x)}-\frac {1}{32 a^8 (a x+1)}-\frac {59}{32 a^8 (1-a x)^2}+\frac {1}{2 a^8 (1-a x)^3}-\frac {1}{16 a^8 (1-a x)^4}+\frac {201 \log (1-a x)}{64 a^8}-\frac {9 \log (a x+1)}{64 a^8}+\frac {x}{a^7}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(7/2),x]

[Out]

(a^7*(1 - 1/(a^2*x^2))^(7/2)*(x/a^7 - 1/(16*a^8*(1 - a*x)^4) + 1/(2*a^8*(1 - a*x)^3) - 59/(32*a^8*(1 - a*x)^2)
 + 75/(16*a^8*(1 - a*x)) - 1/(32*a^8*(1 + a*x)) + (201*Log[1 - a*x])/(64*a^8) - (9*Log[1 + a*x])/(64*a^8)))/(c
 - c/(a^2*x^2))^(7/2)

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fricas [A]  time = 0.52, size = 207, normalized size = 0.58 \[ \frac {{\left (64 \, a^{6} x^{6} - 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} + 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} - 414 \, a x - 9 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x + 1\right ) + 201 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

1/64*(64*a^6*x^6 - 192*a^5*x^5 - 174*a^4*x^4 + 618*a^3*x^3 - 118*a^2*x^2 - 414*a*x - 9*(a^5*x^5 - 3*a^4*x^4 +
2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1)*log(a*x + 1) + 201*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1
)*log(a*x - 1) + 208)*sqrt(a^2*c)/(a^7*c^4*x^5 - 3*a^6*c^4*x^4 + 2*a^5*c^4*x^3 + 2*a^4*c^4*x^2 - 3*a^3*c^4*x +
 a^2*c^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="giac")

[Out]

integrate(1/((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)

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maple [A]  time = 0.07, size = 247, normalized size = 0.69 \[ \frac {\left (a x -1\right ) \left (a x +1\right ) \left (64 x^{6} a^{6}+201 \ln \left (a x -1\right ) x^{5} a^{5}-9 \ln \left (a x +1\right ) x^{5} a^{5}-192 x^{5} a^{5}-603 \ln \left (a x -1\right ) x^{4} a^{4}+27 \ln \left (a x +1\right ) x^{4} a^{4}-174 x^{4} a^{4}+402 \ln \left (a x -1\right ) x^{3} a^{3}-18 a^{3} x^{3} \ln \left (a x +1\right )+618 x^{3} a^{3}+402 \ln \left (a x -1\right ) x^{2} a^{2}-18 \ln \left (a x +1\right ) x^{2} a^{2}-118 a^{2} x^{2}-603 \ln \left (a x -1\right ) x a +27 a x \ln \left (a x +1\right )-414 a x +201 \ln \left (a x -1\right )-9 \ln \left (a x +1\right )+208\right )}{64 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(7/2),x)

[Out]

1/64/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)*(a*x+1)*(64*x^6*a^6+201*ln(a*x-1)*x^5*a^5-9*ln(a*x+1)*x^5*a^5-192*x^5*a^5
-603*ln(a*x-1)*x^4*a^4+27*ln(a*x+1)*x^4*a^4-174*x^4*a^4+402*ln(a*x-1)*x^3*a^3-18*a^3*x^3*ln(a*x+1)+618*x^3*a^3
+402*ln(a*x-1)*x^2*a^2-18*ln(a*x+1)*x^2*a^2-118*a^2*x^2-603*ln(a*x-1)*x*a+27*a*x*ln(a*x+1)-414*a*x+201*ln(a*x-
1)-9*ln(a*x+1)+208)/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 \[ \int \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="maxima")

[Out]

integrate(1/((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]

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))^(3/2)),x)

[Out]

int(1/((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(3/2)), 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))**(3/2)/(c-c/a**2/x**2)**(7/2),x)

[Out]

Timed out

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