3.9.62 \(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [862]

Optimal. Leaf size=358 \[ \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {5 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{24 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^3}+\frac {11 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {19 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {51 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 358, 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 = {6332, 6328, 90} \begin {gather*} \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {5 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\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 {11 \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 (a x+1)^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}}}-\frac {51 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(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)]/(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)) - Sqrt[1 - 1/(a^2*x^2)]/(2
4*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)) + (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1
- a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*
x^2)])

Rule 90

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 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u/x^(
2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), 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 6332

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)^3 (1+a x)^4} \, 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}{16 a^7 (-1+a x)^3}+\frac {5}{16 a^7 (-1+a x)^2}+\frac {19}{32 a^7 (-1+a x)}+\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)}\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}}}{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 {\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 {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}}}-\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}}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 104, normalized size = 0.29 \begin {gather*} \frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (96 a x+\frac {30}{1-a x}-\frac {3}{(-1+a x)^2}-\frac {4}{(1+a x)^3}+\frac {33}{(1+a x)^2}-\frac {144}{1+a x}+57 \log (1-a x)-153 \log (1+a x)\right )}{96 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A]
time = 0.04, size = 247, normalized size = 0.69

method	result	size
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (a x -1\right ) \left (-96 a^{6} x^{6}+153 \ln \left (a x +1\right ) a^{5} x^{5}-57 x^{5} \ln \left (a x -1\right ) a^{5}-96 a^{5} x^{5}+153 \ln \left (a x +1\right ) a^{4} x^{4}-57 x^{4} \ln \left (a x -1\right ) a^{4}+366 a^{4} x^{4}-306 \ln \left (a x +1\right ) a^{3} x^{3}+114 x^{3} \ln \left (a x -1\right ) a^{3}+222 a^{3} x^{3}-306 \ln \left (a x +1\right ) a^{2} x^{2}+114 x^{2} \ln \left (a x -1\right ) a^{2}-338 a^{2} x^{2}+153 \ln \left (a x +1\right ) a x -57 x \ln \left (a x -1\right ) a -122 a x +153 \ln \left (a x +1\right )-57 \ln \left (a x -1\right )+88\right )}{96 a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}}\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(a*x-1)*(-96*a^6*x^6+153*ln(a*x+1)*a^5*x^5-57*x^5*ln(a*x-1)*a^5-96*a^5*x
^5+153*ln(a*x+1)*a^4*x^4-57*x^4*ln(a*x-1)*a^4+366*a^4*x^4-306*ln(a*x+1)*a^3*x^3+114*x^3*ln(a*x-1)*a^3+222*a^3*
x^3-306*ln(a*x+1)*a^2*x^2+114*x^2*ln(a*x-1)*a^2-338*a^2*x^2+153*ln(a*x+1)*a*x-57*x*ln(a*x-1)*a-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 201, normalized size = 0.56 \begin {gather*} \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 - 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 ) + 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 ) - 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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((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 - 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) + 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) - 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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5989 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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