3.8.34 \(\int e^{-3 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{5/2} \, dx\) [734]

Optimal. Leaf size=218 \[ -\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}-\frac {3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 218, 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 = {6295, 6285, 76} \begin {gather*} \frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {3 a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^(5/2)/E^(3*ArcTanh[a*x]),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rule 6285

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

Rule 6295

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 90, normalized size = 0.41 \begin {gather*} \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (-1+4 a x-4 a^2 x^2-8 a^3 x^3-5 a^4 x^4+4 a^5 x^5-12 a^4 x^4 \log (x)\right )}{4 a^4 x^3 \sqrt {1-a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))^(5/2)/E^(3*ArcTanh[a*x]),x]

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)]*(-1 + 4*a*x - 4*a^2*x^2 - 8*a^3*x^3 - 5*a^4*x^4 + 4*a^5*x^5 - 12*a^4*x^4*Log[x]))/(
4*a^4*x^3*Sqrt[1 - a^2*x^2])

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Maple [A]
time = 0.06, size = 86, normalized size = 0.39

method	result	size
default	\(\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-4 x^{5} a^{5}+12 a^{4} \ln \left (x \right ) x^{4}+8 a^{3} x^{3}+4 a^{2} x^{2}-4 a x +1\right )}{4 \left (a^{2} x^{2}-1\right )^{3}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a^2/x^2)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(c*(a^2*x^2-1)/a^2/x^2)^(5/2)*x/(a^2*x^2-1)^3*(-a^2*x^2+1)^(1/2)*(-4*x^5*a^5+12*a^4*ln(x)*x^4+8*a^3*x^3+4*
a^2*x^2-4*a*x+1)

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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((c-c/a^2/x^2)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(5/2)/(a*x + 1)^3, x)

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Fricas [A]
time = 0.41, size = 480, normalized size = 2.20 \begin {gather*} \left [\frac {6 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left (4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right )} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, \frac {12 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - {\left (4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left (4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right )} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2
+ 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (4*a^5*c^2*x^5 - 8*a^3*c^2*x^3 - (4*a^5
- 8*a^3 - 4*a^2 + 4*a - 1)*c^2*x^4 - 4*a^2*c^2*x^2 + 4*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/
(a^2*x^2)))/(a^6*x^5 - a^4*x^3), 1/4*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3
+ a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) - (4*a^5*c^2*x^5 - 8*a^3*c^2
*x^3 - (4*a^5 - 8*a^3 - 4*a^2 + 4*a - 1)*c^2*x^4 - 4*a^2*c^2*x^2 + 4*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a
^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}}}{\left (a x + 1\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)

[Out]

Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)/(a*x + 1)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(5/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(5/2)/(a*x + 1)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - c/(a^2*x^2))^(5/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)

[Out]

int(((c - c/(a^2*x^2))^(5/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)

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