3.729 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ \frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6159, 6129, 98, 143, 41, 216} \[ -\frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2)),x]

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 98

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

Rule 143

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 6129

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

Rule 6159

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 95, normalized size = 0.77 \[ \frac {-3 a^3 x^3-11 a^2 x^2+6 (a x+1) \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )+4 a x+10}{3 a^2 c x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2)),x]

[Out]

(10 + 4*a*x - 11*a^2*x^2 - 3*a^3*x^3 + 6*(1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c*
Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))

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fricas [A]  time = 0.70, size = 280, normalized size = 2.26 \[ \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {6 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(a^2*x^2 + 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)
- (3*a^3*x^3 + 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2), -1/
3*(6*(a^2*x^2 + 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) +
(3*a^3*x^3 + 14*a^2*x^2 + 10*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 + 2*a^2*c^2*x + a*c^2)]

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giac [A]  time = 0.22, size = 127, normalized size = 1.02 \[ -\frac {6 \, {\left (a x + 1\right )} a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} + \frac {24 \, a^{2} c \arctan \left (\frac {\sqrt {c - \frac {2 \, c}{a x + 1}}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {a^{2} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} c^{2} + 15 \, a^{2} \sqrt {c - \frac {2 \, c}{a x + 1}} c^{3}}{c^{3}}}{6 \, a^{3} c^{2} \mathrm {sgn}\left (-\frac {1}{a x + 1} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(a*x + 1)*a^2*sqrt(c - 2*c/(a*x + 1)) + 24*a^2*c*arctan(sqrt(c - 2*c/(a*x + 1))/sqrt(-c))/sqrt(-c) + (
a^2*(c - 2*c/(a*x + 1))^(3/2)*c^2 + 15*a^2*sqrt(c - 2*c/(a*x + 1))*c^3)/c^3)/(a^3*c^2*sgn(-1/(a*x + 1) + 1))

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maple [B]  time = 0.05, size = 326, normalized size = 2.63 \[ -\frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{3}+15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}-4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2}-6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2} c -4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x a -6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a c -12 c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}\right ) \left (a x -1\right )}{3 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c^(3/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^3*a^3+15*x^2*a^2*c^(3/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)-4*c^(3/
2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^2*a^2-6*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*(
c*(a^2*x^2-1)/a^2)^(1/2)*x*a^2*c-4*c^(3/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x*a-6*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1
/2))*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a*c-12*c^(3/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+2*(c
*(a^2*x^2-1)/a^2)^(1/2)*c^(3/2))*(a*x-1)/((a*x-1)*(a*x+1)*c/a^2)^(1/2)/x^3/(c*(a^2*x^2-1)/a^2/x^2)^(3/2)/a^4/c
^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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