3.13.51 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\) [1251]

Optimal. Leaf size=98 \[ -\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}} \]

[Out]

-2/7*(-a*x+1)/a/(-a^2*c*x^2+c)^(7/2)+1/7*x/c/(-a^2*c*x^2+c)^(5/2)+4/21*x/c^2/(-a^2*c*x^2+c)^(3/2)+8/21*x/c^3/(
-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6277, 667, 198, 197} \begin {gather*} \frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2)),x]

[Out]

(-2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) + x/(7*c*(c - a^2*c*x^2)^(5/2)) + (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/
2)) + (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 667

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)*((a + c*x^2)^(p + 1)/(c*(p
+ 1))), x] - Dist[e^2*((p + 2)/(c*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 6277

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

Rubi steps

\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {5}{7} \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 96, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \left (6-9 a x-24 a^2 x^2-4 a^3 x^3+16 a^4 x^4+8 a^5 x^5\right )}{21 a c^3 (1-a x)^{3/2} (1+a x)^{7/2} \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2)),x]

[Out]

-1/21*(Sqrt[1 - a^2*x^2]*(6 - 9*a*x - 24*a^2*x^2 - 4*a^3*x^3 + 16*a^4*x^4 + 8*a^5*x^5))/(a*c^3*(1 - a*x)^(3/2)
*(1 + a*x)^(7/2)*Sqrt[c - a^2*c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(267\) vs. \(2(82)=164\).
time = 0.06, size = 268, normalized size = 2.73

method	result	size
gosper	\(-\frac {\left (a x -1\right )^{2} \left (8 x^{5} a^{5}+16 a^{4} x^{4}-4 a^{3} x^{3}-24 a^{2} x^{2}-9 a x +6\right )}{21 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} a}\)	\(64\)
trager	\(-\frac {\left (8 x^{5} a^{5}+16 a^{4} x^{4}-4 a^{3} x^{3}-24 a^{2} x^{2}-9 a x +6\right ) \sqrt {-a^{2} c \,x^{2}+c}}{21 c^{4} \left (a x +1\right )^{4} \left (a x -1\right )^{2} a}\)	\(74\)
default	\(-\frac {x}{5 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}-\frac {4 \left (\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}\right )}{5 c}+\frac {-\frac {2}{7 a c \left (x +\frac {1}{a}\right ) \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}+\frac {12 a \left (-\frac {-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c}{10 a^{2} c^{2} \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{15 a^{2} c^{2} \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right )}{15 a^{2} c^{3} \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}}{c}\right )}{7}}{a}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*x/c/(-a^2*c*x^2+c)^(5/2)-4/5/c*(1/3*x/c/(-a^2*c*x^2+c)^(3/2)+2/3*x/c^2/(-a^2*c*x^2+c)^(1/2))+2/a*(-1/7/a/
c/(x+1/a)/(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(5/2)+6/7*a*(-1/10*(-2*a^2*c*(x+1/a)+2*a*c)/a^2/c^2/(-c*a^2*(x+1/a)
^2+2*a*c*(x+1/a))^(5/2)+4/5/c*(-1/6*(-2*a^2*c*(x+1/a)+2*a*c)/a^2/c^2/(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(3/2)-1/
3/a^2/c^3*(-2*a^2*c*(x+1/a)+2*a*c)/(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(1/2))))

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Maxima [A]
time = 0.26, size = 98, normalized size = 1.00 \begin {gather*} -\frac {2}{7 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c\right )}} + \frac {8 \, x}{21 \, \sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{21 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {x}{7 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7/((-a^2*c*x^2 + c)^(5/2)*a^2*c*x + (-a^2*c*x^2 + c)^(5/2)*a*c) + 8/21*x/(sqrt(-a^2*c*x^2 + c)*c^3) + 4/21*
x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 1/7*x/((-a^2*c*x^2 + c)^(5/2)*c)

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Fricas [A]
time = 0.46, size = 124, normalized size = 1.27 \begin {gather*} -\frac {{\left (8 \, a^{5} x^{5} + 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 24 \, a^{2} x^{2} - 9 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(8*a^5*x^5 + 16*a^4*x^4 - 4*a^3*x^3 - 24*a^2*x^2 - 9*a*x + 6)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 + 2*a^6*
c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) - a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x*
*5*sqrt(-a**2*c*x**2 + c) + 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c)
- 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) + c**3*sqrt(-a**2*c*x**2 + c)), x)
- Integral(-1/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) - a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x
**5*sqrt(-a**2*c*x**2 + c) + 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c)
- 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) + c**3*sqrt(-a**2*c*x**2 + c)), x
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (81) = 162\).
time = 0.44, size = 300, normalized size = 3.06 \begin {gather*} \frac {a^{5} {\left (\frac {14 \, {\left (7 \, c - \frac {15 \, c}{a x + 1}\right )}}{a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )} + \frac {3 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{42} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 21 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{43} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 210 \, a^{30} c^{45} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 70 \, a^{30} c^{44} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6}}{a^{35} c^{49} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{7} \mathrm {sgn}\left (a\right )^{7}}\right )} - \frac {256 \, \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{\sqrt {-c} c^{3}}}{672 \, {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/672*(a^5*(14*(7*c - 15*c/(a*x + 1))/(a^5*(c - 2*c/(a*x + 1))*c^3*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*s
gn(a)) + (3*a^30*(c - 2*c/(a*x + 1))^3*c^42*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 - 21*a^30*(c
- 2*c/(a*x + 1))^2*c^43*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 - 210*a^30*c^45*sqrt(-c + 2*c/(a*
x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 - 70*a^30*c^44*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))^6*sgn(a)^6)/(a^
35*c^49*sgn(1/(a*x + 1))^7*sgn(a)^7)) - 256*sgn(1/(a*x + 1))*sgn(a)/(sqrt(-c)*c^3))/abs(a)

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Mupad [B]
time = 1.12, size = 133, normalized size = 1.36 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}-\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x+1\right )}^3}-\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c - a^2*c*x^2)^(1/2)*((11*x)/(42*c^4) - 5/(28*a*c^4)))/((a*x - 1)^2*(a*x + 1)^2) - (c - a^2*c*x^2)^(1/2)/(28
*a*c^4*(a*x + 1)^4) - (c - a^2*c*x^2)^(1/2)/(14*a*c^4*(a*x + 1)^3) - (8*x*(c - a^2*c*x^2)^(1/2))/(21*c^4*(a*x
- 1)*(a*x + 1))

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