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

Optimal. Leaf size=163 \[ -\frac {x^2 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^2}+\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^3}+\frac {7 x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{24 a^3}+\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^3}+\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt {a x+1} \sqrt {1-a x}} \]

[Out]

7/8*x*(c-c/a^2/x^2)^(1/2)/a^3+7/24*x*(-a*x+1)*(c-c/a^2/x^2)^(1/2)/a^3+1/6*x*(-a*x+1)^2*(c-c/a^2/x^2)^(1/2)/a^3
-1/4*x^2*(-a*x+1)^2*(c-c/a^2/x^2)^(1/2)/a^2+7/8*x*arcsin(a*x)*(c-c/a^2/x^2)^(1/2)/a^3/(-a*x+1)^(1/2)/(a*x+1)^(
1/2)

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Rubi [A]  time = 0.40, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6159, 6129, 90, 80, 50, 41, 216} \[ -\frac {x^2 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^2}+\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^3}+\frac {7 x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{24 a^3}+\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^3}+\frac {7 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt {a x+1} \sqrt {1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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 e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt {1-a x} \sqrt {1+a x} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x^2 (1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2} (-1+2 a x)}{\sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{12 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}+\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}+\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {\left (7 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x}{8 a^3}+\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{24 a^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{6 a^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (1-a x)^2}{4 a^2}+\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{8 a^3 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 93, normalized size = 0.57 \[ -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (21 \log \left (\sqrt {a^2 x^2-1}+a x\right )+\sqrt {a^2 x^2-1} \left (6 a^3 x^3-16 a^2 x^2+21 a x-32\right )\right )}{24 a^3 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/24*(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(-32 + 21*a*x - 16*a^2*x^2 + 6*a^3*x^3) + 21*Log[a*x + Sqrt
[-1 + a^2*x^2]]))/(a^3*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 3.20, size = 222, normalized size = 1.36 \[ \left [-\frac {2 \, {\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{48 \, a^{4}}, -\frac {{\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{24 \, a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(2*(6*a^4*x^4 - 16*a^3*x^3 + 21*a^2*x^2 - 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*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))/a^4, -1/24*((6*a^4*x^4 - 16*a^3*x^3 + 21*a^2
*x^2 - 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*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)))/a^4]

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giac [A]  time = 0.27, size = 128, normalized size = 0.79 \[ -\frac {1}{48} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left ({\left (2 \, x {\left (\frac {3 \, x \mathrm {sgn}\relax (x)}{a^{2}} - \frac {8 \, \mathrm {sgn}\relax (x)}{a^{3}}\right )} + \frac {21 \, \mathrm {sgn}\relax (x)}{a^{4}}\right )} x - \frac {32 \, \mathrm {sgn}\relax (x)}{a^{5}}\right )} - \frac {42 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a^{4} {\left | a \right |}} + \frac {{\left (21 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 64 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\relax (x)}{a^{5} {\left | a \right |}}\right )} {\left | a \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(2*sqrt(a^2*c*x^2 - c)*((2*x*(3*x*sgn(x)/a^2 - 8*sgn(x)/a^3) + 21*sgn(x)/a^4)*x - 32*sgn(x)/a^5) - 42*sq
rt(c)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a^4*abs(a)) + (21*a*sqrt(c)*log(abs(c)) + 64*sqrt
(-c)*abs(a))*sgn(x)/(a^5*abs(a)))*abs(a)

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maple [A]  time = 0.04, size = 196, normalized size = 1.20 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-6 x \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4}+16 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3}-27 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2} c +27 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-48 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right )+48 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a c \right )}{24 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*x*(-6*x*(c*(a^2*x^2-1)/a^2)^(3/2)*a^4+16*(c*(a^2*x^2-1)/a^2)^(3/2)*a^3-27*(
c*(a^2*x^2-1)/a^2)^(1/2)*x*a^2*c+27*c^(3/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))-48*c^(3/2)*ln((c^(1/2)*((a
*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))+48*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*a*c)/(c*(a^2*x^2-1)/a^2)^(1/2)/c/a^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {x^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\right )\, dx - \int \frac {a x^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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