3.1122 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac {7 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}+\frac {a^2 (10 a x+9)}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 a^2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

2/3*a^2*(a*x+1)/(-a^2*c*x^2+c)^(3/2)-7/2*a^2*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(3/2)+1/3*a^2*(10*a*x+9)/
c/(-a^2*c*x^2+c)^(1/2)-1/2*(-a^2*c*x^2+c)^(1/2)/c^2/x^2-2*a*(-a^2*c*x^2+c)^(1/2)/c^2/x

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Rubi [A]  time = 0.42, antiderivative size = 142, 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 = {6151, 1805, 1807, 807, 266, 63, 208} \[ -\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {7 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {a^2 (10 a x+9)}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 a^2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (a^2*(9 + 10*a*x))/(3*c*Sqrt[c - a^2*c*x^2]) - Sqrt[c - a^2*c*x^
2]/(2*c^2*x^2) - (2*a*Sqrt[c - a^2*c*x^2])/(c^2*x) - (7*a^2*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/(2*c^(3/2))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

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

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 6151

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

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1+a x)^2}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{3} \int \frac {-3-6 a x-6 a^2 x^2-4 a^3 x^3}{x^3 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}+\frac {\int \frac {3+6 a x+9 a^2 x^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx}{3 c}\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {\int \frac {-12 a c-21 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{6 c^2}\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}+\frac {\left (7 a^2\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx}{2 c}\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}+\frac {\left (7 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{2 c^2}\\ &=\frac {2 a^2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^2 (9+10 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c^2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c^2 x}-\frac {7 a^2 \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 105, normalized size = 0.74 \[ -\frac {7 a^2 \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )}{2 c^{3/2}}+\frac {7 a^2 \log (x)}{2 c^{3/2}}-\frac {\left (32 a^3 x^3-43 a^2 x^2+6 a x+3\right ) \sqrt {c-a^2 c x^2}}{6 c^2 x^2 (a x-1)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(Sqrt[c - a^2*c*x^2]*(3 + 6*a*x - 43*a^2*x^2 + 32*a^3*x^3))/(c^2*x^2*(-1 + a*x)^2) + (7*a^2*Log[x])/(2*c^
(3/2)) - (7*a^2*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/(2*c^(3/2))

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fricas [A]  time = 0.87, size = 266, normalized size = 1.87 \[ \left [\frac {21 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right )} \sqrt {-a^{2} c x^{2} + c}}{12 \, {\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}}, -\frac {21 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (32 \, a^{3} x^{3} - 43 \, a^{2} x^{2} + 6 \, a x + 3\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, {\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(21*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2)
- 2*(32*a^3*x^3 - 43*a^2*x^2 + 6*a*x + 3)*sqrt(-a^2*c*x^2 + c))/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2), -1/6*(
21*(a^4*x^4 - 2*a^3*x^3 + a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (32*a^3*x^
3 - 43*a^2*x^2 + 6*a*x + 3)*sqrt(-a^2*c*x^2 + c))/(a^2*c^2*x^4 - 2*a*c^2*x^3 + c^2*x^2)]

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giac [A]  time = 0.41, size = 204, normalized size = 1.44 \[ a^{4} c^{2} {\left (\frac {7 \, \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{3}} - \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a - 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} \sqrt {-c} {\left | a \right |} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a c + 4 \, \sqrt {-c} c {\left | a \right |}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2} a^{3} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*c^2*(7*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/(a^2*sqrt(-c)*c^3) - ((sqrt(-a^2*c)*x - s
qrt(-a^2*c*x^2 + c))^3*a - 4*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*sqrt(-c)*abs(a) + (sqrt(-a^2*c)*x - sqr
t(-a^2*c*x^2 + c))*a*c + 4*sqrt(-c)*c*abs(a))/(((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^2*a^3*c^3))

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maple [A]  time = 0.05, size = 205, normalized size = 1.44 \[ \frac {7 a^{2}}{2 c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {7 a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}-\frac {2 a}{c x \sqrt {-a^{2} c \,x^{2}+c}}+\frac {4 a^{3} x}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 a}{3 c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}+\frac {4 a^{3} x}{3 c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}-\frac {1}{2 c \,x^{2} \sqrt {-a^{2} c \,x^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/2*a^2/c/(-a^2*c*x^2+c)^(1/2)-7/2*a^2/c^(3/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-2*a/c/x/(-a^2*c*x^2+
c)^(1/2)+4*a^3/c*x/(-a^2*c*x^2+c)^(1/2)-2/3*a/c/(x-1/a)/(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2)+4/3*a^3/c/(-(x-
1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2)*x-1/2/c/x^2/(-a^2*c*x^2+c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{- a^{3} c x^{6} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt {- a^{2} c x^{2} + c} + a c x^{4} \sqrt {- a^{2} c x^{2} + c} - c x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{6} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt {- a^{2} c x^{2} + c} + a c x^{4} \sqrt {- a^{2} c x^{2} + c} - c x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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