3.1151 \(\int \frac {e^{3 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=70 \[ \frac {(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {\sin ^{-1}(a x)}{a^2 c} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 789, 653, 216} \[ \frac {(a x+1)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {\sin ^{-1}(a x)}{a^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 653

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 789

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

Rule 6148

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 70, normalized size = 1.00 \[ \frac {-7 a^2 x^2+3 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-2 a x+5}{3 a^2 c (a x-1) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 96, normalized size = 1.37 \[ -\frac {5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x - 5\right )} + 5}{3 \, {\left (a^{4} c x^{2} - 2 \, a^{3} c x + a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.35, size = 112, normalized size = 1.60 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a c {\left | a \right |}} + \frac {2 \, {\left (\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 5\right )}}{3 \, a c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)*sgn(a)/(a*c*abs(a)) + 2/3*(12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 3*(sqrt(-a^2*x^2 + 1)*abs(
a) + a)^2/(a^4*x^2) - 5)/(a*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^3*abs(a))

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maple [B]  time = 0.04, size = 155, normalized size = 2.21 \[ -\frac {5 x}{c a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c a \sqrt {a^{2}}}-\frac {3}{c \,a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {4}{3 c \,a^{3} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8 x}{3 c a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.52, size = 241, normalized size = 3.44 \[ \frac {1}{3} \, a {\left (\frac {2 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {2 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {7 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} + \frac {3 \, \arcsin \left (a x\right )}{a^{3} c} - \frac {9}{\sqrt {-a^{2} x^{2} + 1} a^{3} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.95, size = 108, normalized size = 1.54 \[ \frac {4}{3\,a^2\,c\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {3}{a^2\,c\,\sqrt {1-a^2\,x^2}}-\frac {7\,x}{3\,a\,c\,\sqrt {1-a^2\,x^2}}+\frac {4\,x}{3\,a\,c\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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