3.12.50 \(\int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx\) [1150]

Optimal. Leaf size=95 \[ \frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \text {ArcSin}(a x)}{a^3 c} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6283, 1649, 21, 683, 655, 222} \begin {gather*} \frac {3 \text {ArcSin}(a x)}{a^3 c}+\frac {(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 222

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 655

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

Rule 683

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

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 6283

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

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Mathematica [A]
time = 0.09, size = 78, normalized size = 0.82 \begin {gather*} \frac {14-5 a x-16 a^2 x^2+3 a^3 x^3+9 (-1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^3 c (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14 - 5*a*x - 16*a^2*x^2 + 3*a^3*x^3 + 9*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^3*c*(-1 + a*x)*Sqrt[1
- a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(87)=174\).
time = 0.08, size = 207, normalized size = 2.18

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{4} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}}{c}\)	\(147\)
default	\(-\frac {a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {7 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {4}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}+\frac {4 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}}{a^{3}}}{c}\)	\(207\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (87) = 174\).
time = 0.54, size = 364, normalized size = 3.83 \begin {gather*} \frac {1}{6} \, {\left (\frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} + \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac {6 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} - \frac {26 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{3} c} + \frac {18 \, \arcsin \left (a x\right )}{a^{4} c} - \frac {36}{\sqrt {-a^{2} x^{2} + 1} a^{4} c}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^8*c^4*x + sqrt(-a^2*x^2 + 1)*a^7*c^4) - a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^8*c^4
*x - sqrt(-a^2*x^2 + 1)*a^7*c^4) + 3*a*c/(sqrt(-a^2*x^2 + 1)*a^6*c^2*x + sqrt(-a^2*x^2 + 1)*a^5*c^2) - 3*a*c/(
sqrt(-a^2*x^2 + 1)*a^6*c^2*x - sqrt(-a^2*x^2 + 1)*a^5*c^2) - 4*c/(sqrt(-a^2*x^2 + 1)*a^5*c^2*x + sqrt(-a^2*x^2
+ 1)*a^4*c^2) - 4*c/(sqrt(-a^2*x^2 + 1)*a^5*c^2*x - sqrt(-a^2*x^2 + 1)*a^4*c^2) + 6*x^2/(sqrt(-a^2*x^2 + 1)*a
^2*c) - 26*x/(sqrt(-a^2*x^2 + 1)*a^3*c) + 18*arcsin(a*x)/(a^4*c) - 36/(sqrt(-a^2*x^2 + 1)*a^4*c))*a

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Fricas [A]
time = 0.34, size = 103, normalized size = 1.08 \begin {gather*} -\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{5} c x^{2} - 2 \, a^{4} c x + a^{3} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*
a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^5*c*x^2 - 2*a^4*c*x + a^3*c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {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 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 {3 a^{2} 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 + \int \frac {a^{3} x^{5}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(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*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(3*a**2*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) + Integral(a**3*x**5/(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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 0.94, size = 133, normalized size = 1.40 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^5\,x^2-2\,c\,a^4\,x+c\,a^3\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (a\,c\,\sqrt {-a^2}-a^2\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,c}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c\,\sqrt {-a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(1 - a^2*x^2)^(1/2))/(3*(a^3*c + a^5*c*x^2 - 2*a^4*c*x)) + (13*(1 - a^2*x^2)^(1/2))/(3*(a*c*(-a^2)^(1/2) -
a^2*c*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a^3*c) + (3*asinh(x*(-a^2)^(1/2)))/(a^2*c*(-a^2)^(1
/2))

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