3.1078 \(\int e^{2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(21 a x+32) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]

[Out]

7/8*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))*c^(1/2)/a^3-2/3*x^2*(-a^2*c*x^2+c)^(1/2)/a-1/4*x^3*(-a^2*c*x^2+c)
^(1/2)-1/24*(21*a*x+32)*(-a^2*c*x^2+c)^(1/2)/a^3

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Rubi [A]  time = 0.29, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6151, 1809, 833, 780, 217, 203} \[ -\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {(21 a x+32) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2*Sqrt[c - a^2*c*x^2])/(3*a) - (x^3*Sqrt[c - a^2*c*x^2])/4 - ((32 + 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3
) + (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 780

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

Rule 833

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -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 e^{2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx &=c \int \frac {x^2 (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x^2 \left (-7 a^2 c-8 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^2}\\ &=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (16 a^3 c^2+21 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{12 a^4 c}\\ &=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32+21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {(7 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a^2}\\ &=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32+21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {(7 c) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}\\ &=-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{3 a}-\frac {1}{4} x^3 \sqrt {c-a^2 c x^2}-\frac {(32+21 a x) \sqrt {c-a^2 c x^2}}{24 a^3}+\frac {7 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 88, normalized size = 0.79 \[ -\frac {21 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\left (6 a^3 x^3+16 a^2 x^2+21 a x+32\right ) \sqrt {c-a^2 c x^2}}{24 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 168, normalized size = 1.50 \[ \left [-\frac {2 \, {\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} - 21 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{48 \, a^{3}}, -\frac {{\left (6 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 21 \, a x + 32\right )} \sqrt {-a^{2} c x^{2} + c} + 21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 84, normalized size = 0.75 \[ -\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, x + \frac {8}{a}\right )} x + \frac {21}{a^{2}}\right )} x + \frac {32}{a^{3}}\right )} - \frac {7 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt {-c} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(-a^2*c*x^2 + c)*((2*(3*x + 8/a)*x + 21/a^2)*x + 32/a^3) - 7/8*c*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2
*c*x^2 + c)))/(a^2*sqrt(-c)*abs(a))

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maple [B]  time = 0.04, size = 186, normalized size = 1.66 \[ \frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}-\frac {9 x \sqrt {-a^{2} c \,x^{2}+c}}{8 a^{2}}-\frac {9 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 a^{2} \sqrt {a^{2} c}}+\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{3} c}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{3}}+\frac {2 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a^{2} \sqrt {a^{2} c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)/a^2/c-9/8/a^2*x*(-a^2*c*x^2+c)^(1/2)-9/8/a^2*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x
/(-a^2*c*x^2+c)^(1/2))+2/3/a^3*(-a^2*c*x^2+c)^(3/2)/c-2/a^3*(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2)+2/a^2*c/(a^
2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-(x-1/a)^2*a^2*c-2*a*c*(x-1/a))^(1/2))

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maxima [A]  time = 0.64, size = 96, normalized size = 0.86 \[ -\frac {1}{24} \, a {\left (\frac {27 \, \sqrt {-a^{2} c x^{2} + c} x}{a^{3}} - \frac {6 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{3} c} - \frac {21 \, \sqrt {c} \arcsin \left (a x\right )}{a^{4}} + \frac {48 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4}} - \frac {16 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{4} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*a*(27*sqrt(-a^2*c*x^2 + c)*x/a^3 - 6*(-a^2*c*x^2 + c)^(3/2)*x/(a^3*c) - 21*sqrt(c)*arcsin(a*x)/a^4 + 48*
sqrt(-a^2*c*x^2 + c)/a^4 - 16*(-a^2*c*x^2 + c)^(3/2)/(a^4*c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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