3.1108 \(\int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{\sqrt{c-a^2 c x^2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{11 \sqrt{c-a^2 c x^2}}{3 a^4 c}+\frac{(a x+1)^2}{a^4 \sqrt{c-a^2 c x^2}}-\frac{3 \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^4 \sqrt{c}} \]

[Out]

(1 + a*x)^2/(a^4*Sqrt[c - a^2*c*x^2]) + (11*Sqrt[c - a^2*c*x^2])/(3*a^4*c) + (x*Sqrt[c - a^2*c*x^2])/(a^3*c) +
(x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*c) - (3*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^4*Sqrt[c])

________________________________________________________________________________________

Rubi [A]  time = 0.355805, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6151, 1635, 1815, 641, 217, 203} \[ \frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{11 \sqrt{c-a^2 c x^2}}{3 a^4 c}+\frac{(a x+1)^2}{a^4 \sqrt{c-a^2 c x^2}}-\frac{3 \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^2/(a^4*Sqrt[c - a^2*c*x^2]) + (11*Sqrt[c - a^2*c*x^2])/(3*a^4*c) + (x*Sqrt[c - a^2*c*x^2])/(a^3*c) +
(x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*c) - (3*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^4*Sqrt[c])

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]

Rule 1635

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 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

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 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 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])

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{\sqrt{c-a^2 c x^2}} \, dx &=c \int \frac{x^3 (1+a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}-\int \frac{(1+a x) \left (\frac{2}{a^3}+\frac{x}{a^2}+\frac{x^2}{a}\right )}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}+\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}+\frac{\int \frac{-\frac{6 c}{a}-11 c x-6 a c x^2}{\sqrt{c-a^2 c x^2}} \, dx}{3 a^2 c}\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}-\frac{\int \frac{18 a c^2+22 a^2 c^2 x}{\sqrt{c-a^2 c x^2}} \, dx}{6 a^4 c^2}\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}+\frac{11 \sqrt{c-a^2 c x^2}}{3 a^4 c}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}-\frac{3 \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{a^3}\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}+\frac{11 \sqrt{c-a^2 c x^2}}{3 a^4 c}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{a^3}\\ &=\frac{(1+a x)^2}{a^4 \sqrt{c-a^2 c x^2}}+\frac{11 \sqrt{c-a^2 c x^2}}{3 a^4 c}+\frac{x \sqrt{c-a^2 c x^2}}{a^3 c}+\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a^2 c}-\frac{3 \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^4 \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.171213, size = 97, normalized size = 0.71 \[ \frac{\frac{\left (a^3 x^3+2 a^2 x^2+5 a x-14\right ) \sqrt{c-a^2 c x^2}}{a x-1}+9 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{3 a^4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[c - a^2*c*x^2]*(-14 + 5*a*x + 2*a^2*x^2 + a^3*x^3))/(-1 + a*x) + 9*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x
^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(3*a^4*c)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 149, normalized size = 1.1 \begin{align*}{\frac{{x}^{2}}{3\,{a}^{2}c}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{8}{3\,{a}^{4}c}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{x}{{a}^{3}c}\sqrt{-{a}^{2}c{x}^{2}+c}}-3\,{\frac{1}{{a}^{3}\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }-2\,{\frac{1}{{a}^{5}c}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \end{align*}

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)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.95172, size = 446, normalized size = 3.26 \begin{align*} \left [-\frac{9 \,{\left (a x - 1\right )} \sqrt{-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \,{\left (a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right )} \sqrt{-a^{2} c x^{2} + c}}{6 \,{\left (a^{5} c x - a^{4} c\right )}}, \frac{9 \,{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right )} \sqrt{-a^{2} c x^{2} + c}}{3 \,{\left (a^{5} c x - a^{4} c\right )}}\right ] \end{align*}

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)^(1/2),x, algorithm="fricas")

[Out]

[-1/6*(9*(a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*(a^3*x^3 + 2*a^2*x^
2 + 5*a*x - 14)*sqrt(-a^2*c*x^2 + c))/(a^5*c*x - a^4*c), 1/3*(9*(a*x - 1)*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*
a*sqrt(c)*x/(a^2*c*x^2 - c)) + (a^3*x^3 + 2*a^2*x^2 + 5*a*x - 14)*sqrt(-a^2*c*x^2 + c))/(a^5*c*x - a^4*c)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{3}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{4}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx \end{align*}

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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

undef