3.12.16 \(\int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{(c-a^2 c x^2)^{3/2}} \, dx\) [1116]

Optimal. Leaf size=117 \[ \frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {2 \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^4 c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6286, 1649, 655, 223, 209} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^4 c^{3/2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {(a x+1)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (a x+1)}{3 a^4 c \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^2/(3*a^4*(c - a^2*c*x^2)^(3/2)) - (8*(1 + a*x))/(3*a^4*c*Sqrt[c - a^2*c*x^2]) - Sqrt[c - a^2*c*x^2]/
(a^4*c^2) + (2*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^4*c^(3/2))

Rule 209

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

Rule 223

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 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 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 6286

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 {x^3 (1+a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{3} \int \frac {(1+a x) \left (\frac {2}{a^3}+\frac {3 x}{a^2}+\frac {3 x^2}{a}\right )}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}+\frac {\int \frac {\frac {6}{a^3}+\frac {3 x}{a^2}}{\sqrt {c-a^2 c x^2}} \, dx}{3 c}\\ &=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {2 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a^3 c}\\ &=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {2 \text {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 c}\\ &=\frac {(1+a x)^2}{3 a^4 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 (1+a x)}{3 a^4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a^4 c^2}+\frac {2 \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^4 c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 90, normalized size = 0.77 \begin {gather*} \frac {\frac {\left (-10+14 a x-3 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{(-1+a x)^2}-6 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{3 a^4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-10 + 14*a*x - 3*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(-1 + a*x)^2 - 6*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(S
qrt[c]*(-1 + a^2*x^2))])/(3*a^4*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(239\) vs. \(2(101)=202\).
time = 0.08, size = 240, normalized size = 2.05

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c}+\frac {\frac {2 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a^{3} \sqrt {c \,a^{2}}}+\frac {8 \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{3 a^{5} c \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}}{c}\)	\(164\)
default	\(\frac {x^{2}}{c \,a^{2} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {4}{c \,a^{4} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {x}{c \,a^{2} \sqrt {-a^{2} c \,x^{2}+c}}-\frac {\arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{c \,a^{2} \sqrt {c \,a^{2}}}\right )}{a}-\frac {2 x}{a^{3} c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 \left (\frac {1}{3 a c \left (x -\frac {1}{a}\right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c}{3 a \,c^{2} \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{a^{4}}\)	\(240\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (101) = 202\).
time = 0.55, size = 260, normalized size = 2.22 \begin {gather*} \frac {1}{3} \, {\left (\frac {a^{3}}{\sqrt {-a^{2} c x^{2} + c} a^{9} c x + \sqrt {-a^{2} c x^{2} + c} a^{8} c} - \frac {a^{3}}{\sqrt {-a^{2} c x^{2} + c} a^{9} c x - \sqrt {-a^{2} c x^{2} + c} a^{8} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{7} c x + \sqrt {-a^{2} c x^{2} + c} a^{6} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{7} c x - \sqrt {-a^{2} c x^{2} + c} a^{6} c} + \frac {3 \, x^{2}}{\sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} a^{4} c} + \frac {6 \, \arcsin \left (a x\right )}{a^{5} c^{\frac {3}{2}}} - \frac {12}{\sqrt {-a^{2} c x^{2} + c} a^{5} c}\right )} a \end {gather*}

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]

1/3*(a^3/(sqrt(-a^2*c*x^2 + c)*a^9*c*x + sqrt(-a^2*c*x^2 + c)*a^8*c) - a^3/(sqrt(-a^2*c*x^2 + c)*a^9*c*x - sqr
t(-a^2*c*x^2 + c)*a^8*c) - a/(sqrt(-a^2*c*x^2 + c)*a^7*c*x + sqrt(-a^2*c*x^2 + c)*a^6*c) - a/(sqrt(-a^2*c*x^2
+ c)*a^7*c*x - sqrt(-a^2*c*x^2 + c)*a^6*c) + 3*x^2/(sqrt(-a^2*c*x^2 + c)*a^3*c) - 8*x/(sqrt(-a^2*c*x^2 + c)*a^
4*c) + 6*arcsin(a*x)/(a^5*c^(3/2)) - 12/(sqrt(-a^2*c*x^2 + c)*a^5*c))*a

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Fricas [A]
time = 0.35, size = 229, normalized size = 1.96 \begin {gather*} \left [-\frac {3 \, {\left (a^{2} x^{2} - 2 \, 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 ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a^{2} x^{2} - 14 \, a x + 10\right )}}{3 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}}, -\frac {6 \, {\left (a^{2} x^{2} - 2 \, 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 ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a^{2} x^{2} - 14 \, a x + 10\right )}}{3 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}}\right ] \end {gather*}

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]

[-1/3*(3*(a^2*x^2 - 2*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) + sqrt(-a^2
*c*x^2 + c)*(3*a^2*x^2 - 14*a*x + 10))/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2), -1/3*(6*(a^2*x^2 - 2*a*x + 1)*sq
rt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(3*a^2*x^2 - 14*a*x + 10
))/(a^6*c^2*x^2 - 2*a^5*c^2*x + a^4*c^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{3}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{4}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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