3.11.35 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^2}{x^3} \, dx\) [1035]

Optimal. Leaf size=17 \[ -\frac {c^2 (1+a x)^4}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 75} \begin {gather*} -\frac {c^2 (a x+1)^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c^2*(1 + a*x)^4)/x^2

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 6285

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

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x^3} \, dx &=c^2 \int \frac {(1-a x) (1+a x)^3}{x^3} \, dx\\ &=-\frac {c^2 (1+a x)^4}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {c^2 (1+a x)^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c^2*(1 + a*x)^4)/x^2

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 31, normalized size = 1.82

method	result	size
gosper	\(-\frac {c^{2} \left (a^{4} x^{4}+4 a^{3} x^{3}+4 a x +1\right )}{2 x^{2}}\)	\(30\)
default	\(c^{2} \left (-\frac {a^{4} x^{2}}{2}-2 a^{3} x -\frac {2 a}{x}-\frac {1}{2 x^{2}}\right )\)	\(31\)
risch	\(-\frac {a^{4} c^{2} x^{2}}{2}-2 a^{3} c^{2} x +\frac {-2 a \,c^{2} x -\frac {1}{2} c^{2}}{x^{2}}\)	\(39\)
norman	\(\frac {-\frac {1}{2} c^{2}-2 a \,c^{2} x -2 a^{3} c^{2} x^{3}-\frac {1}{2} a^{4} c^{2} x^{4}}{x^{2}}\)	\(40\)
meijerg	\(\frac {a^{2} c^{2} \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2}+\frac {a^{2} c^{2} \ln \left (-a^{2} x^{2}+1\right )}{2}-\frac {a^{2} c^{2} \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2}-\frac {a^{3} c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}-4 a^{2} c^{2} \arctanh \left (a x \right )-\frac {a^{3} c^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}-\frac {a^{2} c^{2} \left (\ln \left (-a^{2} x^{2}+1\right )-2 \ln \left (x \right )-\ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{2}\)	\(221\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.26, size = 37, normalized size = 2.18 \begin {gather*} -\frac {1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac {4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.31, size = 37, normalized size = 2.18 \begin {gather*} -\frac {a^{4} c^{2} x^{4} + 4 \, a^{3} c^{2} x^{3} + 4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
time = 0.07, size = 39, normalized size = 2.29 \begin {gather*} - \frac {a^{4} c^{2} x^{2}}{2} - 2 a^{3} c^{2} x - \frac {4 a c^{2} x + c^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.41, size = 37, normalized size = 2.18 \begin {gather*} -\frac {1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac {4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 29, normalized size = 1.71 \begin {gather*} -\frac {c^2\,\left (a^4\,x^4+4\,a^3\,x^3+4\,a\,x+1\right )}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________