3.7.50 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^2} \, dx\) [650]

Optimal. Leaf size=125 \[ \frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {3 \text {ArcSin}(a x)}{a c^2} \]

[Out]

1/5*(a*x+1)^3/a/c^2/(-a^2*x^2+1)^(5/2)-6/5*(a*x+1)^2/a/c^2/(-a^2*x^2+1)^(3/2)-3*arcsin(a*x)/a/c^2+24/5*(a*x+1)
/a/c^2/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a/c^2

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Rubi [A]
time = 0.23, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6292, 6283, 1649, 655, 222} \begin {gather*} \frac {(a x+1)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (a x+1)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {3 \text {ArcSin}(a x)}{a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^3/(5*a*c^2*(1 - a^2*x^2)^(5/2)) - (6*(1 + a*x)^2)/(5*a*c^2*(1 - a^2*x^2)^(3/2)) + (24*(1 + a*x))/(5*
a*c^2*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^2) - (3*ArcSin[a*x])/(a*c^2)

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 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]

Rule 6292

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

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=\frac {a^4 \int \frac {e^{3 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^4 \int \frac {(1+a x)^2 \left (\frac {3}{a^4}+\frac {5 x}{a^3}+\frac {5 x^2}{a^2}+\frac {5 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^2}\\ &=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^4 \int \frac {(1+a x) \left (\frac {27}{a^4}+\frac {30 x}{a^3}+\frac {15 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^2}\\ &=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^2 \sqrt {1-a^2 x^2}}-\frac {a^4 \int \frac {\frac {45}{a^4}+\frac {15 x}{a^3}}{\sqrt {1-a^2 x^2}} \, dx}{15 c^2}\\ &=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {3 \sin ^{-1}(a x)}{a c^2}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 88, normalized size = 0.70 \begin {gather*} \frac {24-33 a x-18 a^2 x^2+34 a^3 x^3-5 a^4 x^4-15 (-1+a x)^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{5 a c^2 (-1+a x)^2 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24 - 33*a*x - 18*a^2*x^2 + 34*a^3*x^3 - 5*a^4*x^4 - 15*(-1 + a*x)^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(5*a*c^2*(
-1 + a*x)^2*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. \(373\) vs. \(2(111)=222\).
time = 0.73, size = 374, normalized size = 2.99

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}+\frac {6 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {24 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4}}{c^{2}}\)	\(193\)
default	\(\frac {a^{4} \left (\frac {-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}}{a}+\frac {\frac {3 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {5}{a^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {7 x}{a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {2}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {6 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{6}}+\frac {\frac {3}{a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {3 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{5}}\right )}{c^{2}}\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4/c^2*(1/a*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+3/a^2*(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a^2
)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))+5/a^5/(-a^2*x^2+1)^(1/2)+7*x/a^4/(-a^2*x^2+1)^(1/2)+2/a^6*(1
/5/a/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-3/5*a*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/3/
a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))+9/a^5*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^
(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)/(c-c/a^2/x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 143, normalized size = 1.14 \begin {gather*} \frac {24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right )} \sqrt {-a^{2} x^{2} + 1} - 24}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\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)/(c-c/a^2/x^2)^2,x, algorithm="fricas")

[Out]

1/5*(24*a^3*x^3 - 72*a^2*x^2 + 72*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(
a*x)) + (5*a^3*x^3 - 39*a^2*x^2 + 57*a*x - 24)*sqrt(-a^2*x^2 + 1) - 24)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c
^2*x - a*c^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{4} \left (\int \frac {x^{4}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \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)/(c-c/a**2/x**2)**2,x)

[Out]

a**4*(Integral(x**4/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**
2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2
*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2

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Giac [A]
time = 0.42, size = 180, normalized size = 1.44 \begin {gather*} -\frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} - \frac {2 \, {\left (\frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 19\right )}}{5 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \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)/(c-c/a^2/x^2)^2,x, algorithm="giac")

[Out]

-3*arcsin(a*x)*sgn(a)/(c^2*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c^2) - 2/5*(80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2
*x) - 120*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 15*(s
qrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 19)/(c^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a))

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Mupad [B]
time = 0.84, size = 272, normalized size = 2.18 \begin {gather*} \frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,c^2\,x^2-2\,a^6\,c^2\,x+a^5\,c^2\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}+\frac {24\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}+a^2\,c^2\,x^3\,\sqrt {-a^2}-3\,a\,c^2\,x^2\,\sqrt {-a^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^4*(1 - a^2*x^2)^(1/2))/(15*(a^5*c^2 - 2*a^6*c^2*x + a^7*c^2*x^2)) - (3*asinh(x*(-a^2)^(1/2)))/(c^2*(-a^2)
^(1/2)) - (4*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) + (1 - a^2*x^2)^(1/2)/(a*c^2) +
(24*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) - (c^2*(-a^2)^(1/2))/a)) + (1 - a^2*x^2)^(1/2)/(5
*(-a^2)^(1/2)*(3*c^2*x*(-a^2)^(1/2) - (c^2*(-a^2)^(1/2))/a + a^2*c^2*x^3*(-a^2)^(1/2) - 3*a*c^2*x^2*(-a^2)^(1/
2)))

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