3.5.70 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) [470]

Optimal. Leaf size=99 \[ \frac {4 \sqrt {1-a^2 x^2}}{a c}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac {4 \text {ArcSin}(a x)}{a c} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6266, 6263, 807, 677, 679, 222} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}+\frac {4 \sqrt {1-a^2 x^2}}{a c}-\frac {4 \text {ArcSin}(a x)}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[1 - a^2*x^2])/(a*c) + (8*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^2) - (1 - a^2*x^2)^(5/2)/(3*a*c*(1 - a*
x)^4) - (4*ArcSin[a*x])/(a*c)

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 677

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

Rule 679

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

Rule 807

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

Rule 6263

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

Rule 6266

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 62, normalized size = 0.63 \begin {gather*} -\frac {(1+a x)^{5/2}+16 \sqrt {2} (-1+a x) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((1 + a*x)^(5/2) + 16*Sqrt[2]*(-1 + a*x)*Hypergeometric2F1[-3/2, -1/2, 1/2, (1 - a*x)/2])/(a*c*(1 - a*x)^
(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(89)=178\).
time = 0.76, size = 230, normalized size = 2.32

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {20 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{3} \left (x -\frac {1}{a}\right )}\right ) a}{c}\)	\(150\)
default	\(\frac {a \left (a^{2} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+4 a \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {7}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8 x}{a \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {8}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{2}}\right )}{c}\)	\(230\)

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/x),x,method=_RETURNVERBOSE)

[Out]

a/c*(a^2*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+4*a*(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)))+7/a^2/(-a^2*x^2+1)^(1/2)+8*x/a/(-a^2*x^2+1)^(1/2)+8/a^2*(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/x),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 101, normalized size = 1.02 \begin {gather*} \frac {19 \, a^{2} x^{2} - 38 \, a x + 24 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 26 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} + 19}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\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/x),x, algorithm="fricas")

[Out]

1/3*(19*a^2*x^2 - 38*a*x + 24*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 26*a
*x + 19)*sqrt(-a^2*x^2 + 1) + 19)/(a^3*c*x^2 - 2*a^2*c*x + a*c)

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

[Out]

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

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Giac [A]
time = 0.42, size = 126, normalized size = 1.27 \begin {gather*} -\frac {4 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} - \frac {8 \, {\left (\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 4\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\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/x),x, algorithm="giac")

[Out]

-4*arcsin(a*x)*sgn(a)/(c*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c) - 8/3*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) -
3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) - 4)/(c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^3*abs(a))

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Mupad [B]
time = 0.06, size = 129, normalized size = 1.30 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {20\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - a^2*x^2)^(1/2)/(a*c) - (20*(1 - a^2*x^2)^(1/2))/(3*((c*(-a^2)^(1/2))/a - c*x*(-a^2)^(1/2))*(-a^2)^(1/2))
- (4*asinh(x*(-a^2)^(1/2)))/(c*(-a^2)^(1/2)) - (4*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c + a^4*c*x^2 - 2*a^3*c*x))

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