3.3.94 \(\int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)}{x^4} \, dx\) [294]

Optimal. Leaf size=22 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6263, 270} \begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x))/x^4,x]

[Out]

-1/3*(c*(1 - a^2*x^2)^(3/2))/x^3

Rule 270

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x))/x^4,x]

[Out]

-1/3*(c*(1 - a^2*x^2)^(3/2))/x^3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).
time = 0.47, size = 42, normalized size = 1.91

method	result	size
trager	\(\frac {c \left (a^{2} x^{2}-1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\)	\(28\)
gosper	\(-\frac {c \left (a x +1\right )^{2} \left (a x -1\right )^{2}}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}\)	\(33\)
risch	\(-\frac {c \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}\)	\(36\)
default	\(-c \left (\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )\)	\(42\)
meijerg	\(\frac {a^{2} c \sqrt {-a^{2} x^{2}+1}}{x}-\frac {c \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
time = 0.47, size = 40, normalized size = 1.82 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c}{3 \, x} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 29, normalized size = 1.32 \begin {gather*} \frac {{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.85, size = 133, normalized size = 6.05 \begin {gather*} - a^{2} c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + c*Piecewi
se((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-
a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (18) = 36\).
time = 0.42, size = 124, normalized size = 5.64 \begin {gather*} \frac {{\left (a^{4} c - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)/x^4,x, algorithm="giac")

[Out]

1/24*(a^4*c - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c/x^2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) +
1/24*(3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c/x - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c/x^3)/(a^2*abs(a))

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Mupad [B]
time = 0.04, size = 18, normalized size = 0.82 \begin {gather*} -\frac {c\,{\left (1-a^2\,x^2\right )}^{3/2}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*(1 - a^2*x^2)^(3/2))/(3*x^3)

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