3.4.97 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [397]

Optimal. Leaf size=61 \[ \frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6312, 283, 201, 222} \begin {gather*} c^3 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 283

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

Rule 6312

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 51, normalized size = 0.84 \begin {gather*} \frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (1+2 a^2 x^2\right )+3 a x \text {ArcSin}\left (\frac {1}{a x}\right )\right )}{2 a^2 x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 105, normalized size = 1.72

method	result	size
default	\(-\frac {\left (a x -1\right )^{2} c^{3} \left (-3 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}-3 a^{2} x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{2}}\)	\(105\)
risch	\(\frac {\left (a x -1\right ) c^{3}}{2 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}+a^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\right ) c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{3} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (53) = 106\).
time = 0.47, size = 151, normalized size = 2.48 \begin {gather*} -{\left (\frac {3 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - (3*c^3*((a*x - 1)/(a*x + 1))^(5/2) + 2*c^3*((a*x - 1)/(a*x + 1
))^(3/2) + 3*c^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x + 1)^2 - (a*x - 1)
^3*a^2/(a*x + 1)^3 + a^2))*a

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Fricas [A]
time = 0.33, size = 85, normalized size = 1.39 \begin {gather*} -\frac {6 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x + c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (2*a^3*c^3*x^3 + 2*a^2*c^3*x^2 + a*c^3*x + c^3)*sqrt((
a*x - 1)/(a*x + 1)))/(a^3*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.40, size = 75, normalized size = 1.23 \begin {gather*} -\frac {3 \, a^{4} c^{3} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) - 2 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} - \frac {\sqrt {a^{2} x^{2} - 1} a^{2} c^{3}}{x^{2}}}{2 \, a^{5} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*a^4*c^3*arctan(sqrt(a^2*x^2 - 1)) - 2*sqrt(a^2*x^2 - 1)*a^4*c^3 - sqrt(a^2*x^2 - 1)*a^2*c^3/x^2)/(a^5*
sgn(a*x + 1))

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Mupad [B]
time = 1.23, size = 119, normalized size = 1.95 \begin {gather*} \frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+c^3\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^2\,x}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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