3.4.80 \(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [380]

Optimal. Leaf size=88 \[ \frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (4 a+\frac {1}{x}\right )}{2 a^2}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {c^3 \csc ^{-1}(a x)}{2 a}-\frac {2 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6312, 1821, 829, 858, 222, 272, 65, 214} \begin {gather*} \frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (4 a+\frac {1}{x}\right )}{2 a^2}+c^3 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {2 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^3 \csc ^{-1}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^3,x]

[Out]

(c^3*Sqrt[1 - 1/(a^2*x^2)]*(4*a + x^(-1)))/(2*a^2) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (c^3*ArcCsc[a*x])/(2*a) -
(2*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 829

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

Rule 858

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

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

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

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Mathematica [A]
time = 0.10, size = 167, normalized size = 1.90 \begin {gather*} \frac {c^3 \left (1-4 a x-3 a^2 x^2+4 a^3 x^3+2 a^4 x^4+2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \text {ArcSin}\left (\frac {1}{a x}\right )-4 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs. \(2(78)=156\).
time = 0.06, size = 200, normalized size = 2.27

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (2 a^{2} x^{2}+4 a x -1\right ) c^{3}}{2 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {2 a^{3} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(140\)
default	\(-\frac {\left (a x -1\right ) c^{3} \left (-4 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+4 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+4 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{2} \sqrt {a^{2}}}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (78) = 156\).
time = 0.50, size = 201, normalized size = 2.28 \begin {gather*} -{\left (\frac {c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {2 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {2 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {3 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 6 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 5 \, 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))^(1/2)*(c-c/a/x)^3,x, algorithm="maxima")

[Out]

-(c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 2*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 2*c^3*log(sqrt((a
*x - 1)/(a*x + 1)) - 1)/a^2 + (3*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 5*c^3*s
qrt((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 = 0.36, size = 146, normalized size = 1.66 \begin {gather*} -\frac {2 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 4 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 4 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 6 \, a^{2} c^{3} x^{2} + 3 \, 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))^(1/2)*(c-c/a/x)^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + 4*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 4*a
^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^3*x^3 + 6*a^2*c^3*x^2 + 3*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 {a^{3}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {3 a}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{2}}{x \sqrt {\frac {a x}{a x + 1} - \frac {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))**(1/2)*(c-c/a/x)**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (78) = 156\).
time = 0.42, size = 221, normalized size = 2.51 \begin {gather*} -\frac {c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} + 4 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} + 4 \, a c^{3}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2} a {\left | a \right |} \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))^(1/2)*(c-c/a/x)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.11, size = 163, normalized size = 1.85 \begin {gather*} \frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+6\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-3\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a+\frac {a\,\left (a\,x-1\right )}{a\,x+1}-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}-\frac {c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {4\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \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))^(1/2),x)

[Out]

(5*c^3*((a*x - 1)/(a*x + 1))^(1/2) + 6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 3*c^3*((a*x - 1)/(a*x + 1))^(5/2))/(a
+ (a*(a*x - 1))/(a*x + 1) - (a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) - (c^3*atan(((a*x - 1)
/(a*x + 1))^(1/2)))/a - (4*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a

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