\(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x}) \, dx\) [399]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {c \csc ^{-1}(a x)}{a}+\frac {2 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6312, 866, 1821, 858, 222, 272, 65, 214} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+c x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {c \csc ^{-1}(a x)}{a} \]

[In]

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

[Out]

c*Sqrt[1 - 1/(a^2*x^2)]*x - (c*ArcCsc[a*x])/a + (2*c*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 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 866

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x-2 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-2 \arcsin \left (\frac {1}{a x}\right )+2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(45)=90\).

Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.96

method	result	size
default	\(-\frac {\left (a x -1\right )^{2} c \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}-2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\)	\(145\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.80 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a} \]

[In]

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

[Out]

(2*c*arctan(sqrt((a*x - 1)/(a*x + 1))) + 2*c*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 2*c*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1) + (a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/a

Sympy [F]

\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int \frac {a}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a} \]

[In]

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

[Out]

c*(Integral(a/(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**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))/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (45) = 90\).

Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.33 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-2 \, a {\left (\frac {c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} - \frac {c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (45) = 90\).

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.86 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {2 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c}{a \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {4\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {2\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}} \]

[In]

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

[Out]

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