3.9.9 \(\int e^{-\coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [809]

Optimal. Leaf size=108 \[ \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6329, 99, 159, 21, 132, 41, 222, 94, 214} \begin {gather*} c x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}+\frac {2 c \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{a}+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))/E^ArcCoth[a*x],x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 99

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

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m
+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo
Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n,
-1]))

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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 6329

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.51 \begin {gather*} \frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (1+a x)+\text {ArcSin}\left (\frac {1}{a x}\right )-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))/E^ArcCoth[a*x],x]

[Out]

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

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Maple [A]
time = 0.06, size = 166, normalized size = 1.54

method	result	size
risch	\(\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (a \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\frac {a^{2} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{2} \left (a x -1\right )}\)	\(133\)
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -\sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x \right )}{\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 117, normalized size = 1.08 \begin {gather*} -a {\left (\frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 107, normalized size = 0.99 \begin {gather*} -\frac {2 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 121, normalized size = 1.12 \begin {gather*} -\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.28, size = 84, normalized size = 0.78 \begin {gather*} \frac {4\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c\,\mathrm {atan}\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^2*x^2))*((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

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

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