3.9.26 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [826]

Optimal. Leaf size=144 \[ \frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \]

[Out]

-3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c+5/3*(1-1/a/x)^(1/2)/a/c/(1+1/a/x)^(3/2)+x*(1-1/a/x)^(1/2)/c/(1
+1/a/x)^(3/2)+14/3*(1-1/a/x)^(1/2)/a/c/(1+1/a/x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 101, 157, 12, 94, 214} \begin {gather*} \frac {x \sqrt {1-\frac {1}{a x}}}{c \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {\frac {1}{a x}+1}}+\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))),x]

[Out]

(5*Sqrt[1 - 1/(a*x)])/(3*a*c*(1 + 1/(a*x))^(3/2)) + (14*Sqrt[1 - 1/(a*x)])/(3*a*c*Sqrt[1 + 1/(a*x)]) + (Sqrt[1
- 1/(a*x)]*x)/(c*(1 + 1/(a*x))^(3/2)) - (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 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 \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}}}{x^2 \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-\frac {3}{a}+\frac {2 x}{a^2}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {9}{a^2}+\frac {5 x}{a^3}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {a^2 \text {Subst}\left (\int -\frac {9}{a^3 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \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 c}\\ &=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 69, normalized size = 0.48 \begin {gather*} \frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (14+19 a x+3 a^2 x^2\right )}{(1+a x)^2}-\frac {9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))),x]

[Out]

((Sqrt[1 - 1/(a^2*x^2)]*x*(14 + 19*a*x + 3*a^2*x^2))/(1 + a*x)^2 - (9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(
3*c)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(124)=248\).
time = 0.11, size = 346, normalized size = 2.40

method	result	size
risch	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {2 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{3 a^{5} \left (x +\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{3 a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c \left (a x -1\right )}\)	\(173\)
default	\(-\frac {\left (-9 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+9 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -27 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+5 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-27 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -9 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+9 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a \sqrt {a^{2}}\, \left (a x +1\right ) c \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )}\)	\(346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-9*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3+9*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1
/2))*a^4*x^3+6*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*a*x-27*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2+27*ln((a
^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^3*x^2+5*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)-27*((a*x+
1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a*x+27*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^2*x-9*(a^2)
^(1/2)*((a*x+1)*(a*x-1))^(1/2)+9*a*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2)))/a*((a*x-1)/(a*
x+1))^(3/2)/(a^2)^(1/2)/(a*x+1)/c/((a*x+1)*(a*x-1))^(1/2)/(a*x-1)

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Maxima [A]
time = 0.26, size = 140, normalized size = 0.97 \begin {gather*} -\frac {1}{3} \, a {\left (\frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 12 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a*(6*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c/(a*x + 1) - a^2*c) - (((a*x - 1)/(a*x + 1))^(3/2) + 12*sq
rt((a*x - 1)/(a*x + 1)))/(a^2*c) + 9*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(sqrt((a*x - 1)/(a*x +
1)) - 1)/(a^2*c))

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Fricas [A]
time = 0.41, size = 96, normalized size = 0.67 \begin {gather*} -\frac {9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (3 \, a^{2} x^{2} + 19 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} c x + a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(9*(a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (3*a^2
*x^2 + 19*a*x + 14)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c*x + a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 59, normalized size = 0.41 \begin {gather*} \frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 114, normalized size = 0.79 \begin {gather*} \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3\,a\,c}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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