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

Optimal. Leaf size=105 \[ -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]

[Out]

-8/3*(a+1/x)/a^2/c/(1-1/a^2/x^2)^(3/2)+4*arctanh((1-1/a^2/x^2)^(1/2))/a/c-4/3*(3*a+4/x)/a^2/c/(1-1/a^2/x^2)^(1
/2)+x*(1-1/a^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.30, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(a + x^(-1)))/(3*a^2*c*(1 - 1/(a^2*x^2))^(3/2)) - (4*(3*a + 4/x))/(3*a^2*c*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[
1 - 1/(a^2*x^2)]*x)/c + (4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c)

Rule 63

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 208

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

Rule 266

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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 852

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 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 6177

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 \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac {c x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^4}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^5}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-3 c^4-\frac {12 c^4 x}{a}-\frac {13 c^4 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {3 c^4+\frac {12 c^4 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c^5}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 70, normalized size = 0.67 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a^2 x^2-26 a x+19\right )}{(a x-1)^2}+12 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{3 a c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 26*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + 12*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(
3*a*c)

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fricas [A]  time = 0.58, size = 128, normalized size = 1.22 \[ \frac {12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - 7 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

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),x, algorithm="fricas")

[Out]

1/3*(12*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 12*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)
/(a*x + 1)) - 1) + (3*a^3*x^3 - 23*a^2*x^2 - 7*a*x + 19)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a
*c)

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giac [A]  time = 0.18, size = 148, normalized size = 1.41 \[ \frac {2}{3} \, a {\left (\frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {6 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac {{\left (a x + 1\right )} {\left (\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]

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),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.06, size = 346, normalized size = 3.30 \[ \frac {12 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-36 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-9 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -36 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+36 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+7 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+36 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -12 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-12 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{3 a \left (a x -1\right ) \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]

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),x)

[Out]

1/3*(12*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4+12*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
1/2)*x^3*a^3-36*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-9*(a^2)^(1/2)*((a*x-1)*(a*
x+1))^(3/2)*x*a-36*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2+36*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2
))/(a^2)^(1/2))*x*a^2+7*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+36*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a-12*a*ln
((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))-12*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/a/(a*x-1)/(a
^2)^(1/2)/c/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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maxima [A]  time = 0.32, size = 133, normalized size = 1.27 \[ \frac {2}{3} \, a {\left (\frac {\frac {8 \, {\left (a x - 1\right )}}{a x + 1} - \frac {12 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]

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),x, algorithm="maxima")

[Out]

2/3*a*((8*(a*x - 1)/(a*x + 1) - 12*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c*((a
*x - 1)/(a*x + 1))^(3/2)) + 6*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 6*log(sqrt((a*x - 1)/(a*x + 1)) - 1
)/(a^2*c))

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mupad [B]  time = 1.24, size = 100, normalized size = 0.95 \[ \frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {16\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {8\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c) - ((16*(a*x - 1))/(3*(a*x + 1)) - (8*(a*x - 1)^2)/(a*x + 1)^2 + 2
/3)/(a*c*((a*x - 1)/(a*x + 1))^(3/2) - a*c*((a*x - 1)/(a*x + 1))^(5/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \int \frac {x}{\frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]

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),x)

[Out]

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

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