\(\int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx\) [97]

Optimal result

Integrand size = 27, antiderivative size = 46 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

-arctan(e*x/(-e^2*x^2+d^2)^(1/2))-arctanh((-e^2*x^2+d^2)^(1/2)/d)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {864, 858, 223, 209, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=-\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[In]

Int[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)),x]

[Out]

-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 864

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {d-e x}{x \sqrt {d^2-e^2 x^2}} \, dx \\ & = d \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2} \\ & = -\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2} \left (-\log (x)+\log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )\right )}{d} \]

[In]

Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)),x]

[Out]

2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + (Sqrt[d^2]*(-Log[x] + Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]
))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs. \(2(42)=84\).

Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.07

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}}{d}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}}{d}\)	\(141\)

[In]

int((-e^2*x^2+d^2)^(1/2)/x/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/d*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x))-1/d*((-(x+d/e)^2*e
^2+2*d*e*(x+d/e))^(1/2)+d*e/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=2 \, \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) \]

[In]

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

[Out]

2*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + log(-(d - sqrt(-e^2*x^2 + d^2))/x)

Sympy [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x \left (d + e x\right )}\, dx \]

[In]

integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x*(d + e*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=-\frac {e {\left (\frac {d \arcsin \left (\frac {e x}{d}\right )}{e} + \frac {d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{e}\right )}}{d} \]

[In]

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

[Out]

-e*(d*arcsin(e*x/d)/e + d*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/e)/d

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=-\frac {e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} \]

[In]

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

[Out]

-e*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - e*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/ab
s(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x\,\left (d+e\,x\right )} \,d x \]

[In]

int((d^2 - e^2*x^2)^(1/2)/(x*(d + e*x)),x)

[Out]

int((d^2 - e^2*x^2)^(1/2)/(x*(d + e*x)), x)