Integrand size = 27, antiderivative size = 88 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
-arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^4+1/3*(-2*e*x+3*d)/d^4/(-e^2*x^2+d^2)^( 1/2)+1/3/d^2/(e*x+d)/(-e^2*x^2+d^2)^(1/2)
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (4 d^2+d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{(d-e x) (d+e x)^2}-3 \sqrt {d^2} \log (x)+3 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{3 d^5} \]
((d*(4*d^2 + d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/((d - e*x)*(d + e*x)^ 2) - 3*Sqrt[d^2]*Log[x] + 3*Sqrt[d^2]*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]] )/(3*d^5)
Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {569, 25, 532, 27, 243, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 569 |
\(\displaystyle \frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {3 d-2 e x}{x \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 d-2 e x}{x \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {3 d-2 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {3 d}{x \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx}{d}+\frac {3 d-2 e x}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {3 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2}{2 d}+\frac {3 d-2 e x}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {3 d-2 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {3 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{d e^2}}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {3 d-2 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}}{3 d^2}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}\) |
1/(3*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]) + ((3*d - 2*e*x)/(d^2*Sqrt[d^2 - e ^2*x^2]) - (3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^2)/(3*d^2)
3.2.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Simp[(-x^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*p*(c + d*x))), x] + Simp[1/(2 *c^2*p) Int[x^m*(a + b*x^2)^p*(c*(m + 2*p + 1) - d*(m + 2*p + 2)*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[m + 2*p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(78)=156\).
Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d}-\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d}\) | \(171\) |
1/d*(1/d^2/(-e^2*x^2+d^2)^(1/2)-1/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*(-1/3/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d /e))^(1/2)-1/3/e/d^3*(-2*(x+d/e)*e^2+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e)) ^(1/2))
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {4 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} - 4 \, d^{2} e x - 4 \, d^{3} + 3 \, {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (2 \, e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} - d^{6} e x - d^{7}\right )}} \]
1/3*(4*e^3*x^3 + 4*d*e^2*x^2 - 4*d^2*e*x - 4*d^3 + 3*(e^3*x^3 + d*e^2*x^2 - d^2*e*x - d^3)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (2*e^2*x^2 - d*e*x - 4*d^2)*sqrt(-e^2*x^2 + d^2))/(d^4*e^3*x^3 + d^5*e^2*x^2 - d^6*e*x - d^7)
\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x} \,d x } \]
\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]