Integrand size = 18, antiderivative size = 100 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {5 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
-5/2*c^2*arctanh((1-1/a^2/x^2)^(1/2))/a+11/3*c^2*x*(1-1/a^2/x^2)^(1/2)-3/2 *a*c^2*x^2*(1-1/a^2/x^2)^(1/2)+1/3*a^2*c^2*x^3*(1-1/a^2/x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.64 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (22-9 a x+2 a^2 x^2\right )-15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]
(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(22 - 9*a*x + 2*a^2*x^2) - 15*Log[a*(1 + S qrt[1 - 1/(a^2*x^2)])*x]))/(6*a)
Time = 0.40 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6724, 25, 27, 540, 2338, 534, 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.
\(\displaystyle \int (c-a c x)^2 e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6724 |
\(\displaystyle \frac {\int -\frac {c^3 \left (a-\frac {1}{x}\right )^3 x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {c^3 \left (a-\frac {1}{x}\right )^3 x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {c^2 \int \frac {\left (a-\frac {1}{x}\right )^3 x^4}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}\) |
\(\Big \downarrow \) 540 |
\(\displaystyle -\frac {c^2 \left (-\frac {1}{3} \int \frac {\left (9 a^2-\frac {11 a}{x}+\frac {3}{x^2}\right ) x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle -\frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (22 a-\frac {15}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {9}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \left (-15 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-22 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {9}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {15}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-22 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {9}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \left (15 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-22 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {9}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c^2 \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-22 a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {9}{2} a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\) |
-((c^2*(-1/3*(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3) + ((9*a^2*Sqrt[1 - 1/(a^2*x^2 )]*x^2)/2 + (-22*a*Sqrt[1 - 1/(a^2*x^2)]*x + 15*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]])/2)/3))/a)
3.2.99.3.1 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_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^n Subst[Int[(d + c*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(p + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0] && IntegerQ[p] && In tegerQ[n]
Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-9 a x +22\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{6 a}-\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(112\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -24 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\) | \(176\) |
1/6*(2*a^2*x^2-9*a*x+22)*(a*x+1)/a*c^2*((a*x-1)/(a*x+1))^(1/2)-5/2*ln(a^2* x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^2*((a*x-1)/(a*x+1))^(1/2)/( a*x-1)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{2} x^{3} - 7 \, a^{2} c^{2} x^{2} + 13 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]
-1/6*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^2*log(sqrt((a*x - 1 )/(a*x + 1)) - 1) - (2*a^3*c^2*x^3 - 7*a^2*c^2*x^2 + 13*a*c^2*x + 22*c^2)* sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- 2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]
c**2*(Integral(-2*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a** 2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x))
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (84) = 168\).
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.81 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{6} \, a {\left (\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {2 \, {\left (33 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 40 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]
-1/6*a*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^2*log(sqrt((a *x - 1)/(a*x + 1)) - 1)/a^2 + 2*(33*c^2*((a*x - 1)/(a*x + 1))^(5/2) - 40*c ^2*((a*x - 1)/(a*x + 1))^(3/2) + 15*c^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)*a^2/(a*x + 1) - 3*(a*x - 1)^2*a^2/(a*x + 1)^2 + (a*x - 1)^3*a^2/(a*x + 1)^3 - a^2))
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {5 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, a c^{2} x \mathrm {sgn}\left (a x + 1\right ) - 9 \, c^{2} \mathrm {sgn}\left (a x + 1\right )\right )} x + \frac {22 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]
5/2*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/6* sqrt(a^2*x^2 - 1)*((2*a*c^2*x*sgn(a*x + 1) - 9*c^2*sgn(a*x + 1))*x + 22*c^ 2*sgn(a*x + 1)/a)
Time = 4.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.40 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {5\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {40\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+11\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}-\frac {5\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]