Integrand size = 16, antiderivative size = 78 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
1/3*a^2*c^2*(1-1/a^2/x^2)^(3/2)*x^3+1/2*c^2*arctanh((1-1/a^2/x^2)^(1/2))/a -1/2*a*c^2*x^2*(1-1/a^2/x^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-2-3 a x+2 a^2 x^2\right )+3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]
(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-2 - 3*a*x + 2*a^2*x^2) + 3*Log[a*(1 + Sq rt[1 - 1/(a^2*x^2)])*x]))/(6*a)
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6724, 25, 27, 534, 243, 51, 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 |
\(\displaystyle a c \int -c \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^4d\frac {1}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a c \int c \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^4d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -a c^2 \int \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^4d\frac {1}{x}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -a c^2 \left (-\int \sqrt {1-\frac {1}{a^2 x^2}} x^3d\frac {1}{x}-\frac {1}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -a c^2 \left (-\frac {1}{2} \int \sqrt {1-\frac {1}{a^2 x^2}} x^2d\frac {1}{x^2}-\frac {1}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle -a c^2 \left (\frac {1}{2} \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}}{2 a^2}+x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -a c^2 \left (\frac {1}{2} \left (x \sqrt {1-\frac {1}{a^2 x^2}}-\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {1}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -a c^2 \left (\frac {1}{2} \left (x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}\right )-\frac {1}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
-(a*c^2*(-1/3*(a*(1 - 1/(a^2*x^2))^(3/2)*x^3) + (Sqrt[1 - 1/(a^2*x^2)]*x - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/a^2)/2))
3.2.60.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[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 = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a x -1\right ) c^{2}}{6 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(112\) |
default | \(-\frac {\left (a x -1\right ) c^{2} \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a}\) | \(121\) |
1/6*(2*a^2*x^2-3*a*x-2)*(a*x-1)/a*c^2/((a*x-1)/(a*x+1))^(1/2)+1/2*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)/(a*x+1))^ (1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.32 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 5 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]
1/6*(3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*c^2*log(sqrt((a*x - 1)/( a*x + 1)) - 1) + (2*a^3*c^2*x^3 - a^2*c^2*x^2 - 5*a*c^2*x - 2*c^2)*sqrt((a *x - 1)/(a*x + 1)))/a
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- \frac {2 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]
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(1/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. \(2 (66) = 132\).
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.32 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} \, a {\left (\frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 8 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]
1/6*a*(3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(3*c^2*((a*x - 1)/(a*x + 1))^(5/2) + 8*c^2*(( a*x - 1)/(a*x + 1))^(3/2) - 3*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 = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.26 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {2 \, a c^{2} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {2 \, c^{2}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
1/6*sqrt(a^2*x^2 - 1)*((2*a*c^2*x/sgn(a*x + 1) - 3*c^2/sgn(a*x + 1))*x - 2 *c^2/(a*sgn(a*x + 1))) - 1/2*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/( abs(a)*sgn(a*x + 1))
Time = 4.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.77 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {\frac {8\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]