Integrand size = 18, antiderivative size = 105 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {3}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {1}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5-\frac {3 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
-1/4*a^3*c^4*(1-1/a^2/x^2)^(3/2)*x^4+1/5*a^4*c^4*(1-1/a^2/x^2)^(5/2)*x^5-3 /8*c^4*arctanh((1-1/a^2/x^2)^(1/2))/a+3/8*a*c^4*x^2*(1-1/a^2/x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {c^4 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (8+25 a x-16 a^2 x^2-10 a^3 x^3+8 a^4 x^4\right )-15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a} \]
(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8* a^4*x^4) - 15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a)
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, 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, 534, 243, 51, 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.
| \(\displaystyle \int (c-a c x)^4 e^{3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle a^3 c^3 \int -c \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right ) x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 c^3 \int c \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right ) x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 c^4 \int \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right ) x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -a^3 c^4 \left (-\int \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5d\frac {1}{x}-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -a^3 c^4 \left (-\frac {1}{2} \int \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3d\frac {1}{x^2}-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -a^3 c^4 \left (\frac {1}{2} \left (\frac {3 \int \sqrt {1-\frac {1}{a^2 x^2}} x^2d\frac {1}{x^2}}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -a^3 c^4 \left (\frac {1}{2} \left (\frac {3 \left (x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}}{2 a^2}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -a^3 c^4 \left (\frac {1}{2} \left (\frac {3 \left (\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-x \sqrt {1-\frac {1}{a^2 x^2}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -a^3 c^4 \left (\frac {1}{2} \left (\frac {3 \left (\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}-x \sqrt {1-\frac {1}{a^2 x^2}}\right )}{4 a^2}+\frac {1}{2} x^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}\right )\) |
-(a^3*c^4*(-1/5*(a*(1 - 1/(a^2*x^2))^(5/2)*x^5) + (((1 - 1/(a^2*x^2))^(3/2 )*x^2)/2 + (3*(-(Sqrt[1 - 1/(a^2*x^2)]*x) + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] /a^2))/(4*a^2))/2))
3.2.78.3.1 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 = 0.47 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {\left (8 a^{4} x^{4}-10 a^{3} x^{3}-16 a^{2} x^{2}+25 a x +8\right ) \left (a x -1\right ) c^{4}}{40 a \sqrt {\frac {a x -1}{a x +1}}}-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(128\) |
| default | \(\frac {\left (a x -1\right )^{2} c^{4} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+45 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -40 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(192\) |
1/40*(8*a^4*x^4-10*a^3*x^3-16*a^2*x^2+25*a*x+8)*(a*x-1)/a*c^4/((a*x-1)/(a* x+1))^(1/2)-3/8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^4/(a *x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (8 \, a^{5} c^{4} x^{5} - 2 \, a^{4} c^{4} x^{4} - 26 \, a^{3} c^{4} x^{3} + 9 \, a^{2} c^{4} x^{2} + 33 \, a c^{4} x + 8 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a} \]
-1/40*(15*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (8*a^5*c^4*x^5 - 2*a^4*c^4*x^4 - 26*a^3*c^4*x^3 + 9*a ^2*c^4*x^2 + 33*a*c^4*x + 8*c^4)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=c^{4} \left (\int \left (- \frac {4 a x}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{2} x^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{3} x^{3}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{4} x^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right ) \]
c**4*(Integral(-4*a*x/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - s qrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(6*a**2*x**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) + Integral(-4*a**3*x**3/(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) + Integral(a**4*x**4/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqr t(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(1/(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))
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (89) = 178\).
Time = 0.21 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.47 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {1}{40} \, {\left (\frac {15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (15 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 70 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 128 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 70 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} a \]
-1/40*(15*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^4*log(sqrt((a* x - 1)/(a*x + 1)) - 1)/a^2 - 2*(15*c^4*((a*x - 1)/(a*x + 1))^(9/2) - 70*c^ 4*((a*x - 1)/(a*x + 1))^(7/2) - 128*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 70*c ^4*((a*x - 1)/(a*x + 1))^(3/2) - 15*c^4*sqrt((a*x - 1)/(a*x + 1)))/(5*(a*x - 1)*a^2/(a*x + 1) - 10*(a*x - 1)^2*a^2/(a*x + 1)^2 + 10*(a*x - 1)^3*a^2/ (a*x + 1)^3 - 5*(a*x - 1)^4*a^2/(a*x + 1)^4 + (a*x - 1)^5*a^2/(a*x + 1)^5 - a^2))*a
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.31 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {3 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {1}{40} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {25 \, c^{4}}{\mathrm {sgn}\left (a x + 1\right )} - 2 \, {\left (\frac {8 \, a c^{4}}{\mathrm {sgn}\left (a x + 1\right )} - {\left (\frac {4 \, a^{3} c^{4} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {5 \, a^{2} c^{4}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x\right )} x\right )} x + \frac {8 \, c^{4}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \]
3/8*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) + 1/ 40*sqrt(a^2*x^2 - 1)*((25*c^4/sgn(a*x + 1) - 2*(8*a*c^4/sgn(a*x + 1) - (4* a^3*c^4*x/sgn(a*x + 1) - 5*a^2*c^4/sgn(a*x + 1))*x)*x)*x + 8*c^4/(a*sgn(a* x + 1)))
Time = 0.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.04 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {\frac {3\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}+\frac {32\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{5}+\frac {7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{2}-\frac {3\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}-\frac {3\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
((3*c^4*((a*x - 1)/(a*x + 1))^(1/2))/4 - (7*c^4*((a*x - 1)/(a*x + 1))^(3/2 ))/2 + (32*c^4*((a*x - 1)/(a*x + 1))^(5/2))/5 + (7*c^4*((a*x - 1)/(a*x + 1 ))^(7/2))/2 - (3*c^4*((a*x - 1)/(a*x + 1))^(9/2))/4)/(a - (5*a*(a*x - 1))/ (a*x + 1) + (10*a*(a*x - 1)^2)/(a*x + 1)^2 - (10*a*(a*x - 1)^3)/(a*x + 1)^ 3 + (5*a*(a*x - 1)^4)/(a*x + 1)^4 - (a*(a*x - 1)^5)/(a*x + 1)^5) - (3*c^4* atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)