Integrand size = 16, antiderivative size = 132 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {7}{8} a c^4 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {17}{15} a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {3}{4} a^3 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{5} a^4 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5+\frac {7 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
17/15*a^2*c^4*(1-1/a^2/x^2)^(3/2)*x^3-3/4*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)^(3/2)*x^5+7/8*c^4*arctanh((1-1/a^2/x^2)^(1/2))/a -7/8*a*c^4*x^2*(1-1/a^2/x^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {c^4 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+105 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{120 a} \]
(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a)
Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6724, 25, 27, 540, 2338, 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.
| \(\displaystyle \int (c-a c x)^4 e^{\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6724 |
| \(\displaystyle a c \int -c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3 x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a c \int c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3 x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a c^4 \int \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3 x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -a c^4 \left (-\frac {1}{5} \int \sqrt {1-\frac {1}{a^2 x^2}} \left (15 a^2-\frac {17 a}{x}+\frac {5}{x^2}\right ) x^5d\frac {1}{x}-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \int \sqrt {1-\frac {1}{a^2 x^2}} \left (68 a-\frac {35}{x}\right ) x^4d\frac {1}{x}+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \left (-35 \int \sqrt {1-\frac {1}{a^2 x^2}} x^3d\frac {1}{x}-\frac {68}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {35}{2} \int \sqrt {1-\frac {1}{a^2 x^2}} x^2d\frac {1}{x^2}-\frac {68}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {35}{2} \left (x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}}{2 a^2}\right )-\frac {68}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {35}{2} \left (\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {68}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -a c^4 \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {35}{2} \left (\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2}-x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {68}{3} a x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )+\frac {15}{4} a^2 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )-\frac {1}{5} a^3 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
-(a*c^4*(-1/5*(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^5) + ((15*a^2*(1 - 1/(a^2*x^2 ))^(3/2)*x^4)/4 + ((-68*a*(1 - 1/(a^2*x^2))^(3/2)*x^3)/3 - (35*(-(Sqrt[1 - 1/(a^2*x^2)]*x) + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/a^2))/2)/4)/5))
3.2.58.3.1 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[(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 = 128, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a x -1\right ) c^{4}}{120 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {7 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(128\) |
| default | \(\frac {\left (a x -1\right ) c^{4} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-90 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +120 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(183\) |
1/120*(24*a^4*x^4-90*a^3*x^3+112*a^2*x^2-15*a*x-136)*(a*x-1)/a*c^4/((a*x-1 )/(a*x+1))^(1/2)+7/8*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 = 125, normalized size of antiderivative = 0.95 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (24 \, a^{5} c^{4} x^{5} - 66 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 97 \, a^{2} c^{4} x^{2} - 151 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a} \]
1/120*(105*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (24*a^5*c^4*x^5 - 66*a^4*c^4*x^4 + 22*a^3*c^4*x^3 + 97*a^2*c^4*x^2 - 151*a*c^4*x - 136*c^4)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=c^{4} \left (\int \left (- \frac {4 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{3} x^{3}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{4} x^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]
c**4*(Integral(-4*a*x/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(6*a **2*x**2/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(-4*a**3*x**3/sqr t(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a**4*x**4/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. 259 vs. \(2 (112) = 224\).
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.96 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {1}{120} \, {\left (\frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (105 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 790 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 896 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 490 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} a \]
1/120*(105*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^4*log(sqrt(( a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(105*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 79 0*c^4*((a*x - 1)/(a*x + 1))^(7/2) - 896*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 490*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 105*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 = 1.05 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=-\frac {7 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} - \frac {1}{120} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {15 \, c^{4}}{\mathrm {sgn}\left (a x + 1\right )} - 2 \, {\left (\frac {56 \, a c^{4}}{\mathrm {sgn}\left (a x + 1\right )} + 3 \, {\left (\frac {4 \, a^{3} c^{4} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {15 \, a^{2} c^{4}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x\right )} x\right )} x + \frac {136 \, c^{4}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \]
-7/8*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) - 1 /120*sqrt(a^2*x^2 - 1)*((15*c^4/sgn(a*x + 1) - 2*(56*a*c^4/sgn(a*x + 1) + 3*(4*a^3*c^4*x/sgn(a*x + 1) - 15*a^2*c^4/sgn(a*x + 1))*x)*x)*x + 136*c^4/( a*sgn(a*x + 1)))
Time = 4.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.62 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx=\frac {\frac {49\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}-\frac {7\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {224\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{15}+\frac {79\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{6}+\frac {7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}+\frac {7\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
((49*c^4*((a*x - 1)/(a*x + 1))^(3/2))/6 - (7*c^4*((a*x - 1)/(a*x + 1))^(1/ 2))/4 - (224*c^4*((a*x - 1)/(a*x + 1))^(5/2))/15 + (79*c^4*((a*x - 1)/(a*x + 1))^(7/2))/6 + (7*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) + (7* c^4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)