Integrand size = 27, antiderivative size = 105 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {11 \sqrt {c-\frac {c}{a x}} x}{8 a^2}+\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {11 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3} \]
11/8*arctanh((c-c/a/x)^(1/2)/c^(1/2))*c^(1/2)/a^3+11/8*x*(c-c/a/x)^(1/2)/a ^2+11/12*x^2*(c-c/a/x)^(1/2)/a+1/3*x^3*(c-c/a/x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.48 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (a^3 x^3+11 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-\frac {1}{a x}\right )\right )}{3 a^3} \]
Time = 0.69 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6717, 6683, 1070, 281, 948, 87, 52, 52, 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.
| \(\displaystyle \int x^2 \sqrt {c-\frac {c}{a x}} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {\sqrt {c-\frac {c}{a x}} x^2 (a x+1)}{1-a x}dx\) |
| \(\Big \downarrow \) 1070 |
| \(\displaystyle -\int \frac {\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}} x^2}{\frac {1}{x}-a}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\sqrt {c-\frac {c}{a x}}}dx}{a}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^4}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {c \left (\frac {11}{6} \int \frac {x^3}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-\frac {a x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{a}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {c \left (\frac {11}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )-\frac {a x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{a}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {c \left (\frac {11}{6} \left (\frac {3 \left (\frac {\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )-\frac {a x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c \left (\frac {11}{6} \left (\frac {3 \left (-\frac {\int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )-\frac {a x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c \left (\frac {11}{6} \left (\frac {3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )-\frac {a x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{a}\) |
-((c*(-1/3*(a*Sqrt[c - c/(a*x)]*x^3)/c + (11*(-1/2*(Sqrt[c - c/(a*x)]*x^2) /c + (3*(-((Sqrt[c - c/(a*x)]*x)/c) - ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/( a*Sqrt[c])))/(4*a)))/6))/a)
3.5.99.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^ (p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(m + n*(p + r))*( b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.50 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\left (8 a^{2} x^{2}+22 a x +33\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2}}+\frac {11 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{16 a^{2} \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(119\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {5}{2}}+60 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} x -30 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}}+96 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}+48 a \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \right )}{48 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}}\) | \(155\) |
1/24*(8*a^2*x^2+22*a*x+33)/a^2*x*(c*(a*x-1)/a/x)^(1/2)+11/16/a^2*ln((-1/2* a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)/(a*x-1)* (c*(a*x-1)/a/x)^(1/2)*(c*(a*x-1)*a*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.55 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {2 \, {\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 33 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{48 \, a^{3}}, \frac {{\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 33 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\right ] \]
[1/48*(2*(8*a^3*x^3 + 22*a^2*x^2 + 33*a*x)*sqrt((a*c*x - c)/(a*x)) + 33*sq rt(c)*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^3, 1/24 *((8*a^3*x^3 + 22*a^2*x^2 + 33*a*x)*sqrt((a*c*x - c)/(a*x)) - 33*sqrt(-c)* arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c))/a^3]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{a x - 1} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {1}{24} \, \sqrt {a^{2} c x^{2} - a c x} {\left (2 \, x {\left (\frac {4 \, x {\left | a \right |}}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {11 \, {\left | a \right |}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} + \frac {33 \, {\left | a \right |}}{a^{4} \mathrm {sgn}\left (x\right )}\right )} + \frac {11 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{3}} - \frac {11 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{16 \, a^{3} \mathrm {sgn}\left (x\right )} \]
1/24*sqrt(a^2*c*x^2 - a*c*x)*(2*x*(4*x*abs(a)/(a^2*sgn(x)) + 11*abs(a)/(a^ 3*sgn(x))) + 33*abs(a)/(a^4*sgn(x))) + 11/16*sqrt(c)*log(abs(a)*abs(c))*sg n(x)/a^3 - 11/16*sqrt(c)*log(abs(-2*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c* x))*sqrt(c)*abs(a) + a*c))/(a^3*sgn(x))
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]