Integrand size = 24, antiderivative size = 118 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}+\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {3 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
c^2*(c-c/a/x)^(3/2)/a+3/5*c*(c-c/a/x)^(5/2)/a+(c-c/a/x)^(7/2)*x-3*c^(7/2)* arctanh((c-c/a/x)^(1/2)/c^(1/2))/a+3*c^3*(c-c/a/x)^(1/2)/a
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (-2+4 a x+8 a^2 x^2+5 a^3 x^3\right )}{5 a^3 x^2}-\frac {3 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
(c^3*Sqrt[c - c/(a*x)]*(-2 + 4*a*x + 8*a^2*x^2 + 5*a^3*x^3))/(5*a^3*x^2) - (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a
Time = 0.46 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6717, 6683, 1035, 281, 899, 87, 60, 60, 60, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^{7/2} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2}dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (a x+1)}{1-a x}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle -\int \frac {\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}{\frac {1}{x}-a}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2}dx}{a}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle -\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} x^2d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \int \left (c-\frac {c}{a x}\right )^{5/2} xd\frac {1}{x}-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \int \left (c-\frac {c}{a x}\right )^{3/2} xd\frac {1}{x}+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \int \sqrt {c-\frac {c}{a x}} xd\frac {1}{x}+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (c \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+2 \sqrt {c-\frac {c}{a x}}\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (2 \sqrt {c-\frac {c}{a x}}-2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\) |
-((c*(-((a*(c - c/(a*x))^(7/2)*x)/c) - (3*((2*(c - c/(a*x))^(5/2))/5 + c*( (2*(c - c/(a*x))^(3/2))/3 + c*(2*Sqrt[c - c/(a*x)] - 2*Sqrt[c]*ArcTanh[Sqr t[c - c/(a*x)]/Sqrt[c]]))))/2))/a)
3.5.47.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((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[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(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, 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 = 0.55 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (-30 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x^{4}+20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}+4 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{10 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}}\) | \(144\) |
| risch | \(\frac {\left (5 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +2\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{5 x^{2} a^{3} \left (a x -1\right )}-\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(147\) |
-1/10*(c*(a*x-1)/a/x)^(1/2)/x^3*c^3*(-30*(a*x^2-x)^(1/2)*a^(7/2)*x^4+20*a^ (5/2)*(a*x^2-x)^(3/2)*x^2+15*ln(1/2*(2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1)/a^ (1/2))*a^3*x^4+4*a^(3/2)*(a*x^2-x)^(3/2)*x-4*(a*x^2-x)^(3/2)*a^(1/2))/((a* x-1)*x)^(1/2)/a^(7/2)
Time = 0.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.80 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {15 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, a^{3} x^{2}}, \frac {15 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, a^{3} x^{2}}\right ] \]
[1/10*(15*a^2*c^(7/2)*x^2*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a *x)) + c) + 2*(5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a* c*x - c)/(a*x)))/(a^3*x^2), 1/5*(15*a^2*sqrt(-c)*c^3*x^2*arctan(sqrt(-c)*s qrt((a*c*x - c)/(a*x))/c) + (5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2 *c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2)]
Result contains complex when optimal does not.
Time = 7.54 (sec) , antiderivative size = 740, normalized size of antiderivative = 6.27 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=c^{3} \left (\begin {cases} - \frac {\sqrt {c} \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {\sqrt {c} \sqrt {x} \sqrt {a x - 1}}{\sqrt {a}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {i \sqrt {a} \sqrt {c} x^{\frac {3}{2}}}{\sqrt {- a x + 1}} + \frac {i \sqrt {c} \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {i \sqrt {c} \sqrt {x}}{\sqrt {a} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \frac {c^{3} \left (\begin {cases} \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {c - \frac {c}{a x}} & \text {for}\: \frac {c}{a} \neq 0 \\- \sqrt {c} \log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c^{3} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {2 a \left (c - \frac {c}{a x}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {c^{3} \left (\begin {cases} - \frac {4 a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{5} \sqrt {c} x^{3} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {2 a^{4} \sqrt {c} x^{2} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {8 a^{3} \sqrt {c} x \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {6 a^{2} \sqrt {c} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {4 a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 i a^{5} \sqrt {c} x^{3} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {2 i a^{4} \sqrt {c} x^{2} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {8 i a^{3} \sqrt {c} x \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {6 i a^{2} \sqrt {c} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right )}{a^{3}} \]
c**3*Piecewise((-sqrt(c)*acosh(sqrt(a)*sqrt(x))/a + sqrt(c)*sqrt(x)*sqrt(a *x - 1)/sqrt(a), Abs(a*x) > 1), (-I*sqrt(a)*sqrt(c)*x**(3/2)/sqrt(-a*x + 1 ) + I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(-a *x + 1)), True)) + c**3*Piecewise((2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sq rt(-c) + 2*sqrt(c - c/(a*x)), Ne(c/a, 0)), (-sqrt(c)*log(x), True))/a - c* *3*Piecewise((0, Eq(c, 0)), (2*a*(c - c/(a*x))**(3/2)/(3*c), True))/a**2 + c**3*Piecewise((-4*a**(11/2)*sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2) - 15* a**(5/2)*x**(5/2)) + 4*a**(9/2)*sqrt(c)*x**(5/2)/(15*a**(7/2)*x**(7/2) - 1 5*a**(5/2)*x**(5/2)) + 4*a**5*sqrt(c)*x**3*sqrt(a*x - 1)/(15*a**(7/2)*x**( 7/2) - 15*a**(5/2)*x**(5/2)) - 2*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(15*a**(7 /2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) - 8*a**3*sqrt(c)*x*sqrt(a*x - 1)/(15* a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 6*a**2*sqrt(c)*sqrt(a*x - 1)/( 15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*a**(11/2) *sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*a**(9/ 2)*sqrt(c)*x**(5/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*I*a* *5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2 )) - 2*I*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5 /2)*x**(5/2)) - 8*I*a**3*sqrt(c)*x*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 6*I*a**2*sqrt(c)*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7 /2) - 15*a**(5/2)*x**(5/2)), True))/a**3
\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{a x - 1} \,d x } \]
Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]