Integrand size = 24, antiderivative size = 322 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 \sqrt {1-\frac {1}{a^2 x^2}} x^6}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \] Output:
-1/6*c^3*(c-c/a^2/x^2)^(1/2)/a^7/(1-1/a^2/x^2)^(1/2)/x^6+1/5*c^3*(c-c/a^2/ x^2)^(1/2)/a^6/(1-1/a^2/x^2)^(1/2)/x^5+3/4*c^3*(c-c/a^2/x^2)^(1/2)/a^5/(1- 1/a^2/x^2)^(1/2)/x^4-c^3*(c-c/a^2/x^2)^(1/2)/a^4/(1-1/a^2/x^2)^(1/2)/x^3-3 /2*c^3*(c-c/a^2/x^2)^(1/2)/a^3/(1-1/a^2/x^2)^(1/2)/x^2+3*c^3*(c-c/a^2/x^2) ^(1/2)/a^2/(1-1/a^2/x^2)^(1/2)/x+c^3*(c-c/a^2/x^2)^(1/2)*x/(1-1/a^2/x^2)^( 1/2)-c^3*(c-c/a^2/x^2)^(1/2)*ln(x)/a/(1-1/a^2/x^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (-\frac {1}{6 a^7 x^6}+\frac {1}{5 a^6 x^5}+\frac {3}{4 a^5 x^4}-\frac {1}{a^4 x^3}-\frac {3}{2 a^3 x^2}+\frac {3}{a^2 x}+x-\frac {\log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \] Input:
Integrate[(c - c/(a^2*x^2))^(7/2)/E^ArcCoth[a*x],x]
Output:
((c - c/(a^2*x^2))^(7/2)*(-1/6*1/(a^7*x^6) + 1/(5*a^6*x^5) + 3/(4*a^5*x^4) - 1/(a^4*x^3) - 3/(2*a^3*x^2) + 3/(a^2*x) + x - Log[x]/a))/(1 - 1/(a^2*x^ 2))^(7/2)
Time = 0.70 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.31, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6751, 6747, 99, 2009}
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.
\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^{7/2} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2}dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^4 (a x+1)^3}{x^7}dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \int \left (a^7-\frac {a^6}{x}-\frac {3 a^5}{x^2}+\frac {3 a^4}{x^3}+\frac {3 a^3}{x^4}-\frac {3 a^2}{x^5}-\frac {a}{x^6}+\frac {1}{x^7}\right )dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (a^7 x-a^6 \log (x)+\frac {3 a^5}{x}-\frac {3 a^4}{2 x^2}-\frac {a^3}{x^3}+\frac {3 a^2}{4 x^4}+\frac {a}{5 x^5}-\frac {1}{6 x^6}\right )}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[(c - c/(a^2*x^2))^(7/2)/E^ArcCoth[a*x],x]
Output:
(c^3*Sqrt[c - c/(a^2*x^2)]*(-1/6*1/x^6 + a/(5*x^5) + (3*a^2)/(4*x^4) - a^3 /x^3 - (3*a^4)/(2*x^2) + (3*a^5)/x + a^7*x - a^6*Log[x]))/(a^7*Sqrt[1 - 1/ (a^2*x^2)])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart [p]) Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.35
method | result | size |
default | \(-\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \sqrt {\frac {a x -1}{a x +1}}\, x \left (-60 a^{7} x^{7}+60 a^{6} \ln \left (x \right ) x^{6}-180 a^{5} x^{5}+90 a^{4} x^{4}+60 a^{3} x^{3}-45 a^{2} x^{2}-12 a x +10\right )}{60 \left (a x -1\right ) \left (a^{2} x^{2}-1\right )^{3}}\) | \(112\) |
Input:
int((c-c/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/60*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2)*x*(-60*a^7*x^7 +60*a^6*ln(x)*x^6-180*a^5*x^5+90*a^4*x^4+60*a^3*x^3-45*a^2*x^2-12*a*x+10)/ (a*x-1)/(a^2*x^2-1)^3
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.30 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {{\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{6} c^{3} x^{6} \log \left (x\right ) + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x - 10 \, c^{3}\right )} \sqrt {a^{2} c}}{60 \, a^{8} x^{6}} \] Input:
integrate((c-c/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas ")
Output:
1/60*(60*a^7*c^3*x^7 - 60*a^6*c^3*x^6*log(x) + 180*a^5*c^3*x^5 - 90*a^4*c^ 3*x^4 - 60*a^3*c^3*x^3 + 45*a^2*c^3*x^2 + 12*a*c^3*x - 10*c^3)*sqrt(a^2*c) /(a^8*x^6)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((c-c/a**2/x**2)**(7/2)*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Timed out
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima ")
Output:
integrate((c - c/(a^2*x^2))^(7/2)*sqrt((a*x - 1)/(a*x + 1)), x)
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.48 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {1}{60} \, {\left (\frac {60 \, c^{3} x \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right )}{a} - \frac {60 \, c^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {180 \, a^{5} c^{3} x^{5} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 90 \, a^{4} c^{3} x^{4} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 60 \, a^{3} c^{3} x^{3} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) + 45 \, a^{2} c^{3} x^{2} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) + 12 \, a c^{3} x \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 10 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right )}{a^{8} x^{6}}\right )} \sqrt {c} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
Output:
1/60*(60*c^3*x*sgn(a*x + 1)*sgn(x)/a - 60*c^3*log(abs(x))*sgn(a*x + 1)*sgn (x)/a^2 + (180*a^5*c^3*x^5*sgn(a*x + 1)*sgn(x) - 90*a^4*c^3*x^4*sgn(a*x + 1)*sgn(x) - 60*a^3*c^3*x^3*sgn(a*x + 1)*sgn(x) + 45*a^2*c^3*x^2*sgn(a*x + 1)*sgn(x) + 12*a*c^3*x*sgn(a*x + 1)*sgn(x) - 10*c^3*sgn(a*x + 1)*sgn(x))/( a^8*x^6))*sqrt(c)*abs(a)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int {\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \] Input:
int((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - c/(a^2*x^2))^(7/2)*((a*x - 1)/(a*x + 1))^(1/2), x)
Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.25 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (-60 \,\mathrm {log}\left (a x \right ) a^{6} x^{6}+60 a^{7} x^{7}-240 a^{6} x^{6}+180 a^{5} x^{5}-90 a^{4} x^{4}-60 a^{3} x^{3}+45 a^{2} x^{2}+12 a x -10\right )}{60 a^{7} x^{6}} \] Input:
int((c-c/a^2/x^2)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(sqrt(c)*c**3*( - 60*log(a*x)*a**6*x**6 + 60*a**7*x**7 - 240*a**6*x**6 + 1 80*a**5*x**5 - 90*a**4*x**4 - 60*a**3*x**3 + 45*a**2*x**2 + 12*a*x - 10))/ (60*a**7*x**6)