Optimal. Leaf size=322 \[ \frac {c^4 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 x^8 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 x^7 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6197, 6193, 88} \[ \frac {c^4 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 x^7 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 x^8 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^6 (1+a x)^3}{x^9} \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^9+\frac {1}{x^9}-\frac {3 a}{x^8}+\frac {8 a^3}{x^6}-\frac {6 a^4}{x^5}-\frac {6 a^5}{x^4}+\frac {8 a^6}{x^3}-\frac {3 a^8}{x}\right ) \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=-\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 \sqrt {1-\frac {1}{a^2 x^2}} x^8}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 \sqrt {1-\frac {1}{a^2 x^2}} x^7}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 97, normalized size = 0.30 \[ \frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (a^9 x-3 a^8 \log (x)-\frac {4 a^6}{x^2}+\frac {2 a^5}{x^3}+\frac {3 a^4}{2 x^4}-\frac {8 a^3}{5 x^5}+\frac {3 a}{7 x^7}-\frac {1}{8 x^8}\right )}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 96, normalized size = 0.30 \[ \frac {{\left (280 \, a^{9} c^{4} x^{9} - 840 \, a^{8} c^{4} x^{8} \log \relax (x) - 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} + 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x - 35 \, c^{4}\right )} \sqrt {a^{2} c}}{280 \, a^{10} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 112, normalized size = 0.35 \[ -\frac {\left (-280 a^{9} x^{9}+840 a^{8} \ln \relax (x ) x^{8}+1120 x^{6} a^{6}-560 x^{5} a^{5}-420 x^{4} a^{4}+448 x^{3} a^{3}-120 a x +35\right ) x \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {9}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{280 \left (a x -1\right )^{3} \left (a^{2} x^{2}-1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________