Optimal. Leaf size=218 \[ \frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {3 a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 218, 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 = {6160, 6150, 75} \[ \frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 a^4 x^5 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2}}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^4 (1+a x)}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (a^5+\frac {1}{x^5}-\frac {3 a}{x^4}+\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}-\frac {3 a^4}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}{\left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}-\frac {3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.41 \[ \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a^5 x^5-5 a^4 x^4-12 a^4 x^4 \log (x)-8 a^3 x^3-4 a^2 x^2+4 a x-1\right )}{4 a^4 x^3 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 480, normalized size = 2.20 \[ \left [\frac {6 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left (4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right )} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, \frac {12 \, {\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - {\left (4 \, a^{5} c^{2} x^{5} - 8 \, a^{3} c^{2} x^{3} - {\left (4 \, a^{5} - 8 \, a^{3} - 4 \, a^{2} + 4 \, a - 1\right )} c^{2} x^{4} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 86, normalized size = 0.39 \[ \frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-4 x^{5} a^{5}+12 a^{4} \ln \relax (x ) x^{4}+8 x^{3} a^{3}+4 a^{2} x^{2}-4 a x +1\right )}{4 \left (a^{2} x^{2}-1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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