Optimal. Leaf size=301 \[ \frac {3 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 x^7 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac {5 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 301, 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, 88} \[ -\frac {a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{2 \left (1-a^2 x^2\right )^{7/2}}-\frac {5 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{4 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 x^7 \log (x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2}}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^5 (1+a x)^2}{x^7} \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (-a^7+\frac {1}{x^7}-\frac {3 a}{x^6}+\frac {a^2}{x^5}+\frac {5 a^3}{x^4}-\frac {5 a^4}{x^3}-\frac {a^5}{x^2}+\frac {3 a^6}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{7/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac {5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 \left (1-a^2 x^2\right )^{7/2}}+\frac {5 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 98, normalized size = 0.33 \[ \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (60 a^7 x^7-180 a^6 x^6 \log (x)-60 a^5 x^5-150 a^4 x^4+100 a^3 x^3+15 a^2 x^2-36 a x+10\right )}{60 a^6 x^5 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 544, normalized size = 1.81 \[ \left [\frac {90 \, {\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left (a^{8} x^{7} - a^{6} x^{5}\right )}}, \frac {180 \, {\left (a^{7} c^{3} x^{7} - a^{5} c^{3} x^{5}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - {\left (60 \, a^{7} c^{3} x^{7} - 60 \, a^{5} c^{3} x^{5} - 150 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} - 60 \, a^{5} - 150 \, a^{4} + 100 \, a^{3} + 15 \, a^{2} - 36 \, a + 10\right )} c^{3} x^{6} + 100 \, a^{3} c^{3} x^{3} + 15 \, a^{2} c^{3} x^{2} - 36 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{60 \, {\left (a^{8} x^{7} - a^{6} x^{5}\right )}}\right ] \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 0.34 \[ \frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-60 a^{7} x^{7}+180 a^{6} \ln \relax (x ) x^{6}+60 x^{5} a^{5}+150 x^{4} a^{4}-100 x^{3} a^{3}-15 a^{2} x^{2}+36 a x -10\right )}{60 \left (a^{2} x^{2}-1\right )^{4}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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