Optimal. Leaf size=299 \[ -\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}{6 \left (1-a^2 x^2\right )^{7/2}}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{5 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{\left (1-a^2 x^2\right )^{7/2}}-\frac {3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{2 \left (1-a^2 x^2\right )^{7/2}}+\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{\left (1-a^2 x^2\right )^{7/2}}+\frac {a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^8}{\left (1-a^2 x^2\right )^{7/2}}-\frac {a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \log (x)}{\left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 299, 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 = {6295, 6285, 90}
\begin {gather*} \frac {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^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 {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 {3 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 {3 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 {a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{\left (1-a^2 x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6295
Rubi steps
\begin {align*} \int e^{-\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^{-\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)^4 (1+a x)^3}{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 {a}{x^6}-\frac {3 a^2}{x^5}+\frac {3 a^3}{x^4}+\frac {3 a^4}{x^3}-\frac {3 a^5}{x^2}-\frac {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 {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 {3 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 {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{\left (1-a^2 x^2\right )^{7/2}}-\frac {3 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 {3 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 {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 = 0.04, size = 98, normalized size = 0.33 \begin {gather*} -\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (-10+12 a x+45 a^2 x^2-60 a^3 x^3-90 a^4 x^4+180 a^5 x^5+60 a^7 x^7-60 a^6 x^6 \log (x)\right )}{60 a^6 x^5 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 102, normalized size = 0.34
method | result | size |
default | \(-\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \sqrt {-a^{2} x^{2}+1}\, \left (-60 a^{7} x^{7}+60 \ln \left (x \right ) a^{6} x^{6}-180 a^{5} x^{5}+90 a^{4} x^{4}+60 a^{3} x^{3}-45 a^{2} x^{2}-12 a x +10\right )}{60 \left (a^{2} x^{2}-1\right )^{4}}\) | \(102\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 542, normalized size = 1.81 \begin {gather*} \left [\frac {30 \, {\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} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right )} c^{3} x^{6} - 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} 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 {60 \, {\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} + 180 \, a^{5} c^{3} x^{5} - 90 \, a^{4} c^{3} x^{4} - {\left (60 \, a^{7} + 180 \, a^{5} - 90 \, a^{4} - 60 \, a^{3} + 45 \, a^{2} + 12 \, a - 10\right )} c^{3} x^{6} - 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} 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 ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}}}{a x + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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