Optimal. Leaf size=450 \[ \frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \text {ArcSin}(a x)}{(1-a x)^{9/2} (1+a x)^{9/2}}+\frac {245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}} \]
[Out]
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Rubi [A]
time = 0.37, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6294, 6264,
99, 154, 159, 163, 41, 222, 94, 214} \begin {gather*} -\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{28 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{8 (1-a x)}+\frac {71 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{336 (1-a x)^3}+\frac {2 a^8 x^9 \text {ArcSin}(a x) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{(1-a x)^{9/2} (a x+1)^{9/2}}-\frac {501 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{128 (1-a x)^4 (a x+1)^4}+\frac {245 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{128 (1-a x)^{9/2} (a x+1)^{9/2}}+\frac {373 a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{192 (1-a x)^4 (a x+1)^3}+\frac {501 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{640 (1-a x)^4 (a x+1)^2}+\frac {661 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1680 (1-a x)^4 (a x+1)}+\frac {295 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1344 (1-a x)^4}-\frac {127 a^3 x^4 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{420 (1-a x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 94
Rule 99
Rule 154
Rule 159
Rule 163
Rule 214
Rule 222
Rule 6264
Rule 6294
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{9/2} (1+a x)^{9/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{11/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{9/2} \left (2 a-9 a^2 x\right )}{x^8} \, dx}{8 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{9/2} \left (-71 a^2+61 a^3 x\right )}{x^7} \, dx}{56 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {\sqrt {1-a x} (1+a x)^{9/2} \left (508 a^3-295 a^4 x\right )}{x^6} \, dx}{336 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{9/2} \left (-1475 a^4+967 a^5 x\right )}{x^5 \sqrt {1-a x}} \, dx}{1680 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{7/2} \left (-7932 a^5+5343 a^6 x\right )}{x^4 \sqrt {1-a x}} \, dx}{6720 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{5/2} \left (-31563 a^6+23961 a^7 x\right )}{x^3 \sqrt {1-a x}} \, dx}{20160 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{3/2} \left (-78330 a^7+79485 a^8 x\right )}{x^2 \sqrt {1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {\sqrt {1+a x} \left (-77175 a^8+157815 a^9 x\right )}{x \sqrt {1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {77175 a^9-80640 a^{10} x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{40320 a (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac {\left (245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac {\left (2 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (245 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac {\left (2 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \sin ^{-1}(a x)}{(1-a x)^{9/2} (1+a x)^{9/2}}+\frac {245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 166, normalized size = 0.37 \begin {gather*} -\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (1680+3840 a x-4760 a^2 x^2-16896 a^3 x^3+770 a^4 x^4+31232 a^5 x^5+14595 a^6 x^6-45056 a^7 x^7+13440 a^8 x^8\right )+25725 a^8 x^8 \text {ArcTan}\left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+26880 a^8 x^8 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{13440 a^8 x^7 \sqrt {-1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(964\) vs.
\(2(396)=792\).
time = 1.60, size = 965, normalized size = 2.14
method | result | size |
risch | \(\frac {\left (45056 a^{9} x^{9}-14595 a^{8} x^{8}-76288 a^{7} x^{7}+13825 a^{6} x^{6}+48128 a^{5} x^{5}+5530 a^{4} x^{4}-20736 a^{3} x^{3}-6440 a^{2} x^{2}+3840 a x +1680\right ) c^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{13440 x^{7} a^{8} \left (a^{2} x^{2}-1\right )}-\frac {\left (\frac {a^{8} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c}+\frac {2 a^{9} \ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {245 a^{8} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{128 \sqrt {-c}}\right ) c^{4} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{8} \left (a^{2} x^{2}-1\right )}\) | \(267\) |
default | \(-\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {9}{2}} x \left (-5040 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} \sqrt {-\frac {c}{a^{2}}}+23808 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{11} x^{7}-17535 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{10} x^{6}-13056 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{9} x^{5}-6510 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{8} x^{4}-6912 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{7} x^{3}-10920 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{6} x^{2}-11520 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {11}{2}} a^{5} x +58590 c^{\frac {11}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{8}+22050 c^{\frac {11}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{8}-23808 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{11} c \,x^{9}+8575 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} a^{10} c \,x^{8}+8960 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {9}{2}} a^{10} c \,x^{8}+26784 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} a^{9} c^{2} x^{9}+10080 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} a^{9} c^{2} x^{9}-11025 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} a^{8} c^{2} x^{8}-31248 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{7} c^{3} x^{9}-11760 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} a^{7} c^{3} x^{9}+15435 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} a^{6} c^{3} x^{8}+39060 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{4} x^{9}+14700 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} a^{5} c^{4} x^{9}-25725 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4} c^{4} x^{8}-58590 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c^{5} x^{9}-22050 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a^{3} c^{5} x^{9}+77175 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c^{5} x^{8}+77175 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{6} x^{8}\right )}{40320 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} c}\) | \(965\) |
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 = 0.47, size = 482, normalized size = 1.07 \begin {gather*} \left [\frac {53760 \, a^{7} \sqrt {-c} c^{4} x^{7} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 25725 \, a^{7} \sqrt {-c} c^{4} x^{7} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac {25725 \, a^{7} c^{\frac {9}{2}} x^{7} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 13440 \, a^{7} c^{\frac {9}{2}} x^{7} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 30.08, size = 707, normalized size = 1.57 \begin {gather*} \frac {1}{6720} \, {\left (\frac {25725 \, c^{\frac {9}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {13440 \, c^{\frac {9}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {6720 \, \sqrt {a^{2} c x^{2} - c} c^{4} \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{15} c^{5} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 107520 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{14} a c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{13} c^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 430080 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{12} a c^{\frac {13}{2}} \mathrm {sgn}\left (x\right ) + 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{7} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1111040 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {15}{2}} \mathrm {sgn}\left (x\right ) + 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{8} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1576960 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {17}{2}} \mathrm {sgn}\left (x\right ) - 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{9} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 1412096 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {19}{2}} \mathrm {sgn}\left (x\right ) - 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{10} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 831488 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {21}{2}} \mathrm {sgn}\left (x\right ) - 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{11} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 252928 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {23}{2}} \mathrm {sgn}\left (x\right ) - 14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{12} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 45056 \, a c^{\frac {25}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{8} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \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 )}^{9/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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