Optimal. Leaf size=149 \[ -\frac {2 (1-14 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2048 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \sqrt {1-a^2 x^2}}-\frac {256 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \left (1-a^2 x^2\right )^{3/2}}-\frac {112 (1-10 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6136, 6135} \[ -\frac {2 (1-14 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2048 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \sqrt {1-a^2 x^2}}-\frac {256 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \left (1-a^2 x^2\right )^{3/2}}-\frac {112 (1-10 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{6435 a \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6135
Rule 6136
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a \left (1-a^2 x^2\right )^{7/2}}+\frac {56}{65} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {112 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a \left (1-a^2 x^2\right )^{5/2}}+\frac {896 \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1287}\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {112 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a \left (1-a^2 x^2\right )^{5/2}}-\frac {256 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a \left (1-a^2 x^2\right )^{3/2}}+\frac {1024 \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{2145}\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {112 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a \left (1-a^2 x^2\right )^{5/2}}-\frac {256 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2048 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{6435 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.54 \[ -\frac {2 \left (2048 a^7 x^7-1024 a^6 x^6-6912 a^5 x^5+3200 a^4 x^4+8240 a^3 x^3-3384 a^2 x^2-3838 a x+1241\right )}{6435 a (1-a x)^{15/4} (a x+1)^{13/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.67, size = 126, normalized size = 0.85 \[ -\frac {2 \, {\left (2048 \, a^{7} x^{7} - 1024 \, a^{6} x^{6} - 6912 \, a^{5} x^{5} + 3200 \, a^{4} x^{4} + 8240 \, a^{3} x^{3} - 3384 \, a^{2} x^{2} - 3838 \, a x + 1241\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{6435 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 102, normalized size = 0.68 \[ \frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 x^{5} a^{5}+3200 x^{4} a^{4}+8240 x^{3} a^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 211, normalized size = 1.42 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {2482\,\sqrt {1-a^2\,x^2}}{6435\,a^9}-\frac {7676\,x\,\sqrt {1-a^2\,x^2}}{6435\,a^8}+\frac {4096\,x^7\,\sqrt {1-a^2\,x^2}}{6435\,a^2}-\frac {2048\,x^6\,\sqrt {1-a^2\,x^2}}{6435\,a^3}-\frac {1536\,x^5\,\sqrt {1-a^2\,x^2}}{715\,a^4}+\frac {1280\,x^4\,\sqrt {1-a^2\,x^2}}{1287\,a^5}+\frac {3296\,x^3\,\sqrt {1-a^2\,x^2}}{1287\,a^6}-\frac {752\,x^2\,\sqrt {1-a^2\,x^2}}{715\,a^7}\right )}{\frac {1}{a^8}+x^8-\frac {4\,x^6}{a^2}+\frac {6\,x^4}{a^4}-\frac {4\,x^2}{a^6}} \]
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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