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 = 0.168022, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 6136
Rule 6135
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*}
Mathematica [A] time = 0.0303226, 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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Maple [A] time = 0.031, size = 102, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+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\,ax+1241 \right ) }{6435\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53286, size = 300, normalized size = 2.01 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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