Optimal. Leaf size=75 \[ -\frac {2 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6136, 6135} \[ -\frac {2 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a \sqrt {1-a^2 x^2}} \]
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 )^{5/2}} \, dx &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a \left (1-a^2 x^2\right )^{3/2}}+\frac {24}{35} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{35 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.64 \[ -\frac {2 \left (16 a^3 x^3-8 a^2 x^2-22 a x+9\right )}{35 a (1-a x)^{7/4} (a x+1)^{5/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 78, normalized size = 1.04 \[ -\frac {2 \, {\left (16 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 22 \, a x + 9\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{35 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 70, normalized size = 0.93 \[ \frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (16 x^{3} a^{3}-8 a^{2} x^{2}-22 a x +9\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{35 a \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 115, normalized size = 1.53 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {18\,\sqrt {1-a^2\,x^2}}{35\,a^5}-\frac {44\,x\,\sqrt {1-a^2\,x^2}}{35\,a^4}+\frac {32\,x^3\,\sqrt {1-a^2\,x^2}}{35\,a^2}-\frac {16\,x^2\,\sqrt {1-a^2\,x^2}}{35\,a^3}\right )}{\frac {1}{a^4}+x^4-\frac {2\,x^2}{a^2}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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