Optimal. Leaf size=363 \[ \frac{1334 (a x+1)^3 (1-a x)^5}{315 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{2458 (a x+1)^2 (1-a x)^5}{315 a^7 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{2 (a x+1)^4 (1019 a x+704) (1-a x)^5}{315 a^{10} x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{302 (a x+1)^2 (1-a x)^4}{21 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{646 (a x+1)^2 (1-a x)^3}{315 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{214 (a x+1)^2 (1-a x)^2}{315 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{2 (a x+1)^2 (1-a x)}{7 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{(a x+1)^2}{9 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{2 (a x+1)^{9/2} (1-a x)^{9/2} \sin ^{-1}(a x)}{a^{10} x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}} \]
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Rubi [A] time = 0.475402, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6159, 6129, 98, 150, 143, 41, 216} \[ \frac{1334 (a x+1)^3 (1-a x)^5}{315 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{2458 (a x+1)^2 (1-a x)^5}{315 a^7 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{2 (a x+1)^4 (1019 a x+704) (1-a x)^5}{315 a^{10} x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{302 (a x+1)^2 (1-a x)^4}{21 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{646 (a x+1)^2 (1-a x)^3}{315 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{214 (a x+1)^2 (1-a x)^2}{315 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{2 (a x+1)^2 (1-a x)}{7 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}+\frac{(a x+1)^2}{9 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}-\frac{2 (a x+1)^{9/2} (1-a x)^{9/2} \sin ^{-1}(a x)}{a^{10} x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{9/2}} \, dx &=\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} x^9}{(1-a x)^{9/2} (1+a x)^{9/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^9}{(1-a x)^{11/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^7 (8+10 a x)}{(1-a x)^{9/2} (1+a x)^{7/2}} \, dx}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^6 \left (-126 a-88 a^2 x\right )}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{63 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^5 \left (1284 a^2+654 a^3 x\right )}{(1-a x)^{5/2} (1+a x)^{7/2}} \, dx}{315 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^4 \left (-9690 a^3-3900 a^4 x\right )}{(1-a x)^{3/2} (1+a x)^{7/2}} \, dx}{945 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^3 \left (54360 a^4+17490 a^5 x\right )}{\sqrt{1-a x} (1+a x)^{7/2}} \, dx}{945 a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}+\frac{2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x^2 \left (110610 a^5+50580 a^6 x\right )}{\sqrt{1-a x} (1+a x)^{5/2}} \, dx}{4725 a^{12} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}+\frac{2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}+\frac{1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}-\frac{\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{x \left (120060 a^6+91710 a^7 x\right )}{\sqrt{1-a x} (1+a x)^{3/2}} \, dx}{14175 a^{14} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}+\frac{2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}+\frac{1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}+\frac{2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}-\frac{\left (2 (1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}+\frac{2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}+\frac{1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}+\frac{2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}-\frac{\left (2 (1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac{(1+a x)^2}{9 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}-\frac{2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}+\frac{214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}-\frac{646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}+\frac{302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}+\frac{2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}+\frac{1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}+\frac{2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}-\frac{2 (1-a x)^{9/2} (1+a x)^{9/2} \sin ^{-1}(a x)}{a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9}\\ \end{align*}
Mathematica [A] time = 0.157839, size = 151, normalized size = 0.42 \[ \frac{-315 a^8 x^8+1756 a^7 x^7+268 a^6 x^6-5784 a^5 x^5+2060 a^4 x^4+6200 a^3 x^3-3372 a^2 x^2-630 (a x-1)^4 (a x+1)^2 \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-2186 a x+1408}{315 a^2 c^4 x (a x-1)^4 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.195, size = 682, normalized size = 1.9 \begin{align*} \text{result too large to display} \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{{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\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 = 4.27769, size = 1235, normalized size = 3.4 \begin{align*} \left [\frac{315 \,{\left (a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt{c} \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 (315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \,{\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right )}}, \frac{630 \,{\left (a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt{-c} \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 ) -{\left (315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \,{\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a x}{a c^{4} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx - \int \frac{1}{a c^{4} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx \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{{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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