Optimal. Leaf size=176 \[ -\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a} \]
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Rubi [A] time = 0.165263, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6167, 6141, 671, 641, 195, 217, 203} \[ -\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6141
Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2} \, dx\right )\\ &=\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{10} (11 c) \int (1+a x) \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{10} (11 c) \int \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{80} \left (77 c^2\right ) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{96} \left (77 c^3\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{128} \left (77 c^4\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{256} \left (77 c^5\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{256} \left (77 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}\\ \end{align*}
Mathematica [A] time = 0.15354, size = 167, normalized size = 0.95 \[ \frac{c^4 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (-1152 a^{10} x^{10}-1408 a^9 x^9+5584 a^8 x^8+7216 a^7 x^7-10552 a^6 x^6-15048 a^5 x^5+9210 a^4 x^4+16390 a^3 x^3-2185 a^2 x^2-10615 a x+2560\right )+6930 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{11520 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.059, size = 350, normalized size = 2. \begin{align*}{\frac{x}{10} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}}}+{\frac{9\,cx}{80} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{21\,x{c}^{2}}{160} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{21\,{c}^{3}x}{128} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{63\,{c}^{4}x}{256}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{63\,{c}^{5}}{256}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{9\,a} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{cx}{4} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{7\,x{c}^{2}}{24} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{35\,{c}^{3}x}{96} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{c}^{4}x}{64}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}-{\frac{35\,{c}^{5}}{64}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91798, size = 790, normalized size = 4.49 \begin{align*} \left [\frac{3465 \, \sqrt{-c} c^{4} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt{-a^{2} c x^{2} + c}}{23040 \, a}, \frac{3465 \, c^{\frac{9}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt{-a^{2} c x^{2} + c}}{11520 \, 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 [A] time = 1.17853, size = 221, normalized size = 1.26 \begin{align*} \frac{77 \, c^{5} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{256 \, \sqrt{-c}{\left | a \right |}} + \frac{1}{11520} \, \sqrt{-a^{2} c x^{2} + c}{\left (\frac{2560 \, c^{4}}{a} -{\left (8055 \, c^{4} + 2 \,{\left (5120 \, a c^{4} -{\left (3075 \, a^{2} c^{4} + 4 \,{\left (1920 \, a^{3} c^{4} +{\left (39 \, a^{4} c^{4} - 2 \,{\left (640 \, a^{5} c^{4} +{\left (189 \, a^{6} c^{4} - 8 \,{\left (9 \, a^{8} c^{4} x + 20 \, a^{7} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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