Integrand size = 22, antiderivative size = 229 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {8 (1+a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1+a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1+a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1+a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1+a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \]
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Time = 0.14 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{9/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = \frac {\left (c-a^2 c x^2\right )^{9/2} \int (-1+a x)^4 (1+a x)^5 \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = \frac {\left (c-a^2 c x^2\right )^{9/2} \int \left (16 (1+a x)^5-32 (1+a x)^6+24 (1+a x)^7-8 (1+a x)^8+(1+a x)^9\right ) \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ & = \frac {8 (1+a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1+a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1+a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1+a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1+a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {c^4 (1+a x)^6 \sqrt {c-a^2 c x^2} \left (193-528 a x+588 a^2 x^2-308 a^3 x^3+63 a^4 x^4\right )}{630 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.49
method | result | size |
default | \(\frac {\left (63 a^{9} x^{9}+70 a^{8} x^{8}-315 a^{7} x^{7}-360 a^{6} x^{6}+630 a^{5} x^{5}+756 a^{4} x^{4}-630 a^{3} x^{3}-840 a^{2} x^{2}+315 a x +630\right ) x \,c^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{630 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(113\) |
gosper | \(\frac {x \left (63 a^{9} x^{9}+70 a^{8} x^{8}-315 a^{7} x^{7}-360 a^{6} x^{6}+630 a^{5} x^{5}+756 a^{4} x^{4}-630 a^{3} x^{3}-840 a^{2} x^{2}+315 a x +630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{630 \left (a x +1\right )^{5} \left (a x -1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}}\) | \(116\) |
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Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.51 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\frac {{\left (63 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} - 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} + 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} - 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} + 630 \, c^{4} x\right )} \sqrt {-a^{2} c}}{630 \, a} \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\text {Timed out} \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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