Optimal. Leaf size=239 \[ -\frac{5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.25439, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6185, 6184} \[ -\frac{5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6184
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}+\frac{42 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{c \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{840 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c^2 \left (25-n^2\right ) \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{5040 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac{42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{5040 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.51471, size = 260, normalized size = 1.09 \[ \frac{a x^2 \left (1-\frac{1}{a^2 x^2}\right ) e^{n \coth ^{-1}(a x)} \left (-\frac{63 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )}{n^2-9}+\frac{35 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (5 \coth ^{-1}(a x)\right )}{n^2-25}-\frac{7 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (7 \coth ^{-1}(a x)\right )}{n^2-49}+\frac{21 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (3 \coth ^{-1}(a x)\right )}{n^2-9}-\frac{7 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (5 \coth ^{-1}(a x)\right )}{n^2-25}+\frac{a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left (7 \coth ^{-1}(a x)\right )}{n^2-49}+\frac{35 a x}{n^2-1}-\frac{35 n}{n^2-1}\right )}{64 c^3 \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.045, size = 218, normalized size = 0.9 \begin{align*}{\frac{ \left ( 5040\,{a}^{7}{x}^{7}-5040\,n{a}^{6}{x}^{6}+2520\,{a}^{5}{n}^{2}{x}^{5}-840\,{a}^{4}{n}^{3}{x}^{4}-17640\,{x}^{5}{a}^{5}+210\,{a}^{3}{n}^{4}{x}^{3}+15960\,{a}^{4}n{x}^{4}-42\,{a}^{2}{n}^{5}{x}^{2}-7140\,{a}^{3}{n}^{2}{x}^{3}+7\,a{n}^{6}x+2100\,{a}^{2}{n}^{3}{x}^{2}-{n}^{7}+22050\,{x}^{3}{a}^{3}-455\,a{n}^{4}x-17178\,{a}^{2}n{x}^{2}+77\,{n}^{5}+6433\,a{n}^{2}x-1519\,{n}^{3}-11025\,ax+6483\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{8}-84\,{n}^{6}+1974\,{n}^{4}-12916\,{n}^{2}+11025 \right ) } \left ( -{a}^{2}c{x}^{2}+c \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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\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 = 1.6778, size = 1030, normalized size = 4.31 \begin{align*} -\frac{{\left (5040 \, a^{7} x^{7} + 5040 \, a^{6} n x^{6} + n^{7} + 2520 \,{\left (a^{5} n^{2} - 7 \, a^{5}\right )} x^{5} - 77 \, n^{5} + 840 \,{\left (a^{4} n^{3} - 19 \, a^{4} n\right )} x^{4} + 210 \,{\left (a^{3} n^{4} - 34 \, a^{3} n^{2} + 105 \, a^{3}\right )} x^{3} + 1519 \, n^{3} + 42 \,{\left (a^{2} n^{5} - 50 \, a^{2} n^{3} + 409 \, a^{2} n\right )} x^{2} + 7 \,{\left (a n^{6} - 65 \, a n^{4} + 919 \, a n^{2} - 1575 \, a\right )} x - 6483 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{5} n^{8} - 84 \, a c^{5} n^{6} + 1974 \, a c^{5} n^{4} +{\left (a^{9} c^{5} n^{8} - 84 \, a^{9} c^{5} n^{6} + 1974 \, a^{9} c^{5} n^{4} - 12916 \, a^{9} c^{5} n^{2} + 11025 \, a^{9} c^{5}\right )} x^{8} - 12916 \, a c^{5} n^{2} - 4 \,{\left (a^{7} c^{5} n^{8} - 84 \, a^{7} c^{5} n^{6} + 1974 \, a^{7} c^{5} n^{4} - 12916 \, a^{7} c^{5} n^{2} + 11025 \, a^{7} c^{5}\right )} x^{6} + 11025 \, a c^{5} + 6 \,{\left (a^{5} c^{5} n^{8} - 84 \, a^{5} c^{5} n^{6} + 1974 \, a^{5} c^{5} n^{4} - 12916 \, a^{5} c^{5} n^{2} + 11025 \, a^{5} c^{5}\right )} x^{4} - 4 \,{\left (a^{3} c^{5} n^{8} - 84 \, a^{3} c^{5} n^{6} + 1974 \, a^{3} c^{5} n^{4} - 12916 \, a^{3} c^{5} n^{2} + 11025 \, a^{3} c^{5}\right )} x^{2}} \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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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