Optimal. Leaf size=166 \[ -\frac{120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.176648, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6185, 6184} \[ -\frac{120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{20 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c \left (25-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{120 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^2 \left (9-n^2\right ) \left (25-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{20 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{120 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.46125, size = 299, normalized size = 1.8 \[ -\frac{a^2 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} e^{n \coth ^{-1}(a x)} \left (\frac{10 n^5}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{340 n^3}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{10 \left (n^4-34 n^2+225\right )}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2250 n}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-5 n^4 \cosh \left (5 \coth ^{-1}(a x)\right )+50 n^2 \cosh \left (5 \coth ^{-1}(a x)\right )+15 \left (n^4-26 n^2+25\right ) \cosh \left (3 \coth ^{-1}(a x)\right )-5 n^5 \sinh \left (3 \coth ^{-1}(a x)\right )+n^5 \sinh \left (5 \coth ^{-1}(a x)\right )+130 n^3 \sinh \left (3 \coth ^{-1}(a x)\right )-10 n^3 \sinh \left (5 \coth ^{-1}(a x)\right )-125 n \sinh \left (3 \coth ^{-1}(a x)\right )+9 n \sinh \left (5 \coth ^{-1}(a x)\right )-45 \cosh \left (5 \coth ^{-1}(a x)\right )\right )}{16 c^2 (n-5) (n-3) (n-1) (n+1) (n+3) (n+5) \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.042, size = 140, normalized size = 0.8 \begin{align*}{\frac{ \left ( 120\,{x}^{5}{a}^{5}-120\,n{a}^{4}{x}^{4}+60\,{a}^{3}{n}^{2}{x}^{3}-20\,{a}^{2}{n}^{3}{x}^{2}-300\,{x}^{3}{a}^{3}+5\,a{n}^{4}x+260\,n{a}^{2}{x}^{2}-{n}^{5}-110\,a{n}^{2}x+30\,{n}^{3}+225\,ax-149\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{a \left ({n}^{6}-35\,{n}^{4}+259\,{n}^{2}-225 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69961, size = 622, normalized size = 3.75 \begin{align*} -\frac{{\left (120 \, a^{5} x^{5} + 120 \, a^{4} n x^{4} + n^{5} + 60 \,{\left (a^{3} n^{2} - 5 \, a^{3}\right )} x^{3} - 30 \, n^{3} + 20 \,{\left (a^{2} n^{3} - 13 \, a^{2} n\right )} x^{2} + 5 \,{\left (a n^{4} - 22 \, a n^{2} + 45 \, a\right )} x + 149 \, n\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{4} n^{6} - 35 \, a c^{4} n^{4} + 259 \, a c^{4} n^{2} -{\left (a^{7} c^{4} n^{6} - 35 \, a^{7} c^{4} n^{4} + 259 \, a^{7} c^{4} n^{2} - 225 \, a^{7} c^{4}\right )} x^{6} - 225 \, a c^{4} + 3 \,{\left (a^{5} c^{4} n^{6} - 35 \, a^{5} c^{4} n^{4} + 259 \, a^{5} c^{4} n^{2} - 225 \, a^{5} c^{4}\right )} x^{4} - 3 \,{\left (a^{3} c^{4} n^{6} - 35 \, a^{3} c^{4} n^{4} + 259 \, a^{3} c^{4} n^{2} - 225 \, a^{3} c^{4}\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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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