Optimal. Leaf size=197 \[ -\frac{(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac{30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \]
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Rubi [A] time = 0.180955, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ -\frac{(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )}-\frac{30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (n^4-20 n^2+64\right )} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}+\frac{30 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{c \left (36-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac{360 \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c^2 \left (576-52 n^2+n^4\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}+\frac{720 \int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^3 \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}\\ &=\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (4-n^2\right ) \left (576-52 n^2+n^4\right )}-\frac{e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac{30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac{360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (576-52 n^2+n^4\right ) \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.300616, size = 152, normalized size = 0.77 \[ -\frac{\left (10 n^4 \left (3 a^2 x^2-5\right )-120 a n^3 x \left (a^2 x^2-2\right )+8 n^2 \left (45 a^4 x^4-105 a^2 x^2+68\right )-48 a n x \left (15 a^4 x^4-40 a^2 x^2+33\right )+720 \left (a^2 x^2-1\right )^3-6 a n^5 x+n^6\right ) e^{n \coth ^{-1}(a x)}}{a c^4 n \left (n^2-36\right ) \left (n^2-16\right ) \left (n^2-4\right ) \left (a^2 x^2-1\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 167, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 720\,{x}^{6}{a}^{6}-720\,{x}^{5}{a}^{5}n+360\,{a}^{4}{n}^{2}{x}^{4}-120\,{a}^{3}{n}^{3}{x}^{3}-2160\,{x}^{4}{a}^{4}+30\,{a}^{2}{n}^{4}{x}^{2}+1920\,{x}^{3}{a}^{3}n-6\,a{n}^{5}x-840\,{a}^{2}{n}^{2}{x}^{2}+{n}^{6}+240\,a{n}^{3}x+2160\,{a}^{2}{x}^{2}-50\,{n}^{4}-1584\,nax+544\,{n}^{2}-720 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{ \left ({a}^{2}{x}^{2}-1 \right ) ^{3}{c}^{4}an \left ({n}^{6}-56\,{n}^{4}+784\,{n}^{2}-2304 \right ) }} \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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78191, size = 683, normalized size = 3.47 \begin{align*} -\frac{{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \,{\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} + 120 \,{\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \,{\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \,{\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} -{\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \,{\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \,{\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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