Optimal. Leaf size=110 \[ \frac{x^4 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^2 (5 a x+4)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{(15 a x+16) \sqrt{1-a^2 x^2}}{6 a^6 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^6 c^2} \]
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Rubi [A] time = 0.135965, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 819, 780, 216} \[ \frac{x^4 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^2 (5 a x+4)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{(15 a x+16) \sqrt{1-a^2 x^2}}{6 a^6 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^6 c^2} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 819
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{x^5 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac{x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{x^3 (4+5 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac{x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^2 (4+5 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}+\frac{\int \frac{x (8+15 a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac{x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^2 (4+5 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{(16+15 a x) \sqrt{1-a^2 x^2}}{6 a^6 c^2}+\frac{5 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^5 c^2}\\ &=\frac{x^4 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x^2 (4+5 a x)}{3 a^4 c^2 \sqrt{1-a^2 x^2}}-\frac{(16+15 a x) \sqrt{1-a^2 x^2}}{6 a^6 c^2}+\frac{5 \sin ^{-1}(a x)}{2 a^6 c^2}\\ \end{align*}
Mathematica [A] time = 0.049769, size = 86, normalized size = 0.78 \[ \frac{3 a^4 x^4+3 a^3 x^3-23 a^2 x^2+15 (a x-1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-a x+16}{6 a^6 c^2 (a x-1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 202, normalized size = 1.8 \begin{align*} -{\frac{x}{2\,{c}^{2}{a}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{5}{2\,{c}^{2}{a}^{5}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{{c}^{2}{a}^{6}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{4\,{c}^{2}{a}^{7} \left ( x+{a}^{-1} \right ) }\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{1}{6\,{c}^{2}{a}^{8}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{25}{12\,{c}^{2}{a}^{7}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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*} a \int \frac{x^{6}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} - \frac{\frac{3 \, \sqrt{-a^{2} x^{2} + 1}}{c^{2}} - \frac{6 \, a^{2} x^{2} - 5}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c^{2}}}{3 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56941, size = 323, normalized size = 2.94 \begin{align*} -\frac{16 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 16 \, a x + 30 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - a x + 16\right )} \sqrt{-a^{2} x^{2} + 1} + 16}{6 \,{\left (a^{9} c^{2} x^{3} - a^{8} c^{2} x^{2} - a^{7} c^{2} x + a^{6} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{6}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \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 (a x + 1\right )} x^{5}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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