Optimal. Leaf size=108 \[ \frac{x^4 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^2 (5 a x+4)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{15 a x+8}{15 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)}{a^6 c^3} \]
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Rubi [A] time = 0.129224, antiderivative size = 108, 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, 778, 216} \[ \frac{x^4 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^2 (5 a x+4)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{15 a x+8}{15 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)}{a^6 c^3} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 819
Rule 778
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^5 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{x^3 (4+5 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac{x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{x (8+15 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac{x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{8+15 a x}{15 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^5 c^3}\\ &=\frac{x^4 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x^2 (4+5 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{8+15 a x}{15 a^6 c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)}{a^6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0555427, size = 100, normalized size = 0.93 \[ \frac{23 a^4 x^4-8 a^3 x^3-27 a^2 x^2-15 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+7 a x+8}{15 a^6 c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 243, normalized size = 2.3 \begin{align*} -{\frac{1}{{c}^{3}{a}^{5}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{20\,{c}^{3}{a}^{9}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-{\frac{3}{10\,{c}^{3}{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{91}{80\,{c}^{3}{a}^{7}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{1}{24\,{c}^{3}{a}^{8} \left ( x+{a}^{-1} \right ) ^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{19}{48\,{c}^{3}{a}^{7} \left ( x+{a}^{-1} \right ) }\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \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*} -a \int \frac{x^{6}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} + \frac{10 \, a^{2} x^{2} + 15 \,{\left (a^{2} x^{2} - 1\right )}^{2} - 7}{15 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67021, size = 433, normalized size = 4.01 \begin{align*} \frac{8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x + 30 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (23 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 7 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} - 8}{15 \,{\left (a^{11} c^{3} x^{5} - a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{8} c^{3} x^{2} + a^{7} c^{3} x - a^{6} c^{3}\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^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{6}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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 )}^{3} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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