Optimal. Leaf size=129 \[ -\frac{a^4 x^5 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (6 a x+5)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (8 a x+5)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{16 \sqrt{1-a^2 x^2}}{5 a c^3}+\frac{\sin ^{-1}(a x)}{a c^3} \]
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Rubi [A] time = 0.196946, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6157, 6148, 819, 641, 216} \[ -\frac{a^4 x^5 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (6 a x+5)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (8 a x+5)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{16 \sqrt{1-a^2 x^2}}{5 a c^3}+\frac{\sin ^{-1}(a x)}{a c^3} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6148
Rule 819
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^3} \, dx &=-\frac{a^6 \int \frac{e^{\tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac{a^6 \int \frac{x^6 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=-\frac{a^4 x^5 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 \int \frac{x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^3}\\ &=-\frac{a^4 x^5 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (5+6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 \int \frac{x^2 (15+24 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3}\\ &=-\frac{a^4 x^5 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (5+6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (5+8 a x)}{5 c^3 \sqrt{1-a^2 x^2}}+\frac{\int \frac{15+48 a x}{\sqrt{1-a^2 x^2}} \, dx}{15 c^3}\\ &=-\frac{a^4 x^5 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (5+6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (5+8 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{16 \sqrt{1-a^2 x^2}}{5 a c^3}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac{a^4 x^5 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (5+6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (5+8 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{16 \sqrt{1-a^2 x^2}}{5 a c^3}+\frac{\sin ^{-1}(a x)}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.0761693, size = 108, normalized size = 0.84 \[ \frac{15 a^5 x^5-38 a^4 x^4-52 a^3 x^3+87 a^2 x^2+15 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+33 a x-48}{15 a c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 259, normalized size = 2. \begin{align*} -{\frac{1}{a{c}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{{c}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{20\,{a}^{4}{c}^{3}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{23}{60\,{a}^{3}{c}^{3}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{493}{240\,{a}^{2}{c}^{3}}\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\,{a}^{3}{c}^{3} \left ( x+{a}^{-1} \right ) ^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{25}{48\,{a}^{2}{c}^{3} \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*} \int \frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22016, size = 456, normalized size = 3.53 \begin{align*} -\frac{48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, 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 (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} - 48}{15 \,{\left (a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a 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{a^{6} \int \frac{x^{6}}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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