Optimal. Leaf size=125 \[ \frac{(a x+1)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (a x+1)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{24 (a x+1)}{5 a c^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^2}-\frac{3 \sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.334767, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6157, 6148, 1635, 641, 216} \[ \frac{(a x+1)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (a x+1)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{24 (a x+1)}{5 a c^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^2}-\frac{3 \sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6148
Rule 1635
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=\frac{a^4 \int \frac{e^{3 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac{a^4 \int \frac{x^4 (1+a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac{(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{a^4 \int \frac{(1+a x)^2 \left (\frac{3}{a^4}+\frac{5 x}{a^3}+\frac{5 x^2}{a^2}+\frac{5 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^2}\\ &=\frac{(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^4 \int \frac{(1+a x) \left (\frac{27}{a^4}+\frac{30 x}{a^3}+\frac{15 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^2}\\ &=\frac{(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{24 (1+a x)}{5 a c^2 \sqrt{1-a^2 x^2}}-\frac{a^4 \int \frac{\frac{45}{a^4}+\frac{15 x}{a^3}}{\sqrt{1-a^2 x^2}} \, dx}{15 c^2}\\ &=\frac{(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{24 (1+a x)}{5 a c^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^2}-\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=\frac{(1+a x)^3}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{6 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{24 (1+a x)}{5 a c^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2}}{a c^2}-\frac{3 \sin ^{-1}(a x)}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0989422, size = 88, normalized size = 0.7 \[ \frac{-5 a^4 x^4+34 a^3 x^3-18 a^2 x^2-15 (a x-1)^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-33 a x+24}{5 a c^2 (a x-1)^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 212, normalized size = 1.7 \begin{align*} -{\frac{a{x}^{2}}{{c}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+7\,{\frac{1}{a{c}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}}+10\,{\frac{x}{{c}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}}-3\,{\frac{1}{{c}^{2}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{2}{5\,{a}^{3}{c}^{2}} \left ( x-{a}^{-1} \right ) ^{-2}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{13}{5\,{a}^{2}{c}^{2}} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}-{\frac{26\,x}{5\,{c}^{2}}{\frac{1}{\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{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15279, size = 317, normalized size = 2.54 \begin{align*} \frac{24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right )} \sqrt{-a^{2} x^{2} + 1} - 24}{5 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a 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{a^{4} \left (\int \frac{x^{4}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18864, size = 243, normalized size = 1.94 \begin{align*} -\frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c^{2}{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a c^{2}} - \frac{2 \,{\left (\frac{80 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{120 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{70 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 19\right )}}{5 \, c^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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