Optimal. Leaf size=72 \[ \frac{(a x+1)^2}{a^3 c \sqrt{1-a^2 x^2}}+\frac{(a x+6) \sqrt{1-a^2 x^2}}{2 a^3 c}-\frac{5 \sin ^{-1}(a x)}{2 a^3 c} \]
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Rubi [A] time = 0.200876, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6128, 852, 1635, 780, 216} \[ \frac{(a x+1)^2}{a^3 c \sqrt{1-a^2 x^2}}+\frac{(a x+6) \sqrt{1-a^2 x^2}}{2 a^3 c}-\frac{5 \sin ^{-1}(a x)}{2 a^3 c} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1635
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^2}{c-a c x} \, dx &=c \int \frac{x^2 \sqrt{1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac{\int \frac{x^2 (c+a c x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{(1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}-\frac{\int \frac{\left (\frac{2}{a^2}+\frac{x}{a}\right ) (c+a c x)}{\sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=\frac{(1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}+\frac{(6+a x) \sqrt{1-a^2 x^2}}{2 a^3 c}-\frac{5 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^2 c}\\ &=\frac{(1+a x)^2}{a^3 c \sqrt{1-a^2 x^2}}+\frac{(6+a x) \sqrt{1-a^2 x^2}}{2 a^3 c}-\frac{5 \sin ^{-1}(a x)}{2 a^3 c}\\ \end{align*}
Mathematica [A] time = 0.0437367, size = 64, normalized size = 0.89 \[ \frac{10 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-\frac{\sqrt{a x+1} \left (a^2 x^2+3 a x-8\right )}{\sqrt{1-a x}}}{2 a^3 c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 120, normalized size = 1.7 \begin{align*}{\frac{x}{2\,{a}^{2}c}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{5}{2\,{a}^{2}c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{a}^{3}c}}-2\,{\frac{1}{c{a}^{4}}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57929, size = 178, normalized size = 2.47 \begin{align*} \frac{8 \, a x + 10 \,{\left (a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (a^{2} x^{2} + 3 \, a x - 8\right )} \sqrt{-a^{2} x^{2} + 1} - 8}{2 \,{\left (a^{4} c x - a^{3} c\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^{2}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18892, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a^{2} c} + \frac{4}{a^{3} c}\right )} - \frac{5 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a^{2} c{\left | a \right |}} + \frac{4}{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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