Optimal. Leaf size=146 \[ \frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{11 x \sqrt{1-a^2 x^2}}{8 a^4 c}+\frac{13 \sqrt{1-a^2 x^2}}{3 a^5 c}+\frac{(a x+1)^2}{a^5 c \sqrt{1-a^2 x^2}}-\frac{27 \sin ^{-1}(a x)}{8 a^5 c} \]
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Rubi [A] time = 0.343283, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6128, 852, 1635, 1815, 641, 216} \[ \frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{11 x \sqrt{1-a^2 x^2}}{8 a^4 c}+\frac{13 \sqrt{1-a^2 x^2}}{3 a^5 c}+\frac{(a x+1)^2}{a^5 c \sqrt{1-a^2 x^2}}-\frac{27 \sin ^{-1}(a x)}{8 a^5 c} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1635
Rule 1815
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{c-a c x} \, dx &=c \int \frac{x^4 \sqrt{1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac{\int \frac{x^4 (c+a c x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}-\frac{\int \frac{(c+a c x) \left (\frac{2}{a^4}+\frac{x}{a^3}+\frac{x^2}{a^2}+\frac{x^3}{a}\right )}{\sqrt{1-a^2 x^2}} \, dx}{c^2}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}+\frac{\int \frac{-\frac{8 c}{a^2}-\frac{12 c x}{a}-11 c x^2-8 a c x^3}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2 c^2}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}-\frac{\int \frac{24 c+52 a c x+33 a^2 c x^2}{\sqrt{1-a^2 x^2}} \, dx}{12 a^4 c^2}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}+\frac{11 x \sqrt{1-a^2 x^2}}{8 a^4 c}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}+\frac{\int \frac{-81 a^2 c-104 a^3 c x}{\sqrt{1-a^2 x^2}} \, dx}{24 a^6 c^2}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}+\frac{13 \sqrt{1-a^2 x^2}}{3 a^5 c}+\frac{11 x \sqrt{1-a^2 x^2}}{8 a^4 c}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}-\frac{27 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^4 c}\\ &=\frac{(1+a x)^2}{a^5 c \sqrt{1-a^2 x^2}}+\frac{13 \sqrt{1-a^2 x^2}}{3 a^5 c}+\frac{11 x \sqrt{1-a^2 x^2}}{8 a^4 c}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a^3 c}+\frac{x^3 \sqrt{1-a^2 x^2}}{4 a^2 c}-\frac{27 \sin ^{-1}(a x)}{8 a^5 c}\\ \end{align*}
Mathematica [A] time = 0.0647541, size = 81, normalized size = 0.55 \[ \frac{162 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )-\frac{\sqrt{a x+1} \left (6 a^4 x^4+10 a^3 x^3+17 a^2 x^2+47 a x-128\right )}{\sqrt{1-a x}}}{24 a^5 c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 166, normalized size = 1.1 \begin{align*}{\frac{{x}^{3}}{4\,{a}^{2}c}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{11\,x}{8\,{a}^{4}c}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{27}{8\,{a}^{4}c}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{2\,{x}^{2}}{3\,{a}^{3}c}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{10}{3\,{a}^{5}c}\sqrt{-{a}^{2}{x}^{2}+1}}-2\,{\frac{1}{c{a}^{6}}\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 = 1.56577, size = 228, normalized size = 1.56 \begin{align*} \frac{128 \, a x + 162 \,{\left (a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (6 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 47 \, a x - 128\right )} \sqrt{-a^{2} x^{2} + 1} - 128}{24 \,{\left (a^{6} c x - a^{5} 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^{4}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{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 = 1.21367, size = 154, normalized size = 1.05 \begin{align*} \frac{1}{24} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, x{\left (\frac{3 \, x}{a^{2} c} + \frac{8}{a^{3} c}\right )} + \frac{33}{a^{4} c}\right )} x + \frac{80}{a^{5} c}\right )} - \frac{27 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \, a^{4} c{\left | a \right |}} + \frac{4}{a^{4} 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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