Optimal. Leaf size=187 \[ \frac{a^3 (93 a x+80)}{5 c^3 \sqrt{1-a^2 x^2}}+\frac{4 a^3 (6 a x+5)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 a^3 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{18 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3} \]
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Rubi [A] time = 0.524817, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 852, 1805, 1807, 807, 266, 63, 208} \[ \frac{a^3 (93 a x+80)}{5 c^3 \sqrt{1-a^2 x^2}}+\frac{4 a^3 (6 a x+5)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 a^3 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{18 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 852
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^4 (c-a c x)^3} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^4 (c-a c x)^4} \, dx\\ &=\frac{\int \frac{(c+a c x)^4}{x^4 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 c^4-20 a c^4 x-35 a^2 c^4 x^2-40 a^3 c^4 x^3-32 a^4 c^4 x^4}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{15 c^4+60 a c^4 x+120 a^2 c^4 x^2+180 a^3 c^4 x^3+144 a^4 c^4 x^4}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{-15 c^4-60 a c^4 x-135 a^2 c^4 x^2-240 a^3 c^4 x^3}{x^4 \sqrt{1-a^2 x^2}} \, dx}{15 c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}+\frac{\int \frac{180 a c^4+435 a^2 c^4 x+720 a^3 c^4 x^2}{x^3 \sqrt{1-a^2 x^2}} \, dx}{45 c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{\int \frac{-870 a^2 c^4-1620 a^3 c^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{90 c^7}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}+\frac{\left (18 a^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c^3}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}+\frac{\left (9 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{c^3}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}-\frac{(18 a) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{c^3}\\ &=\frac{8 a^3 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 a^3 (5+6 a x)}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^3 (80+93 a x)}{5 c^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c^3 x^3}-\frac{2 a \sqrt{1-a^2 x^2}}{c^3 x^2}-\frac{29 a^2 \sqrt{1-a^2 x^2}}{3 c^3 x}-\frac{18 a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^3}\\ \end{align*}
Mathematica [A] time = 0.0619358, size = 121, normalized size = 0.65 \[ \frac{424 a^6 x^6-578 a^5 x^5-328 a^4 x^4+604 a^3 x^3-85 a^2 x^2-270 a^3 x^3 (a x-1)^2 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-20 a x-5}{15 c^3 x^3 (a x-1)^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 355, normalized size = 1.9 \begin{align*} -{\frac{1}{{c}^{3}} \left ( 2\,a \left ( 1/5\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-2/5\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) \right ) +{\frac{29\,{a}^{2}}{3\,x}\sqrt{-{a}^{2}{x}^{2}+1}}+16\,{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -7\,{a}^{2} \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) +16\,{{a}^{2}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-4\,a \left ( -1/2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{x}^{2}}}-1/2\,{a}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) +{\frac{1}{3\,{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+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 (a c x - c\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66657, size = 389, normalized size = 2.08 \begin{align*} \frac{324 \, a^{6} x^{6} - 972 \, a^{5} x^{5} + 972 \, a^{4} x^{4} - 324 \, a^{3} x^{3} + 270 \,{\left (a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (424 \, a^{5} x^{5} - 1002 \, a^{4} x^{4} + 674 \, a^{3} x^{3} - 70 \, a^{2} x^{2} - 15 \, a x - 5\right )} \sqrt{-a^{2} x^{2} + 1}}{15 \,{\left (a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} + 3 \, a c^{3} x^{4} - c^{3} x^{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{a x}{a^{3} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \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 [B] time = 1.34403, size = 531, normalized size = 2.84 \begin{align*} -\frac{18 \, a^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{c^{3}{\left | a \right |}} - \frac{{\left (5 \, a^{4} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2}}{x} + \frac{335 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac{7559 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}} + \frac{25195 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{4} x^{4}} - \frac{36035 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{6} x^{5}} + \frac{24225 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{8} x^{6}} - \frac{6585 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{10} x^{7}}\right )} a^{6} x^{3}}{120 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} - \frac{\frac{117 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{6}}{x} + \frac{12 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{6}}{x^{2}} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{6}}{x^{3}}}{24 \, a^{2} c^{9}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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