Optimal. Leaf size=100 \[ \frac {2 a^2 (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \]
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Rubi [A] time = 0.26, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {2 a^2 (a x+1)}{c \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 1807
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 (c-a c x)} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^3 (c-a c x)^2} \, dx\\ &=\frac {\int \frac {(c+a c x)^2}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\int \frac {-c^2-2 a c^2 x-2 a^2 c^2 x^2}{x^3 \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}+\frac {\int \frac {4 a c^2+5 a^2 c^2 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{2 c^3}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}+\frac {\left (5 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{2 c}\\ &=\frac {2 a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {2 a \sqrt {1-a^2 x^2}}{c x}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.83 \[ -\frac {-8 a^3 x^3-5 a^2 x^2+5 a^2 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+4 a x+1}{2 c x^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 99, normalized size = 0.99 \[ \frac {4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 5 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{2} x^{2} - 3 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a c x^{3} - c x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 224, normalized size = 2.24 \[ -\frac {{\left (a^{3} + \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {5 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c {\left | a \right |}} - \frac {\frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 99, normalized size = 0.99 \[ -\frac {\frac {5 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {2 a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {2 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 117, normalized size = 1.17 \[ -\frac {\sqrt {1-a^2\,x^2}}{2\,c\,x^2}-\frac {2\,a\,\sqrt {1-a^2\,x^2}}{c\,x}-\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a x}{a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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