Optimal. Leaf size=105 \[ -\frac {4 a^2 (4 a x+3)}{3 c \sqrt {1-a^2 x^2}}-\frac {8 a^2 (a x+1)}{3 c \left (1-a^2 x^2\right )^{3/2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
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Rubi [A] time = 0.32, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6131, 6128, 852, 1805, 807, 266, 63, 208} \[ -\frac {4 a^2 (4 a x+3)}{3 c \sqrt {1-a^2 x^2}}-\frac {8 a^2 (a x+1)}{3 c \left (1-a^2 x^2\right )^{3/2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right ) x^3} \, dx &=-\frac {a \int \frac {e^{3 \tanh ^{-1}(a x)}}{x^2 (1-a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)^4} \, dx}{c}\\ &=-\frac {a \int \frac {(1+a x)^4}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}+\frac {a \int \frac {-3-12 a x-13 a^2 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}-\frac {a \int \frac {3+12 a x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{3 c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}-\frac {\left (4 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{c}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 \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}\\ &=-\frac {8 a^2 (1+a x)}{3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 a^2 (3+4 a x)}{3 c \sqrt {1-a^2 x^2}}+\frac {a \sqrt {1-a^2 x^2}}{c x}+\frac {4 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 0.88 \[ \frac {a \left (-19 a^3 x^3+7 a^2 x^2+12 a x (a x-1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+23 a x-3\right )}{3 c x (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 120, normalized size = 1.14 \[ -\frac {20 \, a^{4} x^{3} - 40 \, a^{3} x^{2} + 20 \, a^{2} x + 12 \, {\left (a^{4} x^{3} - 2 \, a^{3} x^{2} + a^{2} x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (19 \, a^{3} x^{2} - 26 \, a^{2} x + 3 \, a\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c x^{3} - 2 \, a c x^{2} + c x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 217, normalized size = 2.07 \[ \frac {4 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c {\left | a \right |}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{2 \, c x {\left | a \right |}} + \frac {{\left (3 \, a^{3} - \frac {89 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} + \frac {153 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} - \frac {99 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}}\right )} a^{2} x}{6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 165, normalized size = 1.57 \[ \frac {a \left (-\frac {a^{2} x}{\sqrt {-a^{2} x^{2}+1}}-4 a \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+8 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )\right )}{c} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 138, normalized size = 1.31 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{c\,x}-\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )}+\frac {16\,a^3\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{c} \]
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 \[ \frac {a \left (\int \frac {3 a x}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{3} x^{5} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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