Optimal. Leaf size=162 \[ \frac {a^2 (164 a x+135)}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 a^2 (13 a x+10)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 a^2 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {19 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \]
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Rubi [A] time = 0.43, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {a^2 (164 a x+135)}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 a^2 (13 a x+10)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 a^2 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {19 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3} \]
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)^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^3 (c-a c x)^4} \, dx\\ &=\frac {\int \frac {(c+a c x)^4}{x^3 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac {8 a^2 (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^2-32 a^3 c^4 x^3}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^7}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 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+104 a^3 c^4 x^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^7}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 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}{x^3 \sqrt {1-a^2 x^2}} \, dx}{15 c^7}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}+\frac {\int \frac {120 a c^4+285 a^2 c^4 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{30 c^7}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}+\frac {\left (19 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c^3}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}+\frac {\left (19 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c^3}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {19 \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^3}\\ &=\frac {8 a^2 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a^2 (10+13 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 (135+164 a x)}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c^3 x^2}-\frac {4 a \sqrt {1-a^2 x^2}}{c^3 x}-\frac {19 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 113, normalized size = 0.70 \[ \frac {448 a^5 x^5-611 a^4 x^4-346 a^3 x^3+638 a^2 x^2-285 a^2 x^2 (a x-1)^2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-90 a x-15}{30 c^3 x^2 (a x-1)^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 173, normalized size = 1.07 \[ \frac {398 \, a^{5} x^{5} - 1194 \, a^{4} x^{4} + 1194 \, a^{3} x^{3} - 398 \, a^{2} x^{2} + 285 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (448 \, a^{4} x^{4} - 1059 \, a^{3} x^{3} + 713 \, a^{2} x^{2} - 75 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{3} c^{3} x^{5} - 3 \, a^{2} c^{3} x^{4} + 3 \, a c^{3} x^{3} - c^{3} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 338, normalized size = 2.09 \[ -\frac {19 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c^{3} {\left | a \right |}} - \frac {{\left (15 \, a^{3} + \frac {165 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {4234 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac {14330 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac {20965 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac {14385 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac {4080 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}}\right )} a^{4} x^{2}}{120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} - \frac {\frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{3} {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3} {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 223, normalized size = 1.38 \[ -\frac {\frac {19 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {4 a \sqrt {-a^{2} x^{2}+1}}{x}-\frac {29 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}+\frac {9 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}}{c^{3}} \]
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 \[ -\int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 257, normalized size = 1.59 \[ \frac {29\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^3\,x^2}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{c^3\,x}+\frac {164\,a^3\,\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,19{}\mathrm {i}}{2\,c^3} \]
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 \[ - \frac {\int \frac {a x}{a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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