Optimal. Leaf size=101 \[ \frac {a x+1}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 a x+15}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 a x+5}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]
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Rubi [A] time = 0.14, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6148, 823, 12, 266, 63, 208} \[ \frac {a x+1}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 a x+15}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {4 a x+5}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {1+a x}{x \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {\int \frac {5 a^2+4 a^3 x}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {15 a^4+8 a^5 x}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {15 a^6}{x \sqrt {1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2 c^3}\\ &=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 108, normalized size = 1.07 \[ \frac {8 a^4 x^4+7 a^3 x^3-27 a^2 x^2-15 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-8 a x+23}{15 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 198, normalized size = 1.96 \[ \frac {23 \, a^{5} x^{5} - 23 \, a^{4} x^{4} - 46 \, a^{3} x^{3} + 46 \, a^{2} x^{2} + 23 \, a x + 15 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - 27 \, a^{2} x^{2} - 8 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} - 23}{15 \, {\left (a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x - c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 384, normalized size = 3.80 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a}+\frac {11 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{2}}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a}-\frac {5 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \left (x +\frac {1}{a}\right )}}{c^{3}} \]
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 \[ -\int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 306, normalized size = 3.03 \[ \frac {a^2\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {17\,a\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {71\,a\,\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \]
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 \[ \frac {\int \frac {a}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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