Optimal. Leaf size=134 \[ -\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}+\frac {4 a x+5}{3 c^2 x^2 \sqrt {1-a^2 x^2}}+\frac {a x+1}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \]
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Rubi [A] time = 0.16, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6148, 823, 835, 807, 266, 63, 208} \[ -\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}+\frac {4 a x+5}{3 c^2 x^2 \sqrt {1-a^2 x^2}}+\frac {a x+1}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 835
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1+a x}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {5 a^2+4 a^3 x}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {15 a^4+8 a^5 x}{x^3 \sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {\int \frac {-16 a^5-15 a^6 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{6 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {\left (5 a^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 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^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 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^2}\\ &=\frac {1+a x}{3 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+4 a x}{3 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {8 a \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {5 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 103, normalized size = 0.77 \[ \frac {16 a^4 x^4-a^3 x^3-23 a^2 x^2-15 a^2 x^2 (a x-1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+3 a x+3}{6 c^2 x^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 171, normalized size = 1.28 \[ \frac {14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 15 \, {\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{4} x^{4} - a^{3} x^{3} - 23 \, a^{2} x^{2} + 3 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{3} c^{2} x^{5} - a^{2} c^{2} x^{4} - a c^{2} x^{3} + c^{2} x^{2}\right )}} \]
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 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 214, normalized size = 1.60 \[ \frac {-\frac {5 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{2}-\frac {7 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 x +\frac {4}{a}}}{c^{2}} \]
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 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 221, normalized size = 1.65 \[ \frac {a^4\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^2\,x^2}-\frac {a\,\sqrt {1-a^2\,x^2}}{c^2\,x}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {23\,a^3\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2\,c^2} \]
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 \[ \frac {\int \frac {a}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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