Optimal. Leaf size=97 \[ \frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 866,
1819, 837, 12, 272, 65, 214} \begin {gather*} \frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 a x+5}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {8 (a x+1)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 866
Rule 1819
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x (c-a c x)^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x (c-a c x)^4} \, dx\\ &=\frac {\int \frac {(c+a c x)^4}{x \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 c^4-12 a c^4 x+5 a^2 c^4 x^2}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^7}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {15 c^4+24 a c^4 x}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^7}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {15 a^2 c^4}{x \sqrt {1-a^2 x^2}} \, dx}{15 a^2 c^7}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 c^3 \sqrt {1-a^2 x^2}}-\frac {\text {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 {8 (1+a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 a x}{5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {5+8 a x}{5 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 [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 71, normalized size = 0.73 \begin {gather*} \frac {16+60 a x+5 a^2 x^2-60 a^3 x^3+24 a^5 x^5+3 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-a^2 x^2\right )}{15 c^3 \left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs.
\(2(83)=166\).
time = 1.06, size = 275, normalized size = 2.84
method | result | size |
default | \(-\frac {-\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {4 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{2}}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}}{c^{3}}\) | \(275\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 130, normalized size = 1.34 \begin {gather*} \frac {13 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 39 \, a x + 5 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{2} x^{2} - 19 \, a x + 13\right )} \sqrt {-a^{2} x^{2} + 1} - 13}{5 \, {\left (a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right )}} \end {gather*}
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 \begin {gather*} - \frac {\int \frac {a x}{a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{2} \sqrt {- a^{2} x^{2} + 1} - x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + 3 a x^{2} \sqrt {- a^{2} x^{2} + 1} - x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (83) = 166\).
time = 0.42, size = 189, normalized size = 1.95 \begin {gather*} -\frac {a \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^{3} {\left | a \right |}} + \frac {2 \, {\left (13 \, a - \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a x} + \frac {75 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{3} x^{2}} - \frac {55 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{5} x^{3}} + \frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{7} x^{4}}\right )}}{5 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 209, normalized size = 2.15 \begin {gather*} \frac {3\,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 {8\,a\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {2\,a\,\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 {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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