Optimal. Leaf size=99 \[ \frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}-\frac {3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6263, 866,
1819, 821, 272, 65, 214} \begin {gather*} \frac {4 a (a x+1)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}+\frac {a (11 a x+9)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^2} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^2 (c-a c x)^3} \, dx\\ &=\frac {\int \frac {(c+a c x)^3}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^5}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {-3 c^3-9 a c^3 x-8 a^2 c^3 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^5}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3 c^3+9 a c^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{3 c^5}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}+\frac {(3 a) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^2}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}-\frac {3 \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 c^2}\\ &=\frac {4 a (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a (9+11 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x}-\frac {3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 91, normalized size = 0.92 \begin {gather*} \frac {3-16 a x-5 a^2 x^2+14 a^3 x^3-9 a x (-1+a x) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 c^2 x (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 118, normalized size = 1.19
method | result | size |
default | \(\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {11 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-3 a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2}}\) | \(118\) |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {a \left (-\frac {2 \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{a}+\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}+3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c^{2}}\) | \(178\) |
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 = 0.38, size = 118, normalized size = 1.19 \begin {gather*} \frac {13 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 13 \, a x + 9 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (14 \, a^{2} x^{2} - 19 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\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^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 307, normalized size = 3.10 \begin {gather*} \frac {{\left (9 \, a^{2} \log \left (2\right ) - 18 \, a^{2} \log \left (i + 1\right ) + 28 i \, a^{2}\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + \frac {18 \, a^{2} \log \left (\sqrt {-\frac {2 \, c}{a c x - c} - 1} + 1\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {18 \, a^{2} \log \left ({\left | \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1 \right |}\right )}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {6 \, a^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{{\left (\frac {c}{a c x - c} + 1\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {2 \, {\left (a^{2} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2} + 12 \, a^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} \mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{2} \mathrm {sgn}\left (a\right )^{2} \mathrm {sgn}\left (c\right )^{2}\right )}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right )^{3} \mathrm {sgn}\left (a\right )^{3} \mathrm {sgn}\left (c\right )^{3}}}{6 \, c^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 146, normalized size = 1.47 \begin {gather*} \frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,\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}}{c^2\,x}+\frac {11\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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