Optimal. Leaf size=161 \[ \frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {17 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \]
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Rubi [A]
time = 0.29, antiderivative size = 161, 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 = {6263, 866,
1819, 1821, 821, 272, 65, 214} \begin {gather*} -\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}+\frac {a^3 (23 a x+21)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {4 a^3 (a x+1)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {17 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 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 1821
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 (c-a c x)^2} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^4 (c-a c x)^3} \, dx\\ &=\frac {\int \frac {(c+a c x)^3}{x^4 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^5}\\ &=\frac {4 a^3 (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-12 a^2 c^3 x^2-12 a^3 c^3 x^3-8 a^4 c^3 x^4}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^5}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3 c^3+9 a c^3 x+15 a^2 c^3 x^2+21 a^3 c^3 x^3}{x^4 \sqrt {1-a^2 x^2}} \, dx}{3 c^5}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {\int \frac {-27 a c^3-51 a^2 c^3 x-63 a^3 c^3 x^2}{x^3 \sqrt {1-a^2 x^2}} \, dx}{9 c^5}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}+\frac {\int \frac {102 a^2 c^3+153 a^3 c^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{18 c^5}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {\left (17 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 c^2}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {\left (17 a^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {(17 a) \text {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 {4 a^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 (21+23 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c^2 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c^2 x^2}-\frac {17 a^2 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {17 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 111, normalized size = 0.69 \begin {gather*} \frac {2+7 a x+23 a^2 x^2-91 a^3 x^3-29 a^4 x^4+80 a^5 x^5-51 a^3 x^3 (-1+a x) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{6 c^2 x^3 (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 226, normalized size = 1.40
method | result | size |
risch | \(\frac {34 a^{4} x^{4}+9 a^{3} x^{3}-32 a^{2} x^{2}-9 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {a^{3} \left (\frac {\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {14 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-17 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2 c^{2}}\) | \(202\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {17 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+2 a^{2} \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )+3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )-7 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {7 a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{c^{2}}\) | \(226\) |
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 = 144, normalized size = 0.89 \begin {gather*} \frac {50 \, a^{5} x^{5} - 100 \, a^{4} x^{4} + 50 \, a^{3} x^{3} + 51 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (80 \, a^{4} x^{4} - 109 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 5 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} c^{2} x^{5} - 2 \, a c^{2} x^{4} + c^{2} x^{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^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{4} \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 = 0.44, size = 372, normalized size = 2.31 \begin {gather*} \frac {2 \, {\left (51 \, a^{4} \log \left (2\right ) - 102 \, a^{4} \log \left (i + 1\right ) + 160 i \, a^{4}\right )} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + \frac {204 \, a^{4} \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 {204 \, a^{4} \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 {45 \, a^{4} {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 64 \, a^{4} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 27 \, a^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{{\left (\frac {c}{a c x - c} + 1\right )}^{3} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )} - \frac {8 \, {\left (a^{4} {\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} + 24 \, a^{4} \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}}}{24 \, 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 = 192, normalized size = 1.19 \begin {gather*} \frac {2\,a^5\,\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}}{3\,c^2\,x^3}-\frac {3\,a\,\sqrt {1-a^2\,x^2}}{2\,c^2\,x^2}-\frac {17\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c^2\,x}+\frac {23\,a^4\,\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^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,17{}\mathrm {i}}{2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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