Optimal. Leaf size=110 \[ -\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {1}{3} b c^4 d \log (x)-\frac {7}{24} b c^4 d \log (1-c x)-\frac {1}{24} b c^4 d \log (c x+1)-\frac {b c^3 d}{4 x}-\frac {b c^2 d}{6 x^2}-\frac {b c d}{12 x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {43, 5936, 12, 801} \[ -\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {b c^2 d}{6 x^2}-\frac {b c^3 d}{4 x}+\frac {1}{3} b c^4 d \log (x)-\frac {7}{24} b c^4 d \log (1-c x)-\frac {1}{24} b c^4 d \log (c x+1)-\frac {b c d}{12 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 801
Rule 5936
Rubi steps
\begin {align*} \int \frac {(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d (-3-4 c x)}{12 x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{12} (b c d) \int \frac {-3-4 c x}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{12} (b c d) \int \left (-\frac {3}{x^4}-\frac {4 c}{x^3}-\frac {3 c^2}{x^2}-\frac {4 c^3}{x}+\frac {7 c^4}{2 (-1+c x)}+\frac {c^4}{2 (1+c x)}\right ) \, dx\\ &=-\frac {b c d}{12 x^3}-\frac {b c^2 d}{6 x^2}-\frac {b c^3 d}{4 x}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {1}{3} b c^4 d \log (x)-\frac {7}{24} b c^4 d \log (1-c x)-\frac {1}{24} b c^4 d \log (1+c x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.85 \[ -\frac {d \left (8 a c x+6 a-8 b c^4 x^4 \log (x)+7 b c^4 x^4 \log (1-c x)+b c^4 x^4 \log (c x+1)+6 b c^3 x^3+4 b c^2 x^2+2 b c x+2 b (4 c x+3) \tanh ^{-1}(c x)\right )}{24 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 110, normalized size = 1.00 \[ -\frac {b c^{4} d x^{4} \log \left (c x + 1\right ) + 7 \, b c^{4} d x^{4} \log \left (c x - 1\right ) - 8 \, b c^{4} d x^{4} \log \relax (x) + 6 \, b c^{3} d x^{3} + 4 \, b c^{2} d x^{2} + 2 \, {\left (4 \, a + b\right )} c d x + 6 \, a d + {\left (4 \, b c d x + 3 \, b d\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 401, normalized size = 3.65 \[ \frac {1}{3} \, {\left (b c^{3} d \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - b c^{3} d \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {{\left (\frac {6 \, {\left (c x + 1\right )}^{3} b c^{3} d}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2} b c^{3} d}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )} b c^{3} d}{c x - 1} + b c^{3} d\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {12 \, {\left (c x + 1\right )}^{3} a c^{3} d}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2} a c^{3} d}{{\left (c x - 1\right )}^{2}} + \frac {8 \, {\left (c x + 1\right )} a c^{3} d}{c x - 1} + 2 \, a c^{3} d + \frac {5 \, {\left (c x + 1\right )}^{3} b c^{3} d}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{2} b c^{3} d}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} b c^{3} d}{c x - 1} + 2 \, b c^{3} d}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 0.95 \[ -\frac {c d a}{3 x^{3}}-\frac {d a}{4 x^{4}}-\frac {c d b \arctanh \left (c x \right )}{3 x^{3}}-\frac {d b \arctanh \left (c x \right )}{4 x^{4}}-\frac {b c d}{12 x^{3}}-\frac {b \,c^{2} d}{6 x^{2}}-\frac {b \,c^{3} d}{4 x}+\frac {c^{4} d b \ln \left (c x \right )}{3}-\frac {7 c^{4} d b \ln \left (c x -1\right )}{24}-\frac {b \,c^{4} d \ln \left (c x +1\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 114, normalized size = 1.04 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c d + \frac {1}{24} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b d - \frac {a c d}{3 \, x^{3}} - \frac {a d}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 120, normalized size = 1.09 \[ \frac {b\,c^4\,d\,\ln \relax (x)}{3}-\frac {a\,c\,d}{3\,x^3}-\frac {b\,c\,d}{12\,x^3}-\frac {b\,d\,\mathrm {atanh}\left (c\,x\right )}{4\,x^4}-\frac {b\,c^4\,d\,\ln \left (c^2\,x^2-1\right )}{6}-\frac {b\,c^2\,d}{6\,x^2}-\frac {b\,c^3\,d}{4\,x}-\frac {a\,d}{4\,x^4}-\frac {b\,c^5\,d\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{4\,\sqrt {-c^2}}-\frac {b\,c\,d\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.88, size = 129, normalized size = 1.17 \[ \begin {cases} - \frac {a c d}{3 x^{3}} - \frac {a d}{4 x^{4}} + \frac {b c^{4} d \log {\relax (x )}}{3} - \frac {b c^{4} d \log {\left (x - \frac {1}{c} \right )}}{3} - \frac {b c^{4} d \operatorname {atanh}{\left (c x \right )}}{12} - \frac {b c^{3} d}{4 x} - \frac {b c^{2} d}{6 x^{2}} - \frac {b c d \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {b c d}{12 x^{3}} - \frac {b d \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a d}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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