Optimal. Leaf size=98 \[ -\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{3} b c^3 d \log (x)-\frac {5}{12} b c^3 d \log (1-c x)+\frac {1}{12} b c^3 d \log (c x+1)-\frac {b c^2 d}{2 x}-\frac {b c d}{6 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 98, 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} \[ -\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {b c^2 d}{2 x}+\frac {1}{3} b c^3 d \log (x)-\frac {5}{12} b c^3 d \log (1-c x)+\frac {1}{12} b c^3 d \log (c x+1)-\frac {b c d}{6 x^2} \]
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^4} \, dx &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-(b c) \int \frac {d (-2-3 c x)}{6 x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {1}{6} (b c d) \int \frac {-2-3 c x}{x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}-\frac {1}{6} (b c d) \int \left (-\frac {2}{x^3}-\frac {3 c}{x^2}-\frac {2 c^2}{x}+\frac {5 c^3}{2 (-1+c x)}-\frac {c^3}{2 (1+c x)}\right ) \, dx\\ &=-\frac {b c d}{6 x^2}-\frac {b c^2 d}{2 x}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{3} b c^3 d \log (x)-\frac {5}{12} b c^3 d \log (1-c x)+\frac {1}{12} b c^3 d \log (1+c x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 86, normalized size = 0.88 \[ -\frac {d \left (6 a c x+4 a-4 b c^3 x^3 \log (x)+5 b c^3 x^3 \log (1-c x)-b c^3 x^3 \log (c x+1)+6 b c^2 x^2+2 b c x+2 b (3 c x+2) \tanh ^{-1}(c x)\right )}{12 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 101, normalized size = 1.03 \[ \frac {b c^{3} d x^{3} \log \left (c x + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (c x - 1\right ) + 4 \, b c^{3} d x^{3} \log \relax (x) - 6 \, b c^{2} d x^{2} - 2 \, {\left (3 \, a + b\right )} c d x - 4 \, a d - {\left (3 \, b c d x + 2 \, b d\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 306, normalized size = 3.12 \[ \frac {1}{3} \, {\left (b c^{2} d \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - b c^{2} d \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {{\left (\frac {6 \, {\left (c x + 1\right )}^{2} b c^{2} d}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} b c^{2} d}{c x - 1} + b c^{2} d\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {12 \, {\left (c x + 1\right )}^{2} a c^{2} d}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\left (c x + 1\right )} a c^{2} d}{c x - 1} + 2 \, a c^{2} d + \frac {5 \, {\left (c x + 1\right )}^{2} b c^{2} d}{{\left (c x - 1\right )}^{2}} + \frac {8 \, {\left (c x + 1\right )} b c^{2} d}{c x - 1} + 3 \, b c^{2} d}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 95, normalized size = 0.97 \[ -\frac {d a}{3 x^{3}}-\frac {c d a}{2 x^{2}}-\frac {d b \arctanh \left (c x \right )}{3 x^{3}}-\frac {c d b \arctanh \left (c x \right )}{2 x^{2}}-\frac {b c d}{6 x^{2}}-\frac {b \,c^{2} d}{2 x}+\frac {c^{3} d b \ln \left (c x \right )}{3}-\frac {5 c^{3} d b \ln \left (c x -1\right )}{12}+\frac {b \,c^{3} d \ln \left (c x +1\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 99, normalized size = 1.01 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c d - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b d - \frac {a c d}{2 \, x^{2}} - \frac {a d}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 110, normalized size = 1.12 \[ \frac {b\,c^3\,d\,\ln \relax (x)}{3}-\frac {a\,c\,d}{2\,x^2}-\frac {b\,c\,d}{6\,x^2}-\frac {b\,d\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3}-\frac {b\,c^3\,d\,\ln \left (c^2\,x^2-1\right )}{6}-\frac {b\,c^2\,d}{2\,x}-\frac {a\,d}{3\,x^3}-\frac {b\,c^4\,d\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{2\,\sqrt {-c^2}}-\frac {b\,c\,d\,\mathrm {atanh}\left (c\,x\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.42, size = 117, normalized size = 1.19 \[ \begin {cases} - \frac {a c d}{2 x^{2}} - \frac {a d}{3 x^{3}} + \frac {b c^{3} d \log {\relax (x )}}{3} - \frac {b c^{3} d \log {\left (x - \frac {1}{c} \right )}}{3} + \frac {b c^{3} d \operatorname {atanh}{\left (c x \right )}}{6} - \frac {b c^{2} d}{2 x} - \frac {b c d \operatorname {atanh}{\left (c x \right )}}{2 x^{2}} - \frac {b c d}{6 x^{2}} - \frac {b d \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} & \text {for}\: c \neq 0 \\- \frac {a d}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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