Optimal. Leaf size=81 \[ -\frac {d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {4}{3} b c^3 d^2 \log (x)-\frac {4}{3} b c^3 d^2 \log (1-c x)-\frac {b c^2 d^2}{x}-\frac {b c d^2}{6 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {37, 5936, 12, 88} \[ -\frac {d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {b c^2 d^2}{x}+\frac {4}{3} b c^3 d^2 \log (x)-\frac {4}{3} b c^3 d^2 \log (1-c x)-\frac {b c d^2}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 88
Rule 5936
Rubi steps
\begin {align*} \int \frac {(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {(d+c d x)^2}{3 x^3 (-1+c x)} \, dx\\ &=-\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{3} (b c) \int \frac {(d+c d x)^2}{x^3 (-1+c x)} \, dx\\ &=-\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{3} (b c) \int \left (-\frac {d^2}{x^3}-\frac {3 c d^2}{x^2}-\frac {4 c^2 d^2}{x}+\frac {4 c^3 d^2}{-1+c x}\right ) \, dx\\ &=-\frac {b c d^2}{6 x^2}-\frac {b c^2 d^2}{x}-\frac {d^2 (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {4}{3} b c^3 d^2 \log (x)-\frac {4}{3} b c^3 d^2 \log (1-c x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 103, normalized size = 1.27 \[ -\frac {d^2 \left (6 a c^2 x^2+6 a c x+2 a-8 b c^3 x^3 \log (x)+7 b c^3 x^3 \log (1-c x)+b c^3 x^3 \log (c x+1)+6 b c^2 x^2+2 b \left (3 c^2 x^2+3 c x+1\right ) \tanh ^{-1}(c x)+b c x\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 128, normalized size = 1.58 \[ -\frac {b c^{3} d^{2} x^{3} \log \left (c x + 1\right ) + 7 \, b c^{3} d^{2} x^{3} \log \left (c x - 1\right ) - 8 \, b c^{3} d^{2} x^{3} \log \relax (x) + 6 \, {\left (a + b\right )} c^{2} d^{2} x^{2} + {\left (6 \, a + b\right )} c d^{2} x + 2 \, a d^{2} + {\left (3 \, b c^{2} d^{2} x^{2} + 3 \, b c d^{2} x + b d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 330, normalized size = 4.07 \[ \frac {2}{3} \, {\left (2 \, b c^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 2 \, b c^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {2 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + b c^{2} d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {12 \, {\left (c x + 1\right )}^{2} a c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {12 \, {\left (c x + 1\right )} a c^{2} d^{2}}{c x - 1} + 4 \, a c^{2} d^{2} + \frac {4 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + 3 \, b c^{2} d^{2}}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 141, normalized size = 1.74 \[ -\frac {c^{2} d^{2} a}{x}-\frac {d^{2} a}{3 x^{3}}-\frac {c \,d^{2} a}{x^{2}}-\frac {c^{2} d^{2} b \arctanh \left (c x \right )}{x}-\frac {d^{2} b \arctanh \left (c x \right )}{3 x^{3}}-\frac {c \,d^{2} b \arctanh \left (c x \right )}{x^{2}}-\frac {b c \,d^{2}}{6 x^{2}}-\frac {b \,c^{2} d^{2}}{x}+\frac {4 c^{3} d^{2} b \ln \left (c x \right )}{3}-\frac {7 c^{3} d^{2} b \ln \left (c x -1\right )}{6}-\frac {c^{3} d^{2} b \ln \left (c x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 157, normalized size = 1.94 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b c^{2} d^{2} + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b d^{2} - \frac {a c^{2} d^{2}}{x} - \frac {a c d^{2}}{x^{2}} - \frac {a d^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 116, normalized size = 1.43 \[ \frac {d^2\,\left (6\,b\,c^3\,\mathrm {atanh}\left (c\,x\right )-4\,b\,c^3\,\ln \left (c^2\,x^2-1\right )+8\,b\,c^3\,\ln \relax (x)\right )}{6}-\frac {\frac {d^2\,\left (2\,a+2\,b\,\mathrm {atanh}\left (c\,x\right )\right )}{6}+\frac {d^2\,x\,\left (6\,a\,c+b\,c+6\,b\,c\,\mathrm {atanh}\left (c\,x\right )\right )}{6}+\frac {d^2\,x^2\,\left (6\,a\,c^2+6\,b\,c^2+6\,b\,c^2\,\mathrm {atanh}\left (c\,x\right )\right )}{6}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 158, normalized size = 1.95 \[ \begin {cases} - \frac {a c^{2} d^{2}}{x} - \frac {a c d^{2}}{x^{2}} - \frac {a d^{2}}{3 x^{3}} + \frac {4 b c^{3} d^{2} \log {\relax (x )}}{3} - \frac {4 b c^{3} d^{2} \log {\left (x - \frac {1}{c} \right )}}{3} - \frac {b c^{3} d^{2} \operatorname {atanh}{\left (c x \right )}}{3} - \frac {b c^{2} d^{2} \operatorname {atanh}{\left (c x \right )}}{x} - \frac {b c^{2} d^{2}}{x} - \frac {b c d^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}} - \frac {b c d^{2}}{6 x^{2}} - \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} & \text {for}\: c \neq 0 \\- \frac {a d^{2}}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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