Optimal. Leaf size=93 \[ -\frac {b c d^3}{12 x^3}-\frac {b c^2 d^3}{2 x^2}-\frac {7 b c^3 d^3}{4 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}+2 b c^4 d^3 \log (x)-2 b c^4 d^3 \log (1-c x) \]
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Rubi [A]
time = 0.07, antiderivative size = 93, 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, 6083, 12,
90} \begin {gather*} -\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}+2 b c^4 d^3 \log (x)-2 b c^4 d^3 \log (1-c x)-\frac {7 b c^3 d^3}{4 x}-\frac {b c^2 d^3}{2 x^2}-\frac {b c d^3}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 90
Rule 6083
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-(b c) \int \frac {(d+c d x)^3}{4 x^4 (-1+c x)} \, dx\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {1}{4} (b c) \int \frac {(d+c d x)^3}{x^4 (-1+c x)} \, dx\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {1}{4} (b c) \int \left (-\frac {d^3}{x^4}-\frac {4 c d^3}{x^3}-\frac {7 c^2 d^3}{x^2}-\frac {8 c^3 d^3}{x}+\frac {8 c^4 d^3}{-1+c x}\right ) \, dx\\ &=-\frac {b c d^3}{12 x^3}-\frac {b c^2 d^3}{2 x^2}-\frac {7 b c^3 d^3}{4 x}-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}+2 b c^4 d^3 \log (x)-2 b c^4 d^3 \log (1-c x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 131, normalized size = 1.41 \begin {gather*} -\frac {d^3 \left (6 a+24 a c x+2 b c x+36 a c^2 x^2+12 b c^2 x^2+24 a c^3 x^3+42 b c^3 x^3+6 b \left (1+4 c x+6 c^2 x^2+4 c^3 x^3\right ) \tanh ^{-1}(c x)-48 b c^4 x^4 \log (x)+45 b c^4 x^4 \log (1-c x)+3 b c^4 x^4 \log (1+c x)\right )}{24 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs.
\(2(85)=170\).
time = 0.16, size = 178, normalized size = 1.91
method | result | size |
derivativedivides | \(c^{4} \left (d^{3} a \left (-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}\right )-\frac {d^{3} b \arctanh \left (c x \right )}{c x}-\frac {d^{3} b \arctanh \left (c x \right )}{4 c^{4} x^{4}}-\frac {3 d^{3} b \arctanh \left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \arctanh \left (c x \right )}{c^{3} x^{3}}-\frac {15 d^{3} b \ln \left (c x -1\right )}{8}-\frac {d^{3} b \ln \left (c x +1\right )}{8}-\frac {d^{3} b}{12 c^{3} x^{3}}-\frac {d^{3} b}{2 c^{2} x^{2}}-\frac {7 d^{3} b}{4 c x}+2 d^{3} b \ln \left (c x \right )\right )\) | \(178\) |
default | \(c^{4} \left (d^{3} a \left (-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}\right )-\frac {d^{3} b \arctanh \left (c x \right )}{c x}-\frac {d^{3} b \arctanh \left (c x \right )}{4 c^{4} x^{4}}-\frac {3 d^{3} b \arctanh \left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \arctanh \left (c x \right )}{c^{3} x^{3}}-\frac {15 d^{3} b \ln \left (c x -1\right )}{8}-\frac {d^{3} b \ln \left (c x +1\right )}{8}-\frac {d^{3} b}{12 c^{3} x^{3}}-\frac {d^{3} b}{2 c^{2} x^{2}}-\frac {7 d^{3} b}{4 c x}+2 d^{3} b \ln \left (c x \right )\right )\) | \(178\) |
risch | \(-\frac {d^{3} b \left (4 x^{3} c^{3}+6 c^{2} x^{2}+4 c x +1\right ) \ln \left (c x +1\right )}{8 x^{4}}-\frac {d^{3} \left (3 b \,c^{4} \ln \left (c x +1\right ) x^{4}-48 c^{4} b \ln \left (-x \right ) x^{4}+45 x^{4} b \ln \left (-c x +1\right ) c^{4}-12 x^{3} b \ln \left (-c x +1\right ) c^{3}+24 c^{3} x^{3} a +42 b \,c^{3} x^{3}-18 b \,x^{2} \ln \left (-c x +1\right ) c^{2}+36 a \,c^{2} x^{2}+12 b \,c^{2} x^{2}-12 b c x \ln \left (-c x +1\right )+24 c x a +2 b c x -3 b \ln \left (-c x +1\right )+6 a \right )}{24 x^{4}}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs.
\(2 (85) = 170\).
time = 0.26, size = 228, normalized size = 2.45 \begin {gather*} -\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^{3} d^{3} + \frac {3}{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^{2} d^{3} - \frac {1}{2} \, {\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^{3} - \frac {a c^{3} d^{3}}{x} + \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^{3} - \frac {3 \, a c^{2} d^{3}}{2 \, x^{2}} - \frac {a c d^{3}}{x^{3}} - \frac {a d^{3}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 163, normalized size = 1.75 \begin {gather*} -\frac {3 \, b c^{4} d^{3} x^{4} \log \left (c x + 1\right ) + 45 \, b c^{4} d^{3} x^{4} \log \left (c x - 1\right ) - 48 \, b c^{4} d^{3} x^{4} \log \left (x\right ) + 6 \, {\left (4 \, a + 7 \, b\right )} c^{3} d^{3} x^{3} + 12 \, {\left (3 \, a + b\right )} c^{2} d^{3} x^{2} + 2 \, {\left (12 \, a + b\right )} c d^{3} x + 6 \, a d^{3} + 3 \, {\left (4 \, b c^{3} d^{3} x^{3} + 6 \, b c^{2} d^{3} x^{2} + 4 \, b c d^{3} x + b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (92) = 184\).
time = 0.58, size = 207, normalized size = 2.23 \begin {gather*} \begin {cases} - \frac {a c^{3} d^{3}}{x} - \frac {3 a c^{2} d^{3}}{2 x^{2}} - \frac {a c d^{3}}{x^{3}} - \frac {a d^{3}}{4 x^{4}} + 2 b c^{4} d^{3} \log {\left (x \right )} - 2 b c^{4} d^{3} \log {\left (x - \frac {1}{c} \right )} - \frac {b c^{4} d^{3} \operatorname {atanh}{\left (c x \right )}}{4} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{x} - \frac {7 b c^{3} d^{3}}{4 x} - \frac {3 b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{2 x^{2}} - \frac {b c^{2} d^{3}}{2 x^{2}} - \frac {b c d^{3} \operatorname {atanh}{\left (c x \right )}}{x^{3}} - \frac {b c d^{3}}{12 x^{3}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 431 vs.
\(2 (85) = 170\).
time = 0.41, size = 431, normalized size = 4.63 \begin {gather*} \frac {1}{3} \, {\left (6 \, b c^{3} d^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 6 \, b c^{3} d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {6 \, {\left (\frac {4 \, {\left (c x + 1\right )}^{3} b c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2} b c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )} b c^{3} d^{3}}{c x - 1} + b c^{3} d^{3}\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 {48 \, {\left (c x + 1\right )}^{3} a c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {72 \, {\left (c x + 1\right )}^{2} a c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {48 \, {\left (c x + 1\right )} a c^{3} d^{3}}{c x - 1} + 12 \, a c^{3} d^{3} + \frac {18 \, {\left (c x + 1\right )}^{3} b c^{3} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {45 \, {\left (c x + 1\right )}^{2} b c^{3} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {38 \, {\left (c x + 1\right )} b c^{3} d^{3}}{c x - 1} + 11 \, b c^{3} d^{3}}{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 147, normalized size = 1.58 \begin {gather*} \frac {d^3\,\left (21\,b\,c^4\,\mathrm {atanh}\left (c\,x\right )-12\,b\,c^4\,\ln \left (c^2\,x^2-1\right )+24\,b\,c^4\,\ln \left (x\right )\right )}{12}-\frac {\frac {d^3\,\left (3\,a+3\,b\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x\,\left (12\,a\,c+b\,c+12\,b\,c\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x^2\,\left (18\,a\,c^2+6\,b\,c^2+18\,b\,c^2\,\mathrm {atanh}\left (c\,x\right )\right )}{12}+\frac {d^3\,x^3\,\left (12\,a\,c^3+21\,b\,c^3+12\,b\,c^3\,\mathrm {atanh}\left (c\,x\right )\right )}{12}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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