Optimal. Leaf size=196 \[ -\frac {b c d^3}{30 x^5}-\frac {3 b c^2 d^3}{20 x^4}-\frac {11 b c^3 d^3}{36 x^3}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^5 d^3}{12 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (1+c x) \]
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Rubi [A]
time = 0.13, antiderivative size = 196, 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 = {45, 6083, 12,
1816} \begin {gather*} -\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (c x+1)-\frac {11 b c^5 d^3}{12 x}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^3 d^3}{36 x^3}-\frac {3 b c^2 d^3}{20 x^4}-\frac {b c d^3}{30 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 1816
Rule 6083
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^7} \, dx &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^3 \left (-10-36 c x-45 c^2 x^2-20 c^3 x^3\right )}{60 x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \frac {-10-36 c x-45 c^2 x^2-20 c^3 x^3}{x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \left (-\frac {10}{x^6}-\frac {36 c}{x^5}-\frac {55 c^2}{x^4}-\frac {56 c^3}{x^3}-\frac {55 c^4}{x^2}-\frac {56 c^5}{x}+\frac {111 c^6}{2 (-1+c x)}+\frac {c^6}{2 (1+c x)}\right ) \, dx\\ &=-\frac {b c d^3}{30 x^5}-\frac {3 b c^2 d^3}{20 x^4}-\frac {11 b c^3 d^3}{36 x^3}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^5 d^3}{12 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (1+c x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 149, normalized size = 0.76 \begin {gather*} -\frac {d^3 \left (60 a+216 a c x+12 b c x+270 a c^2 x^2+54 b c^2 x^2+120 a c^3 x^3+110 b c^3 x^3+168 b c^4 x^4+330 b c^5 x^5+6 b \left (10+36 c x+45 c^2 x^2+20 c^3 x^3\right ) \tanh ^{-1}(c x)-336 b c^6 x^6 \log (x)+333 b c^6 x^6 \log (1-c x)+3 b c^6 x^6 \log (1+c x)\right )}{360 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 202, normalized size = 1.03
method | result | size |
derivativedivides | \(c^{6} \left (d^{3} a \left (-\frac {3}{4 c^{4} x^{4}}-\frac {1}{3 c^{3} x^{3}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{6 c^{6} x^{6}}\right )-\frac {3 d^{3} b \arctanh \left (c x \right )}{4 c^{4} x^{4}}-\frac {d^{3} b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 d^{3} b \arctanh \left (c x \right )}{5 c^{5} x^{5}}-\frac {d^{3} b \arctanh \left (c x \right )}{6 c^{6} x^{6}}-\frac {37 d^{3} b \ln \left (c x -1\right )}{40}-\frac {d^{3} b \ln \left (c x +1\right )}{120}-\frac {d^{3} b}{30 c^{5} x^{5}}-\frac {3 d^{3} b}{20 c^{4} x^{4}}-\frac {11 d^{3} b}{36 c^{3} x^{3}}-\frac {7 d^{3} b}{15 c^{2} x^{2}}-\frac {11 d^{3} b}{12 c x}+\frac {14 d^{3} b \ln \left (c x \right )}{15}\right )\) | \(202\) |
default | \(c^{6} \left (d^{3} a \left (-\frac {3}{4 c^{4} x^{4}}-\frac {1}{3 c^{3} x^{3}}-\frac {3}{5 c^{5} x^{5}}-\frac {1}{6 c^{6} x^{6}}\right )-\frac {3 d^{3} b \arctanh \left (c x \right )}{4 c^{4} x^{4}}-\frac {d^{3} b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 d^{3} b \arctanh \left (c x \right )}{5 c^{5} x^{5}}-\frac {d^{3} b \arctanh \left (c x \right )}{6 c^{6} x^{6}}-\frac {37 d^{3} b \ln \left (c x -1\right )}{40}-\frac {d^{3} b \ln \left (c x +1\right )}{120}-\frac {d^{3} b}{30 c^{5} x^{5}}-\frac {3 d^{3} b}{20 c^{4} x^{4}}-\frac {11 d^{3} b}{36 c^{3} x^{3}}-\frac {7 d^{3} b}{15 c^{2} x^{2}}-\frac {11 d^{3} b}{12 c x}+\frac {14 d^{3} b \ln \left (c x \right )}{15}\right )\) | \(202\) |
risch | \(-\frac {d^{3} b \left (20 x^{3} c^{3}+45 c^{2} x^{2}+36 c x +10\right ) \ln \left (c x +1\right )}{120 x^{6}}-\frac {d^{3} \left (3 b \,c^{6} \ln \left (c x +1\right ) x^{6}-336 c^{6} b \ln \left (-x \right ) x^{6}+333 x^{6} b \ln \left (-c x +1\right ) c^{6}+330 c^{5} x^{5} b +168 c^{4} x^{4} b -60 x^{3} b \ln \left (-c x +1\right ) c^{3}+120 c^{3} x^{3} a +110 b \,c^{3} x^{3}-135 b \,x^{2} \ln \left (-c x +1\right ) c^{2}+270 a \,c^{2} x^{2}+54 b \,c^{2} x^{2}-108 b c x \ln \left (-c x +1\right )+216 c x a +12 b c x -30 b \ln \left (-c x +1\right )+60 a \right )}{360 x^{6}}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 273, normalized size = 1.39 \begin {gather*} -\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^{3} d^{3} + \frac {1}{8} \, {\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 c^{2} d^{3} - \frac {3}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b c d^{3} + \frac {1}{180} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} b d^{3} - \frac {a c^{3} d^{3}}{3 \, x^{3}} - \frac {3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac {3 \, a c d^{3}}{5 \, x^{5}} - \frac {a d^{3}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 188, normalized size = 0.96 \begin {gather*} -\frac {3 \, b c^{6} d^{3} x^{6} \log \left (c x + 1\right ) + 333 \, b c^{6} d^{3} x^{6} \log \left (c x - 1\right ) - 336 \, b c^{6} d^{3} x^{6} \log \left (x\right ) + 330 \, b c^{5} d^{3} x^{5} + 168 \, b c^{4} d^{3} x^{4} + 10 \, {\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 54 \, {\left (5 \, a + b\right )} c^{2} d^{3} x^{2} + 12 \, {\left (18 \, a + b\right )} c d^{3} x + 60 \, a d^{3} + 3 \, {\left (20 \, b c^{3} d^{3} x^{3} + 45 \, b c^{2} d^{3} x^{2} + 36 \, b c d^{3} x + 10 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.79, size = 257, normalized size = 1.31 \begin {gather*} \begin {cases} - \frac {a c^{3} d^{3}}{3 x^{3}} - \frac {3 a c^{2} d^{3}}{4 x^{4}} - \frac {3 a c d^{3}}{5 x^{5}} - \frac {a d^{3}}{6 x^{6}} + \frac {14 b c^{6} d^{3} \log {\left (x \right )}}{15} - \frac {14 b c^{6} d^{3} \log {\left (x - \frac {1}{c} \right )}}{15} - \frac {b c^{6} d^{3} \operatorname {atanh}{\left (c x \right )}}{60} - \frac {11 b c^{5} d^{3}}{12 x} - \frac {7 b c^{4} d^{3}}{15 x^{2}} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {11 b c^{3} d^{3}}{36 x^{3}} - \frac {3 b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} - \frac {3 b c^{2} d^{3}}{20 x^{4}} - \frac {3 b c d^{3} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} - \frac {b c d^{3}}{30 x^{5}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{6 x^{6}} & \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. 634 vs.
\(2 (172) = 344\).
time = 0.44, size = 634, normalized size = 3.23 \begin {gather*} \frac {1}{45} \, {\left (42 \, b c^{5} d^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 42 \, b c^{5} d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {6 \, {\left (\frac {60 \, {\left (c x + 1\right )}^{5} b c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {90 \, {\left (c x + 1\right )}^{4} b c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {140 \, {\left (c x + 1\right )}^{3} b c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {105 \, {\left (c x + 1\right )}^{2} b c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {42 \, {\left (c x + 1\right )} b c^{5} d^{3}}{c x - 1} + 7 \, b c^{5} d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {6 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {20 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {720 \, {\left (c x + 1\right )}^{5} a c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {1080 \, {\left (c x + 1\right )}^{4} a c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1680 \, {\left (c x + 1\right )}^{3} a c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {1260 \, {\left (c x + 1\right )}^{2} a c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {504 \, {\left (c x + 1\right )} a c^{5} d^{3}}{c x - 1} + 84 \, a c^{5} d^{3} + \frac {318 \, {\left (c x + 1\right )}^{5} b c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {1119 \, {\left (c x + 1\right )}^{4} b c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1742 \, {\left (c x + 1\right )}^{3} b c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {1464 \, {\left (c x + 1\right )}^{2} b c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {636 \, {\left (c x + 1\right )} b c^{5} d^{3}}{c x - 1} + 113 \, b c^{5} d^{3}}{\frac {{\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {6 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {20 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\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 = 1.07, size = 220, normalized size = 1.12 \begin {gather*} \frac {14\,b\,c^6\,d^3\,\ln \left (x\right )}{15}-\frac {7\,b\,c^6\,d^3\,\ln \left (c^2\,x^2-1\right )}{15}-\frac {3\,a\,c^2\,d^3}{4\,x^4}-\frac {a\,c^3\,d^3}{3\,x^3}-\frac {3\,b\,c^2\,d^3}{20\,x^4}-\frac {11\,b\,c^3\,d^3}{36\,x^3}-\frac {7\,b\,c^4\,d^3}{15\,x^2}-\frac {11\,b\,c^5\,d^3}{12\,x}-\frac {a\,d^3}{6\,x^6}-\frac {3\,a\,c\,d^3}{5\,x^5}-\frac {b\,c\,d^3}{30\,x^5}-\frac {b\,d^3\,\mathrm {atanh}\left (c\,x\right )}{6\,x^6}-\frac {11\,b\,c^7\,d^3\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{12\,\sqrt {-c^2}}-\frac {3\,b\,c\,d^3\,\mathrm {atanh}\left (c\,x\right )}{5\,x^5}-\frac {3\,b\,c^2\,d^3\,\mathrm {atanh}\left (c\,x\right )}{4\,x^4}-\frac {b\,c^3\,d^3\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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