Optimal. Leaf size=108 \[ \frac {b d x}{4 c^3}+\frac {b d x^2}{10 c^2}+\frac {b d x^3}{12 c}+\frac {1}{20} b d x^4+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {9 b d \log (1-c x)}{40 c^4}-\frac {b d \log (1+c x)}{40 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {45, 6083, 12,
815, 647, 31} \begin {gather*} \frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {9 b d \log (1-c x)}{40 c^4}-\frac {b d \log (c x+1)}{40 c^4}+\frac {b d x}{4 c^3}+\frac {b d x^2}{10 c^2}+\frac {b d x^3}{12 c}+\frac {1}{20} b d x^4 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 45
Rule 647
Rule 815
Rule 6083
Rubi steps
\begin {align*} \int x^3 (d+c d x) \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac {d x^4 (5+4 c x)}{20 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{20} (b c d) \int \frac {x^4 (5+4 c x)}{1-c^2 x^2} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{20} (b c d) \int \left (-\frac {5}{c^4}-\frac {4 x}{c^3}-\frac {5 x^2}{c^2}-\frac {4 x^3}{c}+\frac {5+4 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {b d x}{4 c^3}+\frac {b d x^2}{10 c^2}+\frac {b d x^3}{12 c}+\frac {1}{20} b d x^4+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {(b d) \int \frac {5+4 c x}{1-c^2 x^2} \, dx}{20 c^3}\\ &=\frac {b d x}{4 c^3}+\frac {b d x^2}{10 c^2}+\frac {b d x^3}{12 c}+\frac {1}{20} b d x^4+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {(b d) \int \frac {1}{-c-c^2 x} \, dx}{40 c^2}-\frac {(9 b d) \int \frac {1}{c-c^2 x} \, dx}{40 c^2}\\ &=\frac {b d x}{4 c^3}+\frac {b d x^2}{10 c^2}+\frac {b d x^3}{12 c}+\frac {1}{20} b d x^4+\frac {1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {9 b d \log (1-c x)}{40 c^4}-\frac {b d \log (1+c x)}{40 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 97, normalized size = 0.90 \begin {gather*} \frac {d \left (30 b c x+12 b c^2 x^2+10 b c^3 x^3+30 a c^4 x^4+6 b c^4 x^4+24 a c^5 x^5+6 b c^4 x^4 (5+4 c x) \tanh ^{-1}(c x)+27 b \log (1-c x)-3 b \log (1+c x)\right )}{120 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 110, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {d a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+\frac {d b \arctanh \left (c x \right ) c^{5} x^{5}}{5}+\frac {d b \arctanh \left (c x \right ) c^{4} x^{4}}{4}+\frac {d b \,c^{4} x^{4}}{20}+\frac {d b \,c^{3} x^{3}}{12}+\frac {d b \,c^{2} x^{2}}{10}+\frac {d b c x}{4}+\frac {9 d b \ln \left (c x -1\right )}{40}-\frac {d b \ln \left (c x +1\right )}{40}}{c^{4}}\) | \(110\) |
default | \(\frac {d a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+\frac {d b \arctanh \left (c x \right ) c^{5} x^{5}}{5}+\frac {d b \arctanh \left (c x \right ) c^{4} x^{4}}{4}+\frac {d b \,c^{4} x^{4}}{20}+\frac {d b \,c^{3} x^{3}}{12}+\frac {d b \,c^{2} x^{2}}{10}+\frac {d b c x}{4}+\frac {9 d b \ln \left (c x -1\right )}{40}-\frac {d b \ln \left (c x +1\right )}{40}}{c^{4}}\) | \(110\) |
risch | \(\frac {d b \,x^{4} \left (4 c x +5\right ) \ln \left (c x +1\right )}{40}-\frac {d c \,x^{5} b \ln \left (-c x +1\right )}{10}+\frac {d c \,x^{5} a}{5}-\frac {d \,x^{4} b \ln \left (-c x +1\right )}{8}+\frac {d \,x^{4} a}{4}+\frac {b d \,x^{4}}{20}+\frac {b d \,x^{3}}{12 c}+\frac {b d \,x^{2}}{10 c^{2}}+\frac {b d x}{4 c^{3}}+\frac {9 b d \ln \left (-c x +1\right )}{40 c^{4}}-\frac {b d \ln \left (c x +1\right )}{40 c^{4}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 121, normalized size = 1.12 \begin {gather*} \frac {1}{5} \, a c d x^{5} + \frac {1}{4} \, a d x^{4} + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 114, normalized size = 1.06 \begin {gather*} \frac {24 \, a c^{5} d x^{5} + 6 \, {\left (5 \, a + b\right )} c^{4} d x^{4} + 10 \, b c^{3} d x^{3} + 12 \, b c^{2} d x^{2} + 30 \, b c d x - 3 \, b d \log \left (c x + 1\right ) + 27 \, b d \log \left (c x - 1\right ) + 3 \, {\left (4 \, b c^{5} d x^{5} + 5 \, b c^{4} d x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{120 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.37, size = 124, normalized size = 1.15 \begin {gather*} \begin {cases} \frac {a c d x^{5}}{5} + \frac {a d x^{4}}{4} + \frac {b c d x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b d x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {b d x^{4}}{20} + \frac {b d x^{3}}{12 c} + \frac {b d x^{2}}{10 c^{2}} + \frac {b d x}{4 c^{3}} + \frac {b d \log {\left (x - \frac {1}{c} \right )}}{5 c^{4}} - \frac {b d \operatorname {atanh}{\left (c x \right )}}{20 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs.
\(2 (92) = 184\).
time = 0.41, size = 491, normalized size = 4.55 \begin {gather*} \frac {1}{15} \, c {\left (\frac {3 \, {\left (\frac {10 \, {\left (c x + 1\right )}^{4} b d}{{\left (c x - 1\right )}^{4}} - \frac {5 \, {\left (c x + 1\right )}^{3} b d}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} b d}{{\left (c x - 1\right )}^{2}} - \frac {5 \, {\left (c x + 1\right )} b d}{c x - 1} + b d\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} - \frac {5 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} - \frac {10 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} c^{5}}{c x - 1} - c^{5}} + \frac {\frac {60 \, {\left (c x + 1\right )}^{4} a d}{{\left (c x - 1\right )}^{4}} - \frac {30 \, {\left (c x + 1\right )}^{3} a d}{{\left (c x - 1\right )}^{3}} + \frac {90 \, {\left (c x + 1\right )}^{2} a d}{{\left (c x - 1\right )}^{2}} - \frac {30 \, {\left (c x + 1\right )} a d}{c x - 1} + 6 \, a d + \frac {27 \, {\left (c x + 1\right )}^{4} b d}{{\left (c x - 1\right )}^{4}} - \frac {69 \, {\left (c x + 1\right )}^{3} b d}{{\left (c x - 1\right )}^{3}} + \frac {79 \, {\left (c x + 1\right )}^{2} b d}{{\left (c x - 1\right )}^{2}} - \frac {47 \, {\left (c x + 1\right )} b d}{c x - 1} + 10 \, b d}{\frac {{\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} - \frac {5 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} - \frac {10 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} c^{5}}{c x - 1} - c^{5}} - \frac {3 \, b d \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{5}} + \frac {3 \, b d \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.99, size = 103, normalized size = 0.95 \begin {gather*} \frac {\frac {b\,c\,d\,x}{4}-\frac {d\,\left (15\,b\,\mathrm {atanh}\left (c\,x\right )-6\,b\,\ln \left (c^2\,x^2-1\right )\right )}{60}+\frac {b\,c^2\,d\,x^2}{10}+\frac {b\,c^3\,d\,x^3}{12}}{c^4}+\frac {d\,\left (15\,a\,x^4+3\,b\,x^4+15\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{60}+\frac {c\,d\,\left (12\,a\,x^5+12\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________