Optimal. Leaf size=211 \[ \frac {a b \sqrt {x}}{2 c^7}+\frac {71 b^2 x}{420 c^6}+\frac {3 b^2 x^2}{70 c^4}+\frac {b^2 x^3}{84 c^2}+\frac {b^2 \sqrt {x} \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^7}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{6 c^5}+\frac {b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{10 c^3}+\frac {b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{14 c}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^8}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+\frac {44 b^2 \log \left (1-c^2 x\right )}{105 c^8} \]
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Rubi [A]
time = 0.37, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6039, 6037,
6127, 272, 45, 6021, 266, 6095} \begin {gather*} -\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^8}+\frac {a b \sqrt {x}}{2 c^7}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{6 c^5}+\frac {b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{10 c^3}+\frac {b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{14 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2+\frac {b^2 \sqrt {x} \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^7}+\frac {71 b^2 x}{420 c^6}+\frac {3 b^2 x^2}{70 c^4}+\frac {b^2 x^3}{84 c^2}+\frac {44 b^2 \log \left (1-c^2 x\right )}{105 c^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 266
Rule 272
Rule 6021
Rule 6037
Rule 6039
Rule 6095
Rule 6127
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \, dx &=\int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2 \, dx\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 224, normalized size = 1.06 \begin {gather*} \frac {210 a b c \sqrt {x}+71 b^2 c^2 x+70 a b c^3 x^{3/2}+18 b^2 c^4 x^2+42 a b c^5 x^{5/2}+5 b^2 c^6 x^3+30 a b c^7 x^{7/2}+105 a^2 c^8 x^4+2 b c \sqrt {x} \left (105 a c^7 x^{7/2}+b \left (105+35 c^2 x+21 c^4 x^2+15 c^6 x^3\right )\right ) \tanh ^{-1}\left (c \sqrt {x}\right )+105 b^2 \left (-1+c^8 x^4\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+b (105 a+176 b) \log \left (1-c \sqrt {x}\right )-105 a b \log \left (1+c \sqrt {x}\right )+176 b^2 \log \left (1+c \sqrt {x}\right )}{420 c^8} \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. \(372\) vs.
\(2(167)=334\).
time = 0.18, size = 373, normalized size = 1.77
method | result | size |
derivativedivides | \(\frac {\frac {71 b^{2} c^{2} x}{420}+\frac {c^{8} x^{4} a^{2}}{4}+\frac {a b \,c^{8} x^{4} \arctanh \left (c \sqrt {x}\right )}{2}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {a b \ln \left (c \sqrt {x}-1\right )}{4}-\frac {a b \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b^{2} c^{6} x^{3}}{84}+\frac {3 b^{2} c^{4} x^{2}}{70}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{8}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}+\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{16}+\frac {44 b^{2} \ln \left (c \sqrt {x}-1\right )}{105}+\frac {44 b^{2} \ln \left (1+c \sqrt {x}\right )}{105}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c \sqrt {x}}{2}+\frac {b^{2} c^{8} x^{4} \arctanh \left (c \sqrt {x}\right )^{2}}{4}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{7} x^{\frac {7}{2}}}{14}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{5} x^{\frac {5}{2}}}{10}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{3} x^{\frac {3}{2}}}{6}+\frac {c^{7} x^{\frac {7}{2}} a b}{14}+\frac {a b \,c^{3} x^{\frac {3}{2}}}{6}+\frac {a b c \sqrt {x}}{2}+\frac {a b \,c^{5} x^{\frac {5}{2}}}{10}}{c^{8}}\) | \(373\) |
default | \(\frac {\frac {71 b^{2} c^{2} x}{420}+\frac {c^{8} x^{4} a^{2}}{4}+\frac {a b \,c^{8} x^{4} \arctanh \left (c \sqrt {x}\right )}{2}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {a b \ln \left (c \sqrt {x}-1\right )}{4}-\frac {a b \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b^{2} c^{6} x^{3}}{84}+\frac {3 b^{2} c^{4} x^{2}}{70}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{8}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}+\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{16}+\frac {44 b^{2} \ln \left (c \sqrt {x}-1\right )}{105}+\frac {44 b^{2} \ln \left (1+c \sqrt {x}\right )}{105}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c \sqrt {x}}{2}+\frac {b^{2} c^{8} x^{4} \arctanh \left (c \sqrt {x}\right )^{2}}{4}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{7} x^{\frac {7}{2}}}{14}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{5} x^{\frac {5}{2}}}{10}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) c^{3} x^{\frac {3}{2}}}{6}+\frac {c^{7} x^{\frac {7}{2}} a b}{14}+\frac {a b \,c^{3} x^{\frac {3}{2}}}{6}+\frac {a b c \sqrt {x}}{2}+\frac {a b \,c^{5} x^{\frac {5}{2}}}{10}}{c^{8}}\) | \(373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 265, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{420} \, {\left (210 \, x^{4} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (15 \, c^{6} x^{\frac {7}{2}} + 21 \, c^{4} x^{\frac {5}{2}} + 35 \, c^{2} x^{\frac {3}{2}} + 105 \, \sqrt {x}\right )}}{c^{8}} - \frac {105 \, \log \left (c \sqrt {x} + 1\right )}{c^{9}} + \frac {105 \, \log \left (c \sqrt {x} - 1\right )}{c^{9}}\right )}\right )} a b + \frac {1}{1680} \, {\left (4 \, c {\left (\frac {2 \, {\left (15 \, c^{6} x^{\frac {7}{2}} + 21 \, c^{4} x^{\frac {5}{2}} + 35 \, c^{2} x^{\frac {3}{2}} + 105 \, \sqrt {x}\right )}}{c^{8}} - \frac {105 \, \log \left (c \sqrt {x} + 1\right )}{c^{9}} + \frac {105 \, \log \left (c \sqrt {x} - 1\right )}{c^{9}}\right )} \operatorname {artanh}\left (c \sqrt {x}\right ) + \frac {20 \, c^{6} x^{3} + 72 \, c^{4} x^{2} + 284 \, c^{2} x - 2 \, {\left (105 \, \log \left (c \sqrt {x} - 1\right ) - 352\right )} \log \left (c \sqrt {x} + 1\right ) + 105 \, \log \left (c \sqrt {x} + 1\right )^{2} + 105 \, \log \left (c \sqrt {x} - 1\right )^{2} + 704 \, \log \left (c \sqrt {x} - 1\right )}{c^{8}}\right )} b^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 273, normalized size = 1.29 \begin {gather*} \frac {420 \, a^{2} c^{8} x^{4} + 20 \, b^{2} c^{6} x^{3} + 72 \, b^{2} c^{4} x^{2} + 284 \, b^{2} c^{2} x + 105 \, {\left (b^{2} c^{8} x^{4} - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} + 4 \, {\left (105 \, a b c^{8} - 105 \, a b + 176 \, b^{2}\right )} \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (105 \, a b c^{8} - 105 \, a b - 176 \, b^{2}\right )} \log \left (c \sqrt {x} - 1\right ) + 4 \, {\left (105 \, a b c^{8} x^{4} - 105 \, a b c^{8} + {\left (15 \, b^{2} c^{7} x^{3} + 21 \, b^{2} c^{5} x^{2} + 35 \, b^{2} c^{3} x + 105 \, b^{2} c\right )} \sqrt {x}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 8 \, {\left (15 \, a b c^{7} x^{3} + 21 \, a b c^{5} x^{2} + 35 \, a b c^{3} x + 105 \, a b c\right )} \sqrt {x}}{1680 \, c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.87, size = 453, normalized size = 2.15 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {44\,b^2\,\ln \left (c\,\sqrt {x}-1\right )}{105\,c^8}+\frac {44\,b^2\,\ln \left (c\,\sqrt {x}+1\right )}{105\,c^8}+\frac {71\,b^2\,x}{420\,c^6}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{16\,c^8}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{16\,c^8}+\frac {b^2\,x^3}{84\,c^2}+\frac {3\,b^2\,x^2}{70\,c^4}+\frac {b^2\,x^4\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{16}+\frac {b^2\,x^4\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{16}+\frac {b^2\,x^{7/2}\,\ln \left (c\,\sqrt {x}+1\right )}{28\,c}+\frac {b^2\,x^{5/2}\,\ln \left (c\,\sqrt {x}+1\right )}{20\,c^3}+\frac {b^2\,x^{3/2}\,\ln \left (c\,\sqrt {x}+1\right )}{12\,c^5}+\frac {b^2\,\sqrt {x}\,\ln \left (c\,\sqrt {x}+1\right )}{4\,c^7}-\frac {b^2\,x^{7/2}\,\ln \left (1-c\,\sqrt {x}\right )}{28\,c}-\frac {b^2\,x^{5/2}\,\ln \left (1-c\,\sqrt {x}\right )}{20\,c^3}-\frac {b^2\,x^{3/2}\,\ln \left (1-c\,\sqrt {x}\right )}{12\,c^5}-\frac {b^2\,\sqrt {x}\,\ln \left (1-c\,\sqrt {x}\right )}{4\,c^7}+\frac {a\,b\,\ln \left (c\,\sqrt {x}-1\right )}{4\,c^8}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{4\,c^8}+\frac {a\,b\,x^4\,\ln \left (c\,\sqrt {x}+1\right )}{4}-\frac {a\,b\,x^4\,\ln \left (1-c\,\sqrt {x}\right )}{4}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{8\,c^8}+\frac {a\,b\,x^{7/2}}{14\,c}+\frac {a\,b\,x^{5/2}}{10\,c^3}+\frac {a\,b\,x^{3/2}}{6\,c^5}+\frac {a\,b\,\sqrt {x}}{2\,c^7}-\frac {b^2\,x^4\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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