Optimal. Leaf size=43 \[ \frac {b x^2}{4 c}-\frac {b \tanh ^{-1}\left (c x^2\right )}{4 c^2}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6037, 281, 327,
212} \begin {gather*} \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {b \tanh ^{-1}\left (c x^2\right )}{4 c^2}+\frac {b x^2}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rule 327
Rule 6037
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {1}{2} (b c) \int \frac {x^5}{1-c^2 x^4} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {1}{4} (b c) \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=\frac {b x^2}{4 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac {b x^2}{4 c}-\frac {b \tanh ^{-1}\left (c x^2\right )}{4 c^2}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.56 \begin {gather*} \frac {b x^2}{4 c}+\frac {a x^4}{4}+\frac {1}{4} b x^4 \tanh ^{-1}\left (c x^2\right )+\frac {b \log \left (1-c x^2\right )}{8 c^2}-\frac {b \log \left (1+c x^2\right )}{8 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 57, normalized size = 1.33
method | result | size |
default | \(\frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \,x^{2}\right )}{4}+\frac {b \,x^{2}}{4 c}+\frac {b \ln \left (c \,x^{2}-1\right )}{8 c^{2}}-\frac {b \ln \left (c \,x^{2}+1\right )}{8 c^{2}}\) | \(57\) |
risch | \(\frac {x^{4} b \ln \left (c \,x^{2}+1\right )}{8}-\frac {x^{4} b \ln \left (-c \,x^{2}+1\right )}{8}+\frac {x^{4} a}{4}+\frac {b \,x^{2}}{4 c}+\frac {b \ln \left (c \,x^{2}-1\right )}{8 c^{2}}-\frac {b \ln \left (c \,x^{2}+1\right )}{8 c^{2}}+\frac {b^{2}}{16 a \,c^{2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 58, normalized size = 1.35 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \operatorname {artanh}\left (c x^{2}\right ) + c {\left (\frac {2 \, x^{2}}{c^{2}} - \frac {\log \left (c x^{2} + 1\right )}{c^{3}} + \frac {\log \left (c x^{2} - 1\right )}{c^{3}}\right )}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 54, normalized size = 1.26 \begin {gather*} \frac {2 \, a c^{2} x^{4} + 2 \, b c x^{2} + {\left (b c^{2} x^{4} - b\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{8 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.61, size = 48, normalized size = 1.12 \begin {gather*} \begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {atanh}{\left (c x^{2} \right )}}{4} + \frac {b x^{2}}{4 c} - \frac {b \operatorname {atanh}{\left (c x^{2} \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {a x^{4}}{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. 181 vs.
\(2 (37) = 74\).
time = 0.41, size = 181, normalized size = 4.21 \begin {gather*} \frac {1}{2} \, c {\left (\frac {{\left (c x^{2} + 1\right )} b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{{\left (\frac {{\left (c x^{2} + 1\right )}^{2} c^{3}}{{\left (c x^{2} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{2} + 1\right )} c^{3}}{c x^{2} - 1} + c^{3}\right )} {\left (c x^{2} - 1\right )}} + \frac {\frac {2 \, {\left (c x^{2} + 1\right )} a}{c x^{2} - 1} + \frac {{\left (c x^{2} + 1\right )} b}{c x^{2} - 1} - b}{\frac {{\left (c x^{2} + 1\right )}^{2} c^{3}}{{\left (c x^{2} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{2} + 1\right )} c^{3}}{c x^{2} - 1} + c^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 60, normalized size = 1.40 \begin {gather*} \frac {a\,x^4}{4}+\frac {b\,x^2}{4\,c}+\frac {b\,x^4\,\ln \left (c\,x^2+1\right )}{8}-\frac {b\,x^4\,\ln \left (1-c\,x^2\right )}{8}+\frac {b\,\mathrm {atan}\left (c\,x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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