Optimal. Leaf size=63 \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {2 b x}{3 c} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6097, 321, 212, 206, 203} \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {2 b x}{3 c} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 321
Rule 6097
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {1}{3} (2 b c) \int \frac {x^4}{1-c^2 x^4} \, dx\\ &=\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {(2 b) \int \frac {1}{1-c^2 x^4} \, dx}{3 c}\\ &=\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac {b \int \frac {1}{1-c x^2} \, dx}{3 c}-\frac {b \int \frac {1}{1+c x^2} \, dx}{3 c}\\ &=\frac {2 b x}{3 c}-\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 91, normalized size = 1.44 \[ \frac {a x^3}{3}+\frac {b \log \left (1-\sqrt {c} x\right )}{6 c^{3/2}}-\frac {b \log \left (\sqrt {c} x+1\right )}{6 c^{3/2}}-\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c x^2\right )+\frac {2 b x}{3 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 186, normalized size = 2.95 \[ \left [\frac {b c^{2} x^{3} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c^{2} x^{3} + 4 \, b c x - 2 \, b \sqrt {c} \arctan \left (\sqrt {c} x\right ) + b \sqrt {c} \log \left (\frac {c x^{2} - 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right )}{6 \, c^{2}}, \frac {b c^{2} x^{3} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c^{2} x^{3} + 4 \, b c x + 2 \, b \sqrt {-c} \arctan \left (\sqrt {-c} x\right ) - b \sqrt {-c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right )}{6 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 75, normalized size = 1.19 \[ -\frac {1}{3} \, b c^{5} {\left (\frac {\arctan \left (\sqrt {c} x\right )}{c^{\frac {13}{2}}} - \frac {\arctan \left (\frac {c x}{\sqrt {-c}}\right )}{\sqrt {-c} c^{6}}\right )} + \frac {1}{6} \, b x^{3} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + \frac {1}{3} \, a x^{3} + \frac {2 \, b x}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.81 \[ \frac {x^{3} a}{3}+\frac {b \,x^{3} \arctanh \left (c \,x^{2}\right )}{3}+\frac {2 b x}{3 c}-\frac {b \arctan \left (x \sqrt {c}\right )}{3 c^{\frac {3}{2}}}-\frac {b \arctanh \left (x \sqrt {c}\right )}{3 c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 66, normalized size = 1.05 \[ \frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{2}\right ) + c {\left (\frac {4 \, x}{c^{2}} - \frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 70, normalized size = 1.11 \[ \frac {a\,x^3}{3}-\frac {b\,\mathrm {atan}\left (\sqrt {c}\,x\right )}{3\,c^{3/2}}+\frac {2\,b\,x}{3\,c}+\frac {b\,x^3\,\ln \left (c\,x^2+1\right )}{6}-\frac {b\,x^3\,\ln \left (1-c\,x^2\right )}{6}+\frac {b\,\mathrm {atan}\left (\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.54, size = 559, normalized size = 8.87 \[ \begin {cases} \frac {4 a c^{2} x^{3} \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {4 i a c^{2} x^{3} \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {4 b c^{2} x^{3} \sqrt {\frac {1}{c}} \operatorname {atanh}{\left (c x^{2} \right )}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {4 i b c^{2} x^{3} \sqrt {\frac {1}{c}} \operatorname {atanh}{\left (c x^{2} \right )}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {2 i b c^{2} \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{12 c^{4} \sqrt {\frac {1}{c}} + 12 i c^{4} \sqrt {\frac {1}{c}}} + \frac {8 b c x \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {8 i b c x \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} - \frac {6 i b c \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{12 c^{3} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {\frac {1}{c}}} + \frac {4 i b c \log {\left (x - \sqrt {\frac {1}{c}} \right )}}{12 c^{3} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {\frac {1}{c}}} + \frac {4 i b c \operatorname {atanh}{\left (c x^{2} \right )}}{12 c^{3} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {\frac {1}{c}}} - \frac {4 b \log {\left (x - i \sqrt {\frac {1}{c}} \right )}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {4 b \log {\left (x - \sqrt {\frac {1}{c}} \right )}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} + \frac {4 b \operatorname {atanh}{\left (c x^{2} \right )}}{12 c^{2} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {\frac {1}{c}}} & \text {for}\: c \neq 0 \\\frac {a x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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