Optimal. Leaf size=58 \[ \frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (c+d x^4\right )^2}}\right )}{4 d} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6715, 5250, 372, 266, 63, 206} \[ \frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (c+d x^4\right )^2}}\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 372
Rule 5250
Rule 6715
Rubi steps
\begin {align*} \int x^3 \left (a+b \sec ^{-1}\left (c+d x^4\right )\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \left (a+b \sec ^{-1}(c+d x)\right ) \, dx,x,x^4\right )\\ &=\frac {a x^4}{4}+\frac {1}{4} b \operatorname {Subst}\left (\int \sec ^{-1}(c+d x) \, dx,x,x^4\right )\\ &=\frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {1}{4} b \operatorname {Subst}\left (\int \frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}} \, dx,x,x^4\right )\\ &=\frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,c+d x^4\right )}{4 d}\\ &=\frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{\left (c+d x^4\right )^2}\right )}{8 d}\\ &=\frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{\left (c+d x^4\right )^2}}\right )}{4 d}\\ &=\frac {a x^4}{4}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d}-\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{\left (c+d x^4\right )^2}}\right )}{4 d}\\ \end {align*}
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Mathematica [B] time = 0.38, size = 137, normalized size = 2.36 \[ \frac {a x^4}{4}-\frac {b \sqrt {\left (c+d x^4\right )^2-1} \left (\log \left (\frac {c+d x^4}{\sqrt {\left (c+d x^4\right )^2-1}}+1\right )-\log \left (1-\frac {c+d x^4}{\sqrt {\left (c+d x^4\right )^2-1}}\right )\right )}{8 d \left (c+d x^4\right ) \sqrt {1-\frac {1}{\left (c+d x^4\right )^2}}}+\frac {b \left (c+d x^4\right ) \sec ^{-1}\left (c+d x^4\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 96, normalized size = 1.66 \[ \frac {b d x^{4} \operatorname {arcsec}\left (d x^{4} + c\right ) + a d x^{4} + 2 \, b c \arctan \left (-d x^{4} - c + \sqrt {d^{2} x^{8} + 2 \, c d x^{4} + c^{2} - 1}\right ) + b \log \left (-d x^{4} - c + \sqrt {d^{2} x^{8} + 2 \, c d x^{4} + c^{2} - 1}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 100, normalized size = 1.72 \[ \frac {1}{4} \, a x^{4} + \frac {1}{8} \, b d {\left (\frac {2 \, {\left (d x^{4} + c\right )} \arccos \left (-\frac {1}{{\left (d x^{4} + c\right )} {\left (\frac {c}{d x^{4} + c} - 1\right )} - c}\right )}{d^{2}} - \frac {\log \left (\sqrt {-\frac {1}{{\left (d x^{4} + c\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (d x^{4} + c\right )}^{2}} + 1} + 1\right )}{d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 81, normalized size = 1.40 \[ \frac {\mathrm {arcsec}\left (d \,x^{4}+c \right ) x^{4} b}{4}+\frac {x^{4} a}{4}+\frac {\mathrm {arcsec}\left (d \,x^{4}+c \right ) b c}{4 d}-\frac {\ln \left (d \,x^{4}+c +\left (d \,x^{4}+c \right ) \sqrt {1-\frac {1}{\left (d \,x^{4}+c \right )^{2}}}\right ) b}{4 d}+\frac {a c}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 71, normalized size = 1.22 \[ \frac {1}{4} \, a x^{4} + \frac {{\left (2 \, {\left (d x^{4} + c\right )} \operatorname {arcsec}\left (d x^{4} + c\right ) - \log \left (\sqrt {-\frac {1}{{\left (d x^{4} + c\right )}^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{{\left (d x^{4} + c\right )}^{2}} + 1} + 1\right )\right )} b}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 52, normalized size = 0.90 \[ \frac {a\,x^4}{4}-\frac {b\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (d\,x^4+c\right )}^2}}}\right )}{4\,d}+\frac {b\,\mathrm {acos}\left (\frac {1}{d\,x^4+c}\right )\,\left (d\,x^4+c\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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