Optimal. Leaf size=57 \[ \frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac {\tan ^{-1}\left (\sqrt {\frac {-a-b x^4+1}{a+b x^4+1}}\right )}{2 b} \]
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Rubi [A] time = 0.12, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6715, 6313, 1961, 12, 203} \[ \frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac {\tan ^{-1}\left (\sqrt {\frac {-a-b x^4+1}{a+b x^4+1}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 1961
Rule 6313
Rule 6715
Rubi steps
\begin {align*} \int x^3 \text {sech}^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \text {sech}^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {\frac {1-a-b x}{1+a+b x}}}{1-a-b x} \, dx,x,x^4\right )\\ &=\frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}-b \operatorname {Subst}\left (\int \frac {1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a-b x^4}{1+a+b x^4}}\right )\\ &=\frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a-b x^4}{1+a+b x^4}}\right )}{2 b}\\ &=\frac {\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac {\tan ^{-1}\left (\sqrt {\frac {1-a-b x^4}{1+a+b x^4}}\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 84, normalized size = 1.47 \[ \frac {\frac {\sqrt {1-\left (a+b x^4\right )^2} \sin ^{-1}\left (a+b x^4\right )}{\sqrt {-\frac {a+b x^4-1}{a+b x^4+1}} \left (a+b x^4+1\right )}+\left (a+b x^4\right ) \text {sech}^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.75, size = 283, normalized size = 4.96 \[ \frac {2 \, b x^{4} \log \left (\frac {{\left (b x^{4} + a\right )} \sqrt {-\frac {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} + 1}{b x^{4} + a}\right ) + a \log \left (\frac {{\left (b x^{4} + a\right )} \sqrt {-\frac {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} + 1}{x^{4}}\right ) - a \log \left (\frac {{\left (b x^{4} + a\right )} \sqrt {-\frac {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} - 1}{x^{4}}\right ) - 2 \, \arctan \left (\frac {{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt {-\frac {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}\right )}{8 \, b} \]
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 \[ \int x^{3} \operatorname {arsech}\left (b x^{4} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 62, normalized size = 1.09 \[ \frac {\mathrm {arcsech}\left (b \,x^{4}+a \right ) x^{4}}{4}+\frac {\mathrm {arcsech}\left (b \,x^{4}+a \right ) a}{4 b}-\frac {\arctan \left (\sqrt {\frac {1}{b \,x^{4}+a}-1}\, \sqrt {\frac {1}{b \,x^{4}+a}+1}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 38, normalized size = 0.67 \[ \frac {{\left (b x^{4} + a\right )} \operatorname {arsech}\left (b x^{4} + a\right ) - \arctan \left (\sqrt {\frac {1}{{\left (b x^{4} + a\right )}^{2}} - 1}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.99, size = 56, normalized size = 0.98 \[ \frac {\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{b\,x^4+a}-1}\,\sqrt {\frac {1}{b\,x^4+a}+1}}\right )}{4\,b}+\frac {\mathrm {acosh}\left (\frac {1}{b\,x^4+a}\right )\,\left (b\,x^4+a\right )}{4\,b} \]
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 \[ \int x^{3} \operatorname {asech}{\left (a + b x^{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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