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 = 0.0776517, antiderivative size = 58, normalized size of antiderivative = 1., 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 6715
Rule 5250
Rule 372
Rule 266
Rule 63
Rule 206
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*}
Mathematica [B] time = 0.36691, 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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Maple [A] time = 0.265, size = 81, normalized size = 1.4 \begin{align*}{\frac{{\rm arcsec} \left (d{x}^{4}+c\right ){x}^{4}b}{4}}+{\frac{{x}^{4}a}{4}}+{\frac{b{\rm arcsec} \left (d{x}^{4}+c\right )c}{4\,d}}-{\frac{b}{4\,d}\ln \left ( d{x}^{4}+c+ \left ( d{x}^{4}+c \right ) \sqrt{1- \left ( d{x}^{4}+c \right ) ^{-2}} \right ) }+{\frac{ac}{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997651, size = 96, normalized size = 1.66 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49047, size = 227, normalized size = 3.91 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsec}\left (d x^{4} + c\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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