Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^4\right )^2}+1}\right )}{4 b}+\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b} \]
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Rubi [A] time = 0.0582378, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6715, 6314, 372, 266, 63, 207} \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^4\right )^2}+1}\right )}{4 b}+\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 6314
Rule 372
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int x^3 \text{csch}^{-1}\left (a+b x^4\right ) \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \text{csch}^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}} \, dx,x,x^4\right )\\ &=\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{1}{x^2}} x} \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{\left (a+b x^4\right )^2}\right )}{8 b}\\ &=\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\frac{1}{\left (a+b x^4\right )^2}}\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \text{csch}^{-1}\left (a+b x^4\right )}{4 b}+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{\left (a+b x^4\right )^2}}\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.136193, size = 74, normalized size = 1.61 \[ \frac{\frac{\sqrt{\left (a+b x^4\right )^2+1} \sinh ^{-1}\left (a+b x^4\right )}{\sqrt{\frac{1}{\left (a+b x^4\right )^2}+1}}+\left (a+b x^4\right )^2 \text{csch}^{-1}\left (a+b x^4\right )}{4 b \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.25, size = 63, normalized size = 1.4 \begin{align*}{\frac{{\rm arccsch} \left (b{x}^{4}+a\right ){x}^{4}}{4}}+{\frac{{\rm arccsch} \left (b{x}^{4}+a\right )a}{4\,b}}+{\frac{1}{4\,b}\ln \left ( b{x}^{4}+a+ \left ( b{x}^{4}+a \right ) \sqrt{1+ \left ( b{x}^{4}+a \right ) ^{-2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01065, size = 77, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (b x^{4} + a\right )} \operatorname{arcsch}\left (b x^{4} + a\right ) + \log \left (\sqrt{\frac{1}{{\left (b x^{4} + a\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{{\left (b x^{4} + a\right )}^{2}} + 1} - 1\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.81386, size = 570, normalized size = 12.39 \begin{align*} \frac{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 (-b x^{4} +{\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}}} - a + 1\right ) - a \log \left (-b x^{4} +{\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}}} - a - 1\right ) - \log \left (-b x^{4} +{\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}}} - a\right )}{4 \, b} \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 x^{3} \operatorname{arcsch}\left (b x^{4} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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