Optimal. Leaf size=46 \[ \frac {\left (a+b x^4\right ) \text {csch}^{-1}\left (a+b x^4\right )}{4 b}+\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{\left (a+b x^4\right )^2}}\right )}{4 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6847, 6449,
379, 272, 65, 213} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rule 379
Rule 6449
Rule 6847
Rubi steps
\begin {align*} \int x^3 \text {csch}^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \text {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} \text {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 {\text {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 {\text {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 {\text {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*}
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Mathematica [A]
time = 0.10, size = 90, normalized size = 1.96 \begin {gather*} \frac {\left (a+b x^4\right )^2 \text {csch}^{-1}\left (a+b x^4\right )+\frac {\sqrt {1+\left (a+b x^4\right )^2} \tanh ^{-1}\left (\frac {a+b x^4}{\sqrt {1+\left (a+b x^4\right )^2}}\right )}{\sqrt {1+\frac {1}{\left (a+b x^4\right )^2}}}}{4 b \left (a+b x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 52, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {\left (b \,x^{4}+a \right ) \mathrm {arccsch}\left (b \,x^{4}+a \right )+\ln \left (b \,x^{4}+a +\left (b \,x^{4}+a \right ) \sqrt {1+\frac {1}{\left (b \,x^{4}+a \right )^{2}}}\right )}{4 b}\) | \(52\) |
default | \(\frac {\left (b \,x^{4}+a \right ) \mathrm {arccsch}\left (b \,x^{4}+a \right )+\ln \left (b \,x^{4}+a +\left (b \,x^{4}+a \right ) \sqrt {1+\frac {1}{\left (b \,x^{4}+a \right )^{2}}}\right )}{4 b}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 57, normalized size = 1.24 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs.
\(2 (40) = 80\).
time = 0.40, size = 266, normalized size = 5.78 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.71, size = 42, normalized size = 0.91 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {\frac {1}{{\left (b\,x^4+a\right )}^2}+1}\right )}{4\,b}+\frac {\mathrm {asinh}\left (\frac {1}{b\,x^4+a}\right )\,\left (b\,x^4+a\right )}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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