Optimal. Leaf size=46 \[ \frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6847, 6449,
379, 272, 65, 213} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {\frac {1}{\left (a+b x^n\right )^2}+1}\right )}{b n}+\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n} \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^{-1+n} \text {csch}^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \text {csch}^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^2}} x} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{\left (a+b x^n\right )^2}\right )}{2 b n}\\ &=\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \text {csch}^{-1}\left (a+b x^n\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 90, normalized size = 1.96 \begin {gather*} \frac {\left (a+b x^n\right )^2 \text {csch}^{-1}\left (a+b x^n\right )+\frac {\sqrt {1+\left (a+b x^n\right )^2} \tanh ^{-1}\left (\frac {a+b x^n}{\sqrt {1+\left (a+b x^n\right )^2}}\right )}{\sqrt {1+\frac {1}{\left (a+b x^n\right )^2}}}}{b n \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{-1+n} \mathrm {arccsch}\left (a +b \,x^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 60, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arcsch}\left (b x^{n} + a\right ) + \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} - 1\right )}{2 \, b n} \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. 334 vs.
\(2 (44) = 88\).
time = 0.40, size = 334, normalized size = 7.26 \begin {gather*} \frac {a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1\right ) - a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} - 1\right ) + {\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac {\sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a}\right ) - \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt {\frac {2 \, a b + {\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 2.21, size = 40, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {\frac {1}{{\left (a+b\,x^n\right )}^2}+1}\right )+\mathrm {asinh}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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