Optimal. Leaf size=18 \[ \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2718}
\begin {gather*} \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sinh (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
time = 0.01, size = 37, normalized size = 2.06 \begin {gather*} \frac {\cosh (a) \cosh \left (b \log \left (c x^n\right )\right )}{b n}+\frac {\sinh (a) \sinh \left (b \log \left (c x^n\right )\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.44, size = 19, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
default | \(\frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 18, normalized size = 1.00 \begin {gather*} \frac {\cosh \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 19, normalized size = 1.06 \begin {gather*} \frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (14) = 28\).
time = 0.26, size = 37, normalized size = 2.06 \begin {gather*} \begin {cases} \log {\left (x \right )} \sinh {\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \sinh {\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} \sinh {\left (a \right )} & \text {for}\: b = 0 \\\frac {\cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (18) = 36\).
time = 0.41, size = 40, normalized size = 2.22 \begin {gather*} \frac {{\left (c^{2 \, b} x^{b n} e^{\left (2 \, a\right )} + \frac {1}{x^{b n}}\right )} e^{\left (-a\right )}}{2 \, b c^{b} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 18, normalized size = 1.00 \begin {gather*} \frac {\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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